Scale Calculus and M-Polyfolds -- An Introduction

TL;DR

This paper introduces scale calculus and M-polyfolds, providing foundational concepts and methods for advanced mathematical analysis in infinite-dimensional spaces, aimed at graduate students and researchers.

Contribution

It offers an accessible introduction to scale calculus and M-polyfolds, bridging the gap between theory and applications in symplectic field theory and related areas.

Findings

Clarifies the foundational aspects of scale calculus.

Introduces the concept of M-polyfolds for infinite-dimensional analysis.

Provides educational material for graduate courses and seminars.

Abstract

These are lecture notes on scale calculus and M-polyfolds written for a graduate course at UNICAMP March-June 2018 and an advanced mini-course given during the biannual meeting of Brazilian mathematicians, CBM-32, at IMPA in August 2019.

Equations

$$\mathfrak{M}:=\{\mathcal{F}=0\},\mathfrak{m}:=\mathfrak{M}/G$$

$$({\tau }_{\ast }v)(t):=v(t+\tau )$$

$$({\tau }_{\ast }\gamma )(t):=\gamma (t+\tau ).$$

$$\Psi :\mathbb{R}\times C^{k+1}\to C^{k+1},(\tau ,v)\mapsto {\tau }_{\ast }v$$

$$d{\Psi }_{(\tau ,v)}(T,V)=\left({\tau }_{\ast }\dot{v}\right)T+{\tau }_{\ast }V\in C^k,(T,V)\in \mathbb{R}\times C^{k+1}$$

$$D\Psi :(\mathbb{R}\times C^{k+1})\times (\mathbb{R}\times C^k)\to C^k,(\tau ,v,T,V)\mapsto D{\Psi }_{(\tau ,v)}(T,V).$$

$$\text{K⊂E∞}\iff \text{K is an sc-subspace (generates a Banach subscale) of E.}$$

$$\text{A closed in E}\iff \text{A is an sc-subspace of E.}$$

$$E_0$$

$$Tf(x,\xi ):=(f(x),Df(x)\xi )$$

$$TU:=U^1\oplus E^0$$

$$U_{m+1}\to \mathcal{L}(E_{m+1},F_m)$$

$$A=A_0\supset A_1\supset A_2\supset \dots$$

$$A_{\infty }:=\bigcap _{m\in {\mathbb{N}}_0}{A}_m$$

$$A$$

$$(B^{\cap A}{)}_m:=B_m:=B\cap A_m,m\in {\mathbb{N}}_0$$

$$B$$

$$(A^k{)}_m:=A_{k+m},m\in {\mathbb{N}}_0.$$

$$E=E_0$$

$$(E\oplus F{)}_m:=E_m\oplus F_m,m\in {\mathbb{N}}_0.$$

$$E_{m-\ell }\setminus E_m\ne \varnothing$$

$$A_{m-1}$$

$$A_m:=A\cap E_m.$$

$$\text{B is an sc-subspace of E}\iff \text{B⊂E∞.}$$

$${\gamma }_{\delta }(s):=e^{\delta s\beta (s)}.$$

$$W_{\delta }^{k,p}(\mathbb{R},{\mathbb{R}}^n):=\{f\in W^{k,p}(\mathbb{R},{\mathbb{R}}^n)\mid {\gamma }_{\delta }f\in W^{k,p}(\mathbb{R},{\mathbb{R}}^n)\}$$

$${\Vert f\Vert }_{W_{\delta }^{k,p}}:={\Vert {\gamma }_{\delta }f\Vert }_{W^{k,p}}.$$

$$0={\delta }_0<{\delta }_1<\dots$$

$$E_m:=W_{{\delta }_m}^{m,p}(\mathbb{R},{\mathbb{R}}^n),m\in {\mathbb{N}}_0$$

Full text

Scale Calculus and $\mathrm{M}$-Polyfolds

An Introduction

Joa Weber

UNICAMP

See pages 1 of capa-neutral-pr

*Dies ist das Geheimnis der Liebe,

daß sie solche verbinde,

deren jedes für sich sein könnte

und doch nichts ist und sein kann ohne das andere

*Friedrich Wilhelm Joseph von Schelling

1775–1854

Preface

This text originates from lecture notes written during the graduate course “MM805 Tópicos de Análise I” held from March through June 2018 at UNICAMP. The manuscript has then been slightly modified in order to serve as accompanying text for an advanced mini-course during the $32^{\rm nd}$* Colóquio Brasileiro de Matemática*, CBM-32, IMPA, Rio de Janeiro, in August 2019.

Scope

Our aim is to give an introduction to the new calculus, called scale calculus, and the generalized manifolds, called M-polyfolds, that were introduced by Hofer, Wysocki, and Zehnder (2007, 2009a, 2009b, 2010) in their construction of a generalized differential geometry in infinite dimensions, called polyfold theory. In this respect we recall and survey in the appendix the incarnations of the usual (Fréchet) calculus in various contexts - from topological vector spaces (TVS) to complete normed vector spaces, that is Banach spaces.

Recently the construction of abstract polyfold theory has been concluded and made available in the form of a book by Hofer et al. (2017). The door is now open, not only to reformulate and reprove past moduli space problems using the new language and tools, but to approach open or new problems.

Content

There are two parts plus an appendix. Part one introduces scale calculus starting with the linear theory (scale Banach spaces, scale linear maps, in particular, scale Fredholm operators – these are related to scale shifts), then we define scale continuity and scale differentiability. The latter is compared to usual (Fréchet) differentiability, then the chain rule is established for scale calculus. Part one concludes with boundary and, more surprisingly, corner recognition in scale calculus and with the construction of scale manifolds.

Part two is concerned with the construction of M-polyfolds in analogy to manifolds, just locally modeled not only on Banach space (Banach manifolds), neither only on scale Banach space (scale manifolds), but on a generalization of retracts called scale retracts. This choice of local model spaces is motivated by Cartan’s last theorem which we therefore review first. Part two concludes with the construction of the scale version of vector bundles, called strong bundles over M-polyfolds, whose local models are strong trivial-bundle retracts. To accommodate Fredholm sections one introduces a double scale structure from which one then extracts two individual scales.

The appendix recalls and reviews relevant background and results in topology and analysis, particularly standard calculus.

Audience

The intended audience are graduate students. Recommended background is basic knowledge of functional analysis including the definition of Sobolev spaces such as $W^{k,p}({\mathbb{R}},{\mathbb{R}}^{n})$.

Acknowledgements

It is a pleasure to thank Brazilian tax payers for the excellent research and teaching opportunities at UNICAMP and for generous financial support through Fundação de Amparo à Pesquisa do Estado de São Paulo (FAPESP), processo $\mathrm{n}^{\rm o}$ 2017/19725-6. I am indebted to the selection committee of CBM-32 for giving me once more the opportunity to teach an advanced mini-course at these bi-annual meetings.

I’d like to thank Daniel Ferreira Machado, Darwin Gregorio Villar Salinas, and José Lucas Pereira Luiz for their interest and a very pleasant atmosphere in the graduate course “MM805 Tópicos de Análise I” held in the first semester of 2018 at UNICAMP.

Last not least, I am very grateful to Kai Cieliebak for handing me out his fine Lecture Notes Cieliebak (2018) while they were still in progress and to Helmut Hofer for a useful comment.

Campinas, Joa Weber

Contents

1. Preface

2. 1 Introduction

3. 2 Scale calculus

4. 2.1 Scale structures

5. 2.2 Examples

6. 2.3 Scale linear theory

7. 2.3.1 Scale linear operators

8. 2.3.2 Scale Fredholm operators

9. 2.4 Scale differentiability

10. 2.5 Differentiability – Scale vs Fréchet

11. 2.6 Chain rule

12. 2.7 Boundary recognition

13. 2.8 Sc-manifolds

14. 3 Sc-retracts – the local models for M-polyfolds

15. 3.1 Cartan’s last theorem

16. 3.2 Sc-smooth retractions and their images $O$

17. 3.2.1 Special case: Splicings and splicing cores

18. 3.2.2 Splicing core with jumping finite dimension

19. 3.3 M-polyfolds and their tangent bundles

20. 3.3.1 Sub-M-polyfolds

21. 3.3.2 Boundary and corners – tameness

22. 3.4 Strong bundles over M-polyfolds

23. 3.4.1 Trivial-strong-bundle retracts - the local models

24. 3.4.2 Strong bundles

25. A Background from Topology and Functional Analysis

26. A.1 Analysis on topological vector spaces

27. A.1.1 Topological spaces

28. A.1.2 Topological vector spaces

29. A.1.3 Metric spaces

30. A.1.4 Normed vector spaces

31. A.2 Analysis on Banach spaces

32. A.2.1 Banach spaces

33. A.2.2 Linear operators

34. A.2.3 Calculus

35. A.2.4 Banach manifolds

36. A.3 Function spaces

37. References

Chapter 1 Introduction

The central problem in areas of global analysis such as Morse, Floer, or Gromov–Witten theory is to study spaces of solutions to nonlinear ordinary or partial differential equations ${\mathcal{F}}(v)=0$. The so-called moduli spaces

[TABLE]

consist in case of ${\mathfrak{M}}$ of parametrized solutions $v\colon\Sigma\to S$ taking values in a manifold or – after localization – in a vector space $S$, often divided out by a group $G$ that acts on ${\mathfrak{M}}$ by reparametrizing the domain manifold $\Sigma$. The elements of ${\mathfrak{m}}$ are then called unparametrized solutions. In case of Morse and Floer homology an element $\tau$ of the group $G=({\mathbb{R}},+)$ acts on the domain $\Sigma={\mathbb{R}}$ by time-shift

[TABLE]

for $t\in{\mathbb{R}}$. The shift map $\Psi:{\mathbb{R}}\times{\rm Map}({\mathbb{R}},S)\to{\rm Map}({\mathbb{R}},S)$ is defined by $(\tau,v)\mapsto\tau_{*}v$. The peculiar different behavior in $\tau$ and in $v$ of this simple map, namely linearity, hence smoothness, in $v$, whereas differentiation with respect to $t$ causes $v$ to loose a derivative, eventually led to the discovery of a new notion of smoothness in infinite dimensions – scale smoothness due to Hofer, Wysocki, and Zehnder (2007, 2017). Scale smoothness is connected to interpolation theory Triebel (1978). It was the crucial insight of Hofer, Wysocki, and Zehnder that requiring compactness of the scale embeddings causes that scale smoothness satisfies the chain rule and therefore is suitable to patch together pieces of scale Banach spaces to obtain scale manifolds, or more generally M-polyfolds – new spaces in infinite dimensions.

From holomorphic curves to polyfold theory.

In 1985 Gromov (1985) generalized holomorphic curves from complex analysis to symplectic geometry and thereby discovered that there is a symplectic topology. Right after Gromov’s seminal ideas Floer (1986, 1988b, 1989) “morsified” holomorphic curves. He used a perturbed holomorphic curve equation to construct a semi-infinite dimensional Morse homology, called Floer homology, which meanwhile has a huge range of applications, from Hamiltonian and contact dynamics through symplectic topology to topological field theories; cf. the survey Abbondandolo and Schlenk (2018). Floer’s construction also motivated further developments like the discovery of Fukaya $A_{\infty}$-categories (Fukaya, 1993; Fukaya et al., 2009) and Symplectic Field Theory (Eliashberg et al., 2000).

All these applications face difficult transversality and compactness issues largely caused by the fact that one does not find oneself working in a single Banach manifold, but rather in a union of such and one has to deal with each strata individually and even do analysis across neighboring ones. To deal with these problems Fukaya and Ono (1999) discovered the notion of Kuranishi structures based on finite dimensional approximation.

In contrast Hofer, Wysocki, and Zehnder stay in infinite dimension and generalize calculus. Traditionally moduli spaces were studied by cumbersome ad-hoc methods all of whose steps had to be carried out, although rather analogous, for each moduli problem from scratch, usually filling hundreds of pages. Even in one specific setup, the differential operator ${\mathcal{F}}$ might act on maps $\gamma$ whose domains and targets vary, in general. Consequently ${\mathcal{F}}$ cannot be defined on some single Banach manifold ${\mathcal{B}}$ of maps with values in some single Banach bundle ${\mathcal{E}}$ over ${\mathcal{B}}$. Therefore the occurring singular limits, e.g. broken trajectories or bubbling off phenomena, cause difficult compactness/gluing and transversality problems for ${\mathcal{F}}$ when defined on many individual Banach manifolds ${\mathcal{B}}_{\beta}$ that are at most strata of a common ambient space ${\mathcal{B}}$. While in traditional approaches the ambient spaces ${\mathcal{B}}$ itself are usually inaccessible to calculus, in a series of papers Hofer, Wysocki, and Zehnder (2007, 2009a, 2009b, 2017) construct ambient spaces in the form of generalized manifolds, called M-polyfolds,111 The “M” is a reminder that M-polyfolds are constructed in analogy to Banach manifolds, just replace the local model Banach space by some (${\mathrm{sc}}$-retract of a) scale Banach space. The more general polyfolds are useful in problems having local symmetries.

which are accessible to a customized generalized calculus called scale or sc-calculus. Now polyfolds generalize M-polyfolds like orbifolds generalize manifolds.

Roughly speaking, polyfold theory is a mixture of a generalized differential geometry, a generalized non-linear analysis, and some category theory.

Shift map motivates scale calculus.

The discovery of scale calculus was triggered by the properties of the shift map. That map shows up already for one of the simplest non-trivial scenarios, namely, the downward gradient equation ${\mathcal{F}}(\gamma):=\dot{\gamma}+(\nabla f)\circ\gamma=0$ for paths $\gamma\colon{\mathbb{R}}\to M$ and associated to a given Morse function $f\colon M\to{\mathbb{R}}$ on a closed Riemannian manifold. Given critical points $x\not=y$ of $f$, the moduli space ${\mathfrak{M}}_{xy}$ consists of all solutions $\gamma\colon{\mathbb{R}}\to M$ of ${\mathcal{F}}(\gamma)=0$ which asymptotically connect $x$ to $y$, i.e. $\lim_{t\to-/+\infty}\gamma(t)=x/y$. Time shift by $\tau\in{\mathbb{R}}$ produces again a solution

[TABLE]

Having the same image in $M$ one calls $\gamma$ and $\tau_{*}\gamma$ equivalent and denotes the space of equivalence classes by ${\mathfrak{m}}_{xy}:={\mathfrak{M}}_{xy}/{\mathbb{R}}$. While the quotient of a manifold by a free and smooth action inherits a manifold structure, unfortunately, the time shift action is not smooth at all.

To illustrate non-smoothness let us simplify the scenario in that we consider the time shift action of ${\mathbb{R}}$ on the compact222 Compactness of the domain $\SS^{1}$ is crucial that inclusion $C^{k+1}\hookrightarrow C^{k}$ is compact.

domain $\SS^{1}={\mathbb{R}}/{\mathbb{Z}}$ of $v\in C^{k}=C^{k}(\SS^{1},{\mathbb{R}})$ where $k\in{\mathbb{N}}_{0}$. The derivative of the shift map

[TABLE]

taken at $(\tau,v)\in{\mathbb{R}}\times C^{k+1}$ does not respect the target space $C^{k+1}$. Indeed

[TABLE]

takes values only in $C^{k}$, because $\dot{v}:=\frac{d}{dt}v$ does. But then there is no reason to ask the second summand $\tau_{*}V$ to be better than $C^{k}$ and for this the assumption $V\in C^{k}$ suffices. While $\Psi(\tau,v)$ behaves terribly in $\tau$ it is extremely tame in $v$, namely linear.

If one accepts different differentiability classes of domain and target spaces, the shift map has the following still respectable properties for $k\in{\mathbb{N}}_{0}$.

• (a)

The shift map $\Psi\colon{\mathbb{R}}\times C^{k}\to C^{k}$ is continuous.

• (b)

The shift map as a map $\Psi\colon{\mathbb{R}}\times C^{k+1}\to C^{k}$ is pointwise differentiable in the usual sense with (Fréchet) derivative $d\Psi_{(\tau,v)}\in{\mathcal{L}}({\mathbb{R}}\times C^{k+1},C^{k})$.

• (c)

At $(\tau,v)\in{\mathbb{R}}\times C^{k+1}$ the derivative $d\Psi_{(\tau,v)}$ extends uniquely ($C^{k+1}$ is dense in $C^{k}$) from ${\mathbb{R}}\times C^{k+1}$ to a continuous linear map ${\mathbb{R}}\times C^{k}\to C^{k}$, denoted by $D\Psi_{(\tau,v)}\in{\mathcal{L}}({\mathbb{R}}\times C^{k},C^{k})$ and called the scale derivative.

• (d)

The extension $D\Psi\colon{\mathbb{R}}\times C^{k+1}\to\mathcal{L}_{\mathrm{c}}({\mathbb{R}}\times C^{k},C^{k})$ is continuous in the compact-open topology,333 But it is not continuous in the norm topology on ${\mathcal{L}}({\mathbb{R}}\times C^{k},C^{k})$; see Remark 2.4.8.

equivalently, it is continuous as a map

[TABLE]

Properties (a–d) suggest that instead of considering $\Psi$ as a map between one domain and one target, both of the same regularity (the same level), one should use the whole nested sequence (scale) of Banach spaces and consider $\Psi$ as a map $({\mathbb{R}}\times C^{k})_{k\in{\mathbb{N}}_{0}}\to(C^{k})_{k\in{\mathbb{N}}_{0}}$ between scales.

The proof of (a–d) hinges on (i) compactness of the linear operator $C^{k+1}\hookrightarrow C^{k}$ given by inclusion and (ii) on density of the intersection $E_{\infty}:=\bigcap_{k=1}^{\infty}C^{k}$ in each of the Banach spaces (levels) $E_{k}:=C^{k}$. A nested sequence of Banach spaces $E=(E_{k})$ satisfying (i) and (ii) is called a Banach scale or an ${\mathrm{sc}}$-Banach space and $E_{k}$ is called level $k$ of the scale.

Now one turns properties (a–d) into a definition calling maps between ${\mathrm{sc}}$-Banach spaces satisfying them continuously ${\mathrm{sc}}$-differentiable or of class ${\mathrm{sc}}^{1}$; cf. Remark 2.0.1 and Definition 2.4.6. The new class ${\mathrm{sc}}^{1}$ generalizes the usual class $C^{1}$ in the following sense: Suppose that $f\colon E\to F$ is a map between Banach scales whose restriction to any domain level $E_{m}$ actually takes values in the corresponding level $F_{m}$ of the target and all the so-called level maps $f_{m}:=f|_{E_{m}}\colon E_{m}\to F_{m}$ are of class $C^{1}$. Then $f$ is of class ${\mathrm{sc}}^{1}$; see Lemma 2.5.6.

Sc-manifolds are modeled on scale Banach spaces.

In complete analogy to manifolds a scale or ${\mathrm{sc}}$-manifold is a paracompact Hausdorff space $X$ just locally modeled on a scale Banach space $E$, as opposed to an ordinary Banach space, and requiring the transition maps to be ${\mathrm{sc}}$-diffeomorphisms. In finite dimension ${\mathrm{sc}}$-calculus specializes to standard calculus and ${\mathrm{sc}}$-manifolds are manifolds.

M-polyfolds are modeled on sc-retracts.

Motivated by Cartan’s last theorem (1986) M-polyfolds are described locally by retracts in scale Banach spaces, replacing the open sets of Banach spaces in the familiar local description of manifolds. As a consequence M-polyfolds may have locally varying dimensions; see Figure 3.1. Enlarging the class of smooth maps one risks loosing vital analysis tools such as the implicit function theorem – which indeed is not available for sc-smooth maps; see Filippenko et al. (2018). However, for moduli space problems one only needs to work in the subclass of sc-Fredholm maps on which an implicit function theorem is available.

Outlook.

Given abstract polyfold theory (Hofer et al., 2017), it is now up to the scientific community to work out and provide modules, or black boxes, also called LEGO pieces, that uniformly cover large classes of applications, say in Morse and Floer theory. A shift map LEGO has been provided by Frauenfelder and Weber (2018).

Appendix on topology and analysis.

In the appendix we review the incarnations of the usual (Fréchet) calculus in various contexts - from topological vector spaces (TVS) to Banach spaces. For self-consistency of the text we recall many results of standard calculus in topology and analysis which are used in the main body.

Notes to the Reader.

Each of the two chapters begins with a detailed summary and survey of its contents. Read both of these two chapter summaries first to get an idea of what about is this text.

In the end the present lecture notes only grew to two chapters plus an appendix providing some background of calculus – from topology to functional analysis. In class we also treated, though briefly, scale Fredholm theory and, as an application, the shift map LEGO (Frauenfelder and Weber, 2018) for Morse and Floer path spaces. In a planned extension we shall add these topics in the form of two additional chapters.

Unless mentioned differently, we (closely) follow Hofer, Wysocki, and Zehnder (2017). Two other great sources are Fabert, Fish, Golovko, and Wehrheim (2016) and Cieliebak (2018).

Chapter 2 Scale calculus

The ubiquitous “sc” a-priori abbreviates scale, but in the context of scale linear operators and maps it stands for scale continuous. The latter is denoted in the context of general, possibly non-linear, maps by ${\mathrm{sc}^{0}}$ or by ${\mathrm{sc}^{k}}$ for $k$ times scale continuously differentiable maps. In a linear context subspace means linear subspace.

Section 2.1 “Scale structures” introduces the notion of a Banach scale which is a nested sequence of sets $E=E_{0}\supset E_{1}\supset\dots$ called levels – each one being actually a Banach space – and subject to two more axioms. A subset $A\subset E$ of the top level generates, we also say induces, naturally a new nested sequence $A^{\cap E}$ by intersecting $A$ with each level $E_{m}$. The new levels $A_{m}:=A\mathop{\cap}E_{m}$ form the nested sequence $A^{\cap E}=\left(A=A_{0}\supset A_{1}\supset\dots\right)$. Of course, not every nested sequence is of the form $A^{\cap E}$.

The three axioms for a Banach scale $E$, also called a scale Banach space or an ${\mathrm{sc}}$-Banach space, are the following: Each level $E_{m}$ is a Banach space under its own norm $\mathopen{|}\cdot\mathclose{|}_{m}$, all inclusions $E_{m}\hookrightarrow E_{m-1}$ are compact linear operators, and the intersection $E_{\infty}:=\bigcap_{m}E_{m}$ of all levels is dense in every level Banach space $E_{m}$. The points of $E_{m}$ are called points of regularity $m$ and those of $E_{\infty}$ smooth points. A Banach subscale of $E$ is a Banach scale $B$ whose levels are Banach subspaces of the corresponding levels of $E$. Is every Banach subscale $B$ generated by its top level $B_{0}$, i.e. is $B=(B_{0})^{\cap E}$? You bet. However, not every closed subspace $A$ of a scale Banach space $E$ generates a Banach subscale. In general, there is no reason that $A^{\cap E}$ satisfies the density axiom, consider e.g. cases of trivial intersection $A\mathop{\cap}E_{\infty}=\{0\}$. Those closed subspaces that do generate a Banach subscale are of crucial significance, they are called ${\mathrm{sc}}$-subspaces.

Because Fredholm theory is a fundamental tool in the analysis of solution spaces of differential equations, ${\mathrm{sc}}$-subspaces $K$ of finite dimension will be key players, as well as ${\mathrm{sc}}$-subspaces $Y$ of finite codimension. Finite dimensional ${\mathrm{sc}}$-subspaces $K$ of an ${\mathrm{sc}}$-Banach space $E$ are characterized as follows. For finite dimensional subspaces $K$ of $E$ it holds:

[TABLE]

Although simple to prove, this equivalence is far reaching. In particular, since due to finite dimension the generated Banach subscale is constant (all levels $K_{m}=K$ are necessarily equal).

Section 2.2 “Examples” presents a number of examples of Banach scales, e.g. Sobolev scales and weighted Sobolev scales, that arise frequently in the study of solution spaces of differential equations on manifolds. The desire to simplify and, most importantly, to unify the many cumbersome steps of the classical treatment of analyzing solution spaces actually was the motivation to invent scale calculus; see e.g. the introductions to Hofer et al. (2005) and Hofer (2006).

Section 2.3 “Scale linear theory” carries over fundamental notions of linear operators to Banach scales. For example a scale linear operator is a linear operator $T\colon E\to F$ between ${\mathrm{sc}}$-Banach spaces which preserves levels, that is $T(E_{m})\subset F_{m}$ $\forall m$. For such $T$ the restriction to level $E_{m}$ takes values in $F_{m}$. The restriction as a map $T_{m}:=T|_{E_{m}}\colon E_{m}\to F_{m}$ is called a level operator. Now one can carry over (some) standard notions and properties of linear operators, say continuity, compactness, projections, and so on, by requiring each level operator to have that property. For instance, a scale continuous operator, called ${\mathrm{sc}}$-operator, is a scale linear operator $T\colon E\to F$ such that all level operators are continuous, that is $T_{m}\in{\mathcal{L}}(E_{m},F_{m})$ $\forall m$.

However, as soon as it comes to ${\mathrm{sc}}$-Fredholm operators, not level preservation $T(E_{m})\subset F_{m}$, but level – better regularity – improvement $S(E_{m})\subset F_{m+1}$ $\forall m$ becomes a key property. The latter are called ${\mathrm{sc}}^{+}$-operators. They have the property that all their level operators are compact.

Similarly, as mentioned earlier for Fredholm operators in the usual sense, finite dimensional and finite codimensional ${\mathrm{sc}}$-subspaces will enter the definition of ${\mathrm{sc}}$-Fredholm operators. Thus one needs the following two notions:

Firstly, the notion of ${\mathrm{sc}}$-splitting of $E=F\oplus G$ into an ${\mathrm{sc}}$-direct sum of ${\mathrm{sc}}$-subspaces $F$ and $G$ called ${\mathrm{sc}}$-complements of one another. Just as for Banach spaces any finite dimensional ${\mathrm{sc}}$-subspace admits an ${\mathrm{sc}}$-complement.

Secondly, the notion of ${\mathrm{sc}}$-quotient $E/A$. This allows to establish for finite codimensional ${\mathrm{sc}}$-subspaces existence of an ${\mathrm{sc}}$-complement (Proposition 2.3.20) and characterize them as follows (Lemma 2.3.21). For finite codimensional subspaces $A$ of $E$ it holds:

[TABLE]

It seems that so far the literature missed to spell out these two facts explicitly.

An ${\mathrm{sc}}$-Fredholm operator is an ${\mathrm{sc}}$-operator $T\colon E\to F$ such that there are ${\mathrm{sc}}$-splittings $E=K\oplus X$ with levels $E_{m}=K\oplus X_{m}$ and $F=Y\oplus C$ with levels $F_{m}=Y_{m}\oplus C$ where $K=\ker T$ is the kernel and $Y={\rm im\,}T$ is the image of $T$ and both $K$ and $C$ are of finite dimension. Looks fine already? Well, there is one condition missing yet.111 While $T$ as a map $X\to Y:={\rm im\,}T$ is an isomorphism, this is not yet guaranteed for the level operators as maps $T_{m}\colon X_{m}=X\mathop{\cap}E_{m}\to({\rm im\,}T)\mathop{\cap}F_{m}=Y_{m}$. Their images $T(X_{m})\subset Y_{m}$ a-priori are only subspaces. To get isomorphisms one needs to exclude elements of higher levels $X\setminus X_{m}$ getting mapped under $T$ to level $F_{m}$, in symbols $T(X\setminus X_{m})\mathop{\cap}F_{m}=\emptyset$.

The operator $T$ as a map $T\colon X\to Y$ must be an ${\mathrm{sc}}$-isomorphism (a bijective ${\mathrm{sc}}$-operator whose inverse is level preserving). This enforces level regularity of $T$ in the sense that $Te\in F_{m}$ implies $e\in E_{m}$. And it assures that the levels $Y\mathop{\cap}F_{m}$ generated by the ${\mathrm{sc}}$-subspace $Y={\rm im\,}T$ coincide with the images $T_{m}(X_{m})$ of the level operators.

It is then a consequence that the level operators $T_{m}\colon E_{m}\to F_{m}$ are all Fredholm with the same kernel $K$ and the same Fredholm index. Vice versa, if the level operators of an ${\mathrm{sc}}$-operator $T\colon E\to F$ are Fredholm and $T$ is level regular in the above sense, then $T$ is ${\mathrm{sc}}$-Fredholm.

The classical stability result that Fredholm property and index are preserved under addition of a compact linear operator carries over this way: The ${\mathrm{sc}}$-Fredholm property is preserved under addition of ${\mathrm{sc}}^{+}$-operators.

Section 2.4 “Scale differentiability” is where the revolution happens. Free difference quotients! Away with Fréchet mainstream suppression! Hofer, Wysocki, and Zehnder (2017) just did it, at least in infinite dimensions..

Remark 2.0.1**.**

Let $U\subset E$ and $V\subset F$ be open subsets of ${\mathrm{sc}}$-Banach spaces. An open subset of $E_{m}$ is given by $U_{m}:=U\cap E_{m}$. A scale continuous 222 Also called of class ${\mathrm{sc}}^{0}$ which by definition means level preserving and continuity of all restrictions as maps $f_{m}:=f|_{U_{m}}\colon U_{m}\to V_{m}$, called level maps.

map $f\colon U\to V$ is called continuously scale differentiable or of class ${\mathrm{sc}}^{1}$ if

• •

the upmost so-called diagonal map (of height one), namely $f$ as a map $f\colon U_{1}\to V_{0}$ is pointwise differentiable and

• •

its derivative $df(x)\in{\mathcal{L}}(E_{1},F_{0})$ admits a continuous linear extension

[TABLE]

from the dense subset $E_{1}$ to $E_{0}$ itself, called the ${\mathrm{sc}}$-derivative of $f$ at $x\in U_{1}$ and denoted by $Df(x)$. Furthermore, it is required that

• •

the tangent map $Tf\colon TU\to TV$ defined by

[TABLE]

is of class ${\mathrm{sc}}^{0}$. Here the tangent bundle of $U$ is the open subset333 To get the shifted scale $U^{k}$ forget the first $k$ levels: Its $m^{\rm th}$ level is $(U^{k})_{m}:=U_{m+k}$.

[TABLE]

of the Banach scale $E^{1}\oplus E^{0}$.

The third axiom, the one requiring level preservation and continuity of the level maps associated to the tangent map $Tf$, has a lot of consequences caused by the shift in the definition of the tangent bundle $TU:=U^{1}\oplus E^{0}$. For instance $Df(x)\colon E_{0}\to F_{0}$ restricts at points of better regularity, say $x\in U_{m+1}\subset U_{1}$, to (continuous) level operators $D_{\ell}f(x):=Df(x)|_{E_{\ell}}\colon E_{\ell}\to F_{\ell}$ for all levels between [math] and down to level $m$.

In general, the scale derivative only admits level operators $D_{\ell}f(x)$ for all levels $\ell$ down to the level right above the $x$-level!

The ${\mathrm{sc}}$-derivative $U_{m+1}\ni x\mapsto Df(x)$ viewed (horizontally) between equal levels $E_{m}\to F_{m}$ enjoys only continuity with respect to the compact open topology444 If $x_{\nu}\to x$ in $U_{m+1}$, then for each fixed $\zeta\in E_{m}$ one has $Df(x_{\nu})\zeta\to Df(x)\zeta$ in $F_{m}$.

whereas viewed as a diagonal map $Df$ is continuous with respect to the operator norm topology, i.e. $C^{0}$ as a map

[TABLE]

where the target carries the operator norm. But for these domains $Df=df$ pointwise, so ${\mathrm{sc}}^{1}$ implies that all diagonal maps of height one $f\colon U_{m+1}\to V_{m}$ are of class $C^{1}$ in the usual sense and this brings us to

Section 2.5 “Differentiability – Scale vs Fréchet”. Here we will see that higher scale differentiability f\in{\mathrm{sc}}^{\color[rgb]{0,1,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,1,1}\pgfsys@color@cmyk@stroke{1}{0}{0}{0}\pgfsys@color@cmyk@fill{1}{0}{0}{0}k}(U,V) implies that all ($\forall m$) diagonal maps $f\colon U_{m+\ell}\to V_{m}$ of height \ell\in\{0,\dots,{\color[rgb]{0,1,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,1,1}\pgfsys@color@cmyk@stroke{1}{0}{0}{0}\pgfsys@color@cmyk@fill{1}{0}{0}{0}k}\} are of class $C^{\ell}$ in the usual Fréchet sense.555 Note: Level maps $f_{m}:U_{m}\to V_{m}$ ($\ell=0$) of an ${\mathrm{sc}}^{k}$-map are only guaranteed to be continuous ($C^{0}$) no matter what is the value of $k$.

Vice versa, for a map $f\colon U\to V$ there is the following criterion to be of class {\mathrm{sc}}^{k{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}+1}}: For each $\ell\in\{0,\dots,k\}$ restriction produces height $\ell$ diagonal maps $f\colon U_{m+\ell}\to V_{m}$ $\forall m$ that are of class C^{\ell{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}+1}}.

Section 2.6 “Chain rule” proves this building block of calculus. It allows to construct scale manifolds by patching together local pieces of ${\mathrm{sc}}$-Banach spaces. If $f\colon U\to V$ and $g\colon V\to W$ are both of class ${\mathrm{sc}}^{1}$, then the composition $g\circ f$ is, too! The exclamation mark is due to the fact that applying an ${\mathrm{sc}}$-derivative one looses one level (of regularity), so one might expect to loose two levels when composing two ${\mathrm{sc}}^{1}$ maps. One doesn’t! This relies on the compactness axiom for the inclusions $E_{k}\hookrightarrow E_{k+1}$ in a Banach scale.

Section 2.7 “Boundary recognition” introduces the degeneracy index $d_{C}(x)$ of a point $x$ in what is called a partial quadrant $C$ in a Banach scale $E$. It takes the value [math] on interior points $x$, value $1$ on boundary points in the usual sense, and points with $d_{C}(x)\geq 2$ are corner points. We state without proof invariance of $d_{C}(x)$ under ${\mathrm{sc}}^{1}$-diffeomorphisms, that is ${\mathrm{sc}}^{1}$-maps with ${\mathrm{sc}}^{1}$-inverses. It is remarkable that ${\mathrm{sc}}$-smooth diffeomorphisms recognize boundary points and corners. In contrast, homeomorphisms also recognize boundaries, but not corners.

Section 2.8 “${\mathrm{sc}}$-manifolds ” defines an ${\mathrm{sc}}$-manifold as a paracompact Hausdorff space $X$ endowed with an equivalence class of ${\mathrm{sc}}$-smooth atlases. A continuous map $f:X\to Y$ between ${\mathrm{sc}}$-manifolds is called ${\mathrm{sc}}$-smooth if so are all representatives with respect to ${\mathrm{sc}}$-charts of $X$ and $Y$. An ${\mathrm{sc}}$-chart of $X$ takes values in an ${\mathrm{sc}}$-Banach space $E$ and so, due to compatibility of ${\mathrm{sc}}$-charts through ${\mathrm{sc}}$-diffeomorphisms, the level structure of $E$ is inherited by the ${\mathrm{sc}}$-manifold $X$. An important class of ${\mathrm{sc}}$-manifolds consists of loop spaces $X:=W^{1,2}(\SS^{1},M)$ for finite dimensional manifolds $M$. These are even strong ${\mathrm{sc}}$-manifolds, or ${\mathrm{ssc}}^{\infty}$-manifolds, in the sense that already level maps are smooth, as opposed to only the diagonal maps as is required for ${\mathrm{sc}}^{\infty}$. Given an ${\mathrm{sc}}$-manifold $X$, its tangent bundle is a map of the form $p\colon TX\to X^{1}$ that projects on the shifted ${\mathrm{sc}}$-manifold (forget level [math] of $X$).

After this survey of Chapter 2 you could, upon first reading, skip the remainder of Chapter 2 and proceed with the introduction to Chapter 3.

2.1 Scale structures

Scales of sets

Definition 2.1.1** (Scales).**

A scale of sets or a scale structure on a set $A$ is a nested sequence of subsets

[TABLE]

The subset $A_{m}$ is called the level $m$ of the scale and its elements points of regularity $m$. The elements of the intersection

[TABLE]

are called the smooth points of the scale. Given a level $A_{m}$, the enclosing levels $A_{0},\dots,A_{m-1}\supset A_{m}$ are called superlevels, the enclosed levels $A_{m}\supset A_{m+1},A_{m+2},\dots$ sublevels, of $A_{m}$.

Definition 2.1.2** (Subscale).**

A subscale of a scale of sets $A$ is a scale of sets $B$ whose levels are subsets of the corresponding levels of $A$, that is

[TABLE]

Definition 2.1.3** (Constant scale).**

The constant scale structure on a set $A$ is the one whose levels $A_{m}:=A$ are all given by $A$ itself.

Definition 2.1.4** (Induced scale $B^{\cap A}\subset A$).**

A scale structure on a set $A$ induces a scale structure on any subset $B\subset A$, called the induced scale or subscale generated by $B$, denoted by $B^{\cap A}$. By definition the $m^{\rm th}$ level

[TABLE]

is the part of $B$ in level $A_{m}$. Observe that $B_{\infty}:=\mathop{\cap}_{m\geq 0}B_{m}=B\mathop{\cap}A_{\infty}$.

Note that for an induced scale emptiness $B_{\infty}=\emptyset$ is possible, even if $A_{\infty}\not=\emptyset$.

Example 2.1.5** (Not every subscale is an induced scale).**

[TABLE]

Definition 2.1.6** (Shifted scale $A^{k}$).**

Forget the first $k$ levels of a scale $A$ and use $A_{k}$ as the new level zero to obtain the shifted scale $A^{k}$ with levels

[TABLE]

We sometimes abbreviate $A^{k}_{m}:=(A^{k})_{m}$.

Banach scales (${\mathrm{sc}}$-Banach spaces)

Definition 2.1.7** (Scale Banach space).**

A scale structure or an sc-structure on a Banach space $E$, is a nested sequence of linear spaces

[TABLE]

called levels such that the following axioms are satisfied. {labeling}(Banach levels)

Each level $E_{m}$ is a Banach space (coming with a norm $\mathopen{|}\cdot\mathclose{|}_{m}:=\mathopen{|}\cdot\mathclose{|}_{E_{m}}$).

The inclusions $E_{m}\stackrel{{\scriptstyle I_{m}}}{{\hookrightarrow}}E_{m-1}$ are compact linear operators for all $m$.

The set of smooth points $E_{\infty}:=\bigcap_{m\in{\mathbb{N}}_{0}}E_{m}$ is dense in each level $E_{m}$.

An sc-Banach space, also called a scale Banach space or a Banach scale, is a Banach space $E$ endowed with a scale structure.

Exercise 2.1.8** (sc-direct sum).**

The Banach space direct sum $E\oplus F$ of two ${\mathrm{sc}}$-Banach spaces $E$ and $F$ is a Banach scale with respect to the natural levels

[TABLE]

Exercise 2.1.9** (Finite dimensional Banach scales are constant).**

A finite dimensional Banach space $E$ has the unique ${\mathrm{sc}}$-structure $E_{0}=E_{1}=\dots$.

Exercise 2.1.10** (Infinite dimensional Banach scales $E$).**

Note that any inclusion operator $E_{m+\ell}\hookrightarrow E_{m}$ is compact, hence continuous. Show that

• (i)

every level $E_{m}$ is a dense subset of each of its superlevel Banach spaces;

• (ii)

no level $E_{m}$ is a closed subset of any of its superlevel Banach spaces. Equivalently, every level $E_{m}$ has a non-empty set complement in each of its superlevel Banach spaces, in symbols

[TABLE]

whenever $m\in{\mathbb{N}}$ und $\ell\in\{1,\dots,m\}$.

Definition 2.1.11**.**

A Banach scale is called reflexive (resp. separable) if every level is a reflexive (resp. separable) Banach space.

Lemma 2.1.12** (Induced nested sequences).**

Any subset of an ${\mathrm{sc}}$-Banach space $E$ induces via level-wise intersection a scale of sets; see (2.1.1). {labeling}(closed)**

A closed subset $A\subset E$ meets any level $E_{m}$ in a closed set $A_{m}=A\mathop{\cap}E_{m}$. If $A\subset E$ is a closed subspace, then the inclusion $i_{m}\colon A_{m}\hookrightarrow A_{m-1}$ is a compact linear operator between Banach spaces.

If $U\subset E$ is an open subset, then $U_{m}=U\mathop{\cap}E_{m}$ is open in $E_{m}$ and the set $U_{\infty}=U\mathop{\cap}E_{\infty}$ of smooth points is dense in every $U_{m}$.

Proof.

The intersection $A\mathop{\cap}E_{m}=(I_{1}\circ\dots\circ I_{m})^{-1}(A)$ is the pre-image under a continuous map; analogous for $U$. (Compactness): Pick a bounded subset $B$ of $A_{m}$. Then $B$ is a subset of all four spaces in the diagram

[TABLE]

The closure of $B$ in $E_{m-1}$ is compact since $I_{m}\circ\iota_{m}$ is a compact linear operator. But $A_{m-1}$ is a closed subspace of $E_{m-1}$ which contains $B$. Thus the closure of $B$ is contained in $A_{m-1}$ as well. (Density): Pick $p\in U_{\infty}=\bigcap_{k\in{\mathbb{N}}_{0}}(U\mathop{\cap}E_{k})\subset(U_{m}\mathop{\cap}E_{\infty})$. By density of $E_{\infty}$ in $E_{m}$ there is a sequence $E_{m}\ni p_{\nu}\to p$ in $E_{m}$. But $p\in U_{m}$ and $U_{m}\subset E_{m}$ is open. ∎

$\mathrm{Sc}$-subspaces I

As a closed linear subspace of a Banach space is a Banach space itself under the restricted norm, it is natural to call it a Banach subspace. In view of this the following definition seems natural in the setting of Banach scales.

Definition 2.1.13** (Banach subscale).**

A Banach subscale of a Banach scale $E$ is a Banach scale $B$ whose levels $B_{m}$ are Banach subspaces of the corresponding levels $E_{m}$ of $E$.

On the other hand, we just saw in Lemma 2.1.12 that a closed linear subspace $A$ in an ${\mathrm{sc}}$-Banach space $E$ generates a nested sequence $A^{\cap E}=(A\mathop{\cap}E_{m}\subset E_{m})_{m=0}^{\infty}$ of Banach subspaces. So it is natural to ask

Does the intersection sequence $A\mathop{\cap}E_{m}$ always form a Banach scale?

Answer: No. (Even if $\dim A<\infty$; see Lemma 2.1.16.)

• 2)

Is a Banach subscale $B\subset E$ generated by its top level $B$? In symbols, is every level $B_{m}$ given by intersection $B\mathop{\cap}E_{m}$?

Answer: Yes. (See Lemma 2.1.15.)

Definition 2.1.14** (Scale subspaces).**

An ${\mathrm{sc}}$-subspace of an ${\mathrm{sc}}$-Banach space $E$ is a closed subspace $A$ of $E$ whose intersections with the levels of $E$ form the levels of a Banach subscale of $E$.666 The axioms (Banach levels) and (compactness) are automatically satisfied for any closed subspace $B$ of $E$; see Lemma 2.1.12. The problematic axiom is (density).

Speaking of an ${\mathrm{sc}}$-subspace $A$ of $E$ implicitly carries the information that $A$ is the Banach subscale of $E$ whose levels are given by

[TABLE]

Alternatively $A^{\cap E}$ denotes the Banach scale generated by an ${\mathrm{sc}}$-subspace $A$.

Lemma 2.1.15**.**

a) The top level $B_{0}$ of a Banach subscale $B$ of a Banach scale $E$ is, firstly, an ${\mathrm{sc}}$-subspace of $E$ and, secondly, generates $B$ ($={B_{0}}^{\cap E}$). b) Every ${\mathrm{sc}}$-subspace of $E$ arises this way.

Proof.

a) By (density) of the set $B_{m}$ in the Banach space $B_{0}$, the closure ${\macc@depth\char 1\relax\frozen@everymath{\macc@group}\macc@set@skewchar\macc@nested@a 111{B}}^{\,0}$ with respect to the $B_{0}$ norm is the whole space $B_{0}$. Hence

[TABLE]

where identity two, also three, holds since $B_{m}$ itself is a closed subspace of the Banach space $E_{m}$ by axiom (Banach levels). b) By definition an ${\mathrm{sc}}$-subspace generates a Banach subscale. ∎

Lemma 2.1.16** (Finite dimensional ${\mathrm{sc}}$-subspaces).**

Given a scale Banach space $E$ and a finite dimensional linear subspace $B\subset E$. Then

[TABLE]

The ${\mathrm{sc}}$-subspace $B$ generates the constant Banach scale with levels $B_{m}=B$.

Proof.

’$\Rightarrow$’ The finite dimensional linear subspace $B\mathop{\cap}E_{\infty}=\bigcap_{m}(B\mathop{\cap}E_{m})=B_{\infty}$ of $B$ is dense by the (density) axiom for the subspace scale generated by $B$. Thus by finite dimension it is even equal to $B=B\mathop{\cap}E_{\infty}\subset E_{\infty}$. ’$\Leftarrow$’ By assumption $B\subset E_{\infty}\subset E_{m}$, thus $B_{m}:=B\mathop{\cap}E_{m}=B$. So $B$ generates the constant scale which by Exercise 2.1.9 is a Banach scale since $\dim B<\infty$. ∎

Example 2.1.17** (Closed but not ${\mathrm{sc}}$).**

Let $E=L^{2}([0,1])$ with the, even reflexive, Banach scale structure $E_{m}:=W^{m,2}([0,1])$. Then the characteristic function $\chi=\chi_{[0,1/2]}$ generates a 1-dimensional, thus closed, subspace $A$ of $E$. Since $\chi$ lies in $L^{2}$, but not in $W^{m,2}$ for $m\geq 1$, the levels $A_{m}:=A\cap E_{m}=\{0\}$ are trivial for $m\geq 1$, hence $A_{\infty}=\{0\}$ is not dense in $E_{0}=L^{2}$.

Exercise 2.1.18**.**

Infinite dimensional ${\mathrm{sc}}$-subspaces cannot lie inside $E_{\infty}$.

[Hint: Given an ${\mathrm{sc}}$-subspace $A\subset E$, show $A\subset E_{\infty}$ $\Rightarrow$ $A_{\infty}=A$, so $A_{1}=A$. But $A_{1}\subset E_{1}\hookrightarrow E$ embeds compactly in $E$, whereas $A$ is closed in $E$.]

The finding that for finite dimensional linear subspaces “being located in the set of smooth points” is equivalent to “generating a (constant) Banach subscale” is extremely useful. For instance, this enters the proofs of

• •

Prop. 2.3.17: Finite dimensional ${\mathrm{sc}}$-subspaces are ${\mathrm{sc}}$-complemented;

• •

Prop. 2.3.20: Finite codimension ${\mathrm{sc}}$-subspaces are ${\mathrm{sc}}$-complemented;

• •

Le. 2.3.21: Characterization of finite codimensional ${\mathrm{sc}}$-subspaces.

This list shows that certain classes of scale subspaces have properties analogous to the corresponding class of Banach subspaces.

Suppose $A$ and $B$ are ${\mathrm{sc}}$-subspaces of an ${\mathrm{sc}}$-Banach space $E$. How about the sum $A+B$ and the intersection $A\mathop{\cap}B$?

Is it possible, in general, to endow the sum $A+B$ and the intersection $A\mathop{\cap}B$ with the structure of Banach scales? So it is natural to ask the following.

Is the sum $A+B$ of ${\mathrm{sc}}$-subspaces always an ${\mathrm{sc}}$-subspace?

Answer: No. The sum of two closed subspaces, even in Hilbert space, is not even closed in general;777 The Hilbert space $l^{2}$ of square summarizable real sequences contains the closed subspaces $A:=\{a\in l^{2}\mid a_{2n}=0\;\forall n\}$ and $B:=\{b\in l^{2}\mid b_{2n}=b_{2n-1}/2n\,\forall n\}$. The sum $A+B$ cannot be closed, because it is dense in $l^{2}$ (since it contains all sequences of compact support) and $A+B$ is not all of $l^{2}$: Write $(1/n)_{n}\in l^{2}$ in the form $a+b$ with $a\in A$ and $b\in B$. Then $1/2n=a_{2n}+b_{2n}=b_{2n}=b_{2n-1}/2n$. So $b_{2n-1}=1$ for all $n$, hence $b\notin l^{2}$.

cf. Schochetman et al. (2001).

Answer: Yes, if $A$ and $B$ are finite dimensional.

Answer: Yes, if $A$ or $B$ is of finite codimension; see Exercise 2.3.23.

• 4)

Is the intersection $A\mathop{\cap}B$ of ${\mathrm{sc}}$-subspaces an ${\mathrm{sc}}$-subspace?

Answer: Yes, if $A$ or $B$ is finite dimensional.

Answer: Yes, if $A$ and $B$ are of finite codimension; see Exercise 2.3.23.

(General case: In each level $E_{m}$ the intersection $(A\mathop{\cap}B)\mathop{\cap}E_{m}=A_{m}\mathop{\cap}B_{m}$ is closed. How about density of $(A\mathop{\cap}B)_{\infty}$ in $A\mathop{\cap}B$?)

2.2 Examples

Throughout $\SS^{1}$ denotes the unit circle in ${\mathbb{R}}^{2}$ or, likewise, the quotient space ${\mathbb{R}}/{\mathbb{Z}}$. It is convenient to think of functions $f\colon\SS^{1}\to{\mathbb{R}}$ as $1$-periodic functions on the real line, that is $f\colon{\mathbb{R}}\to{\mathbb{R}}$ such that $f(t+1)=f(t)$ for every $t$.

By definition a counter-example is an example with negative sign.

Example 2.2.1** (Not a Banach scale).**

The vector space $C^{k}_{\rm bd}({\mathbb{R}})$ of $k$ times continuously differentiable functions $f\colon{\mathbb{R}}\to{\mathbb{R}}$ which, together with their derivatives up to order $k$, are bounded is a Banach space with respect to the $C^{k}$ norm. However, the scale whose (Banach levels) are $E_{m}:=C^{m}_{\rm bd}({\mathbb{R}})$ satisfies (density) since $E_{\infty}$ is equal to $C^{\infty}_{\rm bd}({\mathbb{R}})$, but it does not satisfy (compactness). A counter-example is provided by a bump running to infinity: Pick a bump, that is a compactly supported function $\chi\geq 0$ on ${\mathbb{R}}$, and set $\chi_{\nu}:=\chi(\cdot-\nu)$. Then the set $C:=\{\chi_{\nu}|\nu\in{\mathbb{N}}\}$ is bounded in $E_{1}$, indeed $\mathopen{\|}\chi_{\nu}\mathclose{\|}_{C^{1}}=\mathopen{\|}\chi\mathclose{\|}_{C^{1}}=:c_{\chi}<\infty$, but there is no convergent subsequence with respect to the $C^{0}$ norm, i.e. in $E_{0}$.

So non-compactness of the domain obstructs the (compactness) axiom. There are two ways to fix this. The obvious one is to use a compact domain; below we illustrate this by choosing the simplest one $\SS^{1}$. Another way is to impose a decay condition when approaching infinity. This works well for domains which are a product of a compact manifold with ${\mathbb{R}}$. Concerning targets, replacing ${\mathbb{R}}$ by ${\mathbb{R}}^{n}$ makes no difference in the arguments.

Exercise 2.2.2** (The non-reflexive Banach scale $C^{k}(\SS^{1})$).**

Show that the Banach space $C^{k}(\SS^{1})$ endowed with the scale structure whose levels are the Banach spaces $E_{m}:=C^{k+m}(\SS^{1})$ is a separable non-reflexive Banach scale.

[Hint: Concerning (compactness) use the Arzelà–Ascoli Theorem A.2.20. For separability see e.g. discussion in Weber (2017a, App. A).]

Example 2.2.3** (Sobolev scales – compact domain).**

Fix an integer $k\in{\mathbb{N}}_{0}$ and a real $p\in[1,\infty)$. The Sobolev space $W^{k,p}(\SS^{1},{\mathbb{R}}^{n})$ endowed with the scale structure whose levels are the Banach spaces $E_{m}:=W^{k+m,p}(\SS^{1},{\mathbb{R}}^{n})$ is a Banach scale. These Sobolev scales are separable ($1\leq p<\infty$) and reflexive ($1<p<\infty$) by Theorem A.3.1.

[Hints: Sobolev embedding theorems and $E_{\infty}=C^{\infty}(\SS^{1},{\mathbb{R}}^{n})$.]

Exercise 2.2.4** (Weighted Sobolev scales – non-compact domain ${\mathbb{R}}$).**

Fix a monotone cutoff function $\beta\in C^{\infty}({\mathbb{R}},[-1,1])$ with $\beta(s)-1$ for $s\leq-1$ and $\beta(s)=1$ for $s\geq 1$, as illustrated by Figure 2.1.

Given a constant $\delta\geq 0$, define an exponential weight function by

[TABLE]

Let $k\in{\mathbb{N}}_{0}$ and pick a constant $p\in(1,\infty)$. Check that the set defined by

[TABLE]

is a real vector space on which

[TABLE]

defines a complete norm. Consider a strictly increasing sequence

[TABLE]

of reals. Prove that the levels defined by

[TABLE]

form a Banach scale structure on the Banach space $L^{p}({\mathbb{R}},{\mathbb{R}}^{n})$.

Exercise 2.2.5** (Strictly increasing is necessary).**

Show that if two weights $\delta_{m-1}=\delta_{m}$ are equal in (2.2.4), then the (compactness) axiom fails.

Exercise 2.2.6** (Reflexivity and separability).**

Show that the weighted Sobolev space $W^{m,p}_{\delta}({\mathbb{R}},{\mathbb{R}}^{n})$ is a closed subspace of $W^{m,p}({\mathbb{R}},{\mathbb{R}}^{n})$. Conclude that the weighted Sobolev scales in the previous example are separable ($1\leq p<\infty$) and reflexive ($1<p<\infty$).

Example 2.2.7** (Completion scale – Hölder spaces are not Banach scales).**

Fix a constant $\mu\in(0,1)$. The sequence of Hölder spaces $E_{m}:=C^{m,\mu}(\SS^{1})$ for $m\in{\mathbb{N}}_{0}$ satisfies the (compactness) axiom by the Arzelà–Ascoli Theorem A.2.20, but the set of smooth points $E_{\infty}=C^{\infty}(\SS^{1})$ is not dense in any level $E_{m}$. However, taking the closure of $E_{\infty}$ in each level produces a Banach scale $\bar{E}_{m}:=\overline{E_{\infty}}^{k,\mu}$ called the completion scale. This works for every nested sequence of Banach spaces that satisfy (compactness) as shown by Fabert et al. (2016, Le. 4.11); they also solve Exercise 2.2.4.

Exercise 2.2.8**.**

For which $p\in[1,\infty]$, if any, is $L^{p}(\SS^{1})$ endowed with the levels $E_{m}:=L^{p+m}(\SS^{1})$ a Banach scale?

Definition 2.2.9** (Weighted Hilbert space valued Sobolev spaces).**

Let $k\in{\mathbb{N}}_{0}$, $p\in(1,\infty)$, and $\delta\geq 0$. Suppose $H$ is a separable Hilbert space and define the space $W^{k,p}_{\delta}({\mathbb{R}},H)$ by (2.2.3) with ${\mathbb{R}}^{n}$ replaced by $H$. This is again a Banach space; see Frauenfelder and Weber (2018, Appendix).

Example 2.2.10** (Path spaces for Floer homology).**

A a monotone unbounded function $f\colon{\mathbb{N}}\to(0,\infty)$ is called a growth function. Common types of Floer homologies provide such $f$, order refers to spatial order:

[TABLE]

Here Periodic and Lagrangian Floer homology refer, respectively, to the elliptic PDEs studied by Floer (1988b, 1989) on the cylinder ${\mathbb{R}}\times\SS^{1}$ and by Floer (1988a) imposing Lagrangian boundary conditions along the strip ${\mathbb{R}}\times[0,1]$. Hyperkähler and Heat flow Floer homology refer to the theories established by Hohloch, Noetzel, and Salamon (2009), respectively, by Weber (2013a, b, 2017b). The heat flow is described by a parabolic PDE that relates to Floer’s elliptic PDE; see Salamon and Weber (2006).

Given a constant $p\in(1,\infty)$, let $\delta_{m}$ for $m\in{\mathbb{N}}_{0}$ be a sequence as in (2.2.4). Given a growth function $f$, let $H_{m}=\ell^{2}_{f^{m}}$ be the fractal Hilbert scale on $H=\ell^{2}$ introduced by Frauenfelder and Weber (2018, Ex. 3.8). Then the Banach space $E_{m}$ is defined as intersection of $m+1$ Banach spaces, namely

[TABLE]

The norm on $E_{m}$ is the maximum of the $m+1$ individual norms. This is a complete norm. This endows $E=L^{p}({\mathbb{R}},H)$ with the structure of a Banach scale; see Frauenfelder and Weber (2018, Thm. 8.6).

2.3 Scale linear theory

2.3.1 Scale linear operators

Definition 2.3.1** **(Scale linear operators $T$ and their level

operators $T_{m}$).

• (i)

A scale linear operator is a linear operator $T\colon E\to F$ between Banach scales which is level preserving, that is $T(E_{m})\subset F_{m}$ for every $m\in{\mathbb{N}}_{0}$.

• (ii)

The restriction of a scale linear operator $T\colon E\longrightarrow F$ to a level of $E$ takes values in the corresponding level of $F$. Hence $T$ viewed as a map between corresponding levels is a linear operator

[TABLE]

between Banach spaces, called the $m^{\rm th}$** level operator**.

If a scale linear operator $T\colon E\to F$ is, in addition, a bijective map, then each level operator $T\colon E_{m}\rightarrowtail F_{m}$ is injective – but not necessarily surjective. It will be surjective if the inverse linear map $T^{-1}\colon F\to E$ is level preserving: In this case each level operator $(T^{-1})_{m}\colon F_{m}\rightarrowtail E_{m}$ is injective. This proves

Lemma 2.3.2**.**

Suppose a scale linear operator $T\colon E\to F$ is bijective and its inverse is level preserving. Then every level operator

[TABLE]

is a bijective linear map between Banach spaces.

Scale continuous operators

Definition 2.3.3** (${\mathrm{sc}}$-operators).**

A scale linear operator $T\colon E\to F$ is called scale continuous or scale bounded or of class ${\mathrm{sc}^{0}}$, if each level operator $T_{m}\in{\mathcal{L}}(E_{m},F_{m})$ is a continuous linear operator between Banach spaces.

[TABLE]

Such $T$ is called an sc-operator between Banach scales. In the realm of scale linear operators ${\mathrm{sc}}$ does not abbreviate scale, but scale continuous. Let $\mathcal{L}_{\mathrm{sc}}(E,F)$ be the set of ${\mathrm{sc}}$-operators between the Banach scales $E$ and $F$.

Exercise 2.3.4**.**

Check that $\mathcal{L}_{\mathrm{sc}}(E,F)$ is a linear space.

Exercise 2.3.5**.**

Given $\mathcal{L}_{\mathrm{sc}}(E,F)$, consider the sequence $\left({\mathcal{L}}_{m}\right)_{m\in{\mathcal{N}}_{0}}$ of Banach spaces ${\mathcal{L}}_{m}:={\mathcal{L}}(E_{m},F_{m})$ under the operator norm. Characterize the case in which one has inclusions ${\mathcal{L}}_{m+1}\subset{\mathcal{L}}_{m}$ as a) sets and b) continuous maps between Banach spaces. In b) characterize the case in which c) the set ${\mathcal{L}}_{\infty}:=\mathop{\cap}_{m}{\mathcal{L}}_{m}$ is dense in each level ${\mathcal{L}}_{m}$ and d) every inclusion operator ${\mathcal{L}}_{m+1}\hookrightarrow{\mathcal{L}}_{m}$ is compact.

Definition 2.3.6** (${\mathrm{sc}}$-projections).**

An sc-projection is a scale continuous operator $P$ whose level operators $P_{m}$ are all projections, i.e. $P_{m}\circ P_{m}=P_{m}$. Equivalently, the ${\mathrm{sc}}$-projections are those ${\mathrm{sc}}$-operators $P$ with $P^{2}=P$.

Lemma 2.3.7** (Image and kernel of ${\mathrm{sc}}$-projections are ${\mathrm{sc}}$-subspaces).**

The image, hence the kernel, of any $P=P^{2}\in\mathcal{L}_{\mathrm{sc}}(E)$ are ${\mathrm{sc}}$-subspaces.

Proof.

As $Q:={{\mathchoice{\rm 1\mskip-4.0mul}{\rm 1\mskip-4.0mul}{\rm 1\mskip-4.5mul}{\rm 1\mskip-5.0mul}}}-P$ is an ${\mathrm{sc}}$-projection whose image is the kernel of $P$, it suffices to show that the images $R_{m}:={\rm im\,}P_{m}={\rm Fix}\,P_{m}$ form a Banach subscale of $E$. The inclusion $R_{m+1}\subset R_{m}$ holds by $E_{m+1}\subset E_{m}$. And $R_{m}={\rm Fix}\,P_{m}$ is a closed (linear) subspace of $E_{m}$ by continuity and linearity of $P_{m}$. (Compactness) of the inclusion $I_{m}\colon E_{m}\hookrightarrow E_{m-1}$, together with $R_{m}\subset E_{m}$ being closed, tells that each inclusion $i_{m}\colon R_{m}\hookrightarrow R_{m-1}$ takes bounded sets into pre-compact ones.

It remains to check (density) of $R_{\infty}=\mathop{\cap}_{\ell}R_{\ell}$ in $R_{m}$. To see this pick $r_{m}\in R_{m}\subset E_{m}$ and, by density of $E_{\infty}$ in $E_{m}$, pick some in $E_{m}$ convergent sequence $E_{\infty}\ni e_{\nu}\to r_{m}$. Since $R_{m}={\rm Fix}\,P_{m}$ and by continuity of $P_{m}$ we get

[TABLE]

For each $e_{\nu}$ it holds that

[TABLE]

Here the first equality holds since $e_{\nu}\in E_{\infty}\subset E_{m}$ and $P_{m}$ is the restriction of $P$ to $E_{m}$, so $P_{m}e_{\nu}=Pe_{\nu}$. But $P$ preserves levels and $e_{\nu}$ lies in every $E_{\ell}$, so $Pe_{\nu}$ lies in every $E_{\ell}$ and $Pe_{\nu}=P_{\ell}e_{\nu}\in{\rm im\,}P_{\ell}=R_{\ell}$. Thus $Pe_{\nu}\in R_{\infty}$. ∎

Definition 2.3.8** (${\mathrm{sc}}$-isomorphisms).**

A (linear) sc-isomorphism is a bijective ${\mathrm{sc}}$-operator whose inverse888 Inverses of bijective ${\mathrm{sc}}$-operators are not automatically level preserving: Consider the identity operator from the forgetful Banach scale $E_{0}\supset E_{2}\supset E_{3}\dots$ to $E_{0}\supset E_{1}\supset E_{2}\dots$

is level preserving.

Exercise 2.3.9**.**

For an ${\mathrm{sc}}$-isomorphism $T\colon E\to F$ all level operators

[TABLE]

are continuous bijections with continuous inverses.

[Hint: Bounded inverse theorem, equivalently, open mapping theorem.]

Scale compact operators include ${\mathrm{sc}}^{+}$-operators

Definition 2.3.10** (Scale compact operators).**

An sc-compact operator is a scale linear operator $S\colon E\to F$ whose level operators $S_{m}\colon E_{m}\to F_{m}$ are all compact (hence bounded) linear operators between Banach spaces.

Scale compact operators are ${\mathrm{sc}}$-operators, i.e. elements of $\mathcal{L}_{\mathrm{sc}}$.

Definition 2.3.11** (${\mathrm{sc}}^{+}$-operators).**

Suppose $E$ and $F$ are scale Banach spaces. Recall that $F^{1}$ denotes the Banach scale that arises from $F$ by forgetting the $1^{\rm st}$ level $F_{0}$ and taking $F_{1}$ as the new level [math]. The elements $S\in\mathcal{L}_{\mathrm{sc}}(E,F^{1})$ are called ${\mathrm{sc}}^{+}$-operators and we use the notation

[TABLE]

Remark 2.3.12** (${\mathrm{sc}}^{+}$-operators are scale compact).**

The (compactness) axiom not only shows $\mathcal{L}_{\mathrm{sc}}^{+}(E,F)\subset\mathcal{L}_{\mathrm{sc}}(E,F)$, but also that any ${\mathrm{sc}}^{+}$-operator $S\colon E\to F$ is ${\mathrm{sc}}$-compact: This follows from the commutative diagram

[TABLE]

since the composition of a bounded and a compact linear operator is compact.

Remark 2.3.13** (Are scale compact operators always ${\mathrm{sc}}^{+}$-operators?).**

No: Let $E$ be an infinite dimensional Banach scale. The inclusion $\iota\colon E^{1}\to E$ has compact level operators $\iota_{m}\colon E_{m+1}\to E_{m}$ and it is an ${\mathrm{sc}}^{+}$-operator, indeed $\iota\in\mathcal{L}_{\mathrm{sc}}(E^{1},E^{1})=:\mathcal{L}_{\mathrm{sc}}^{+}(E^{1},E)$. Now forget level one in $E^{1}$ and in $E$, denote the resulting Banach scales by $E^{1}_{\times 1}$ and $E_{\times 1}$, respectively. All level operators of the inclusion $\iota\colon E^{1}_{\times 1}\to E_{\times 1}$ are still compact, but $\iota$ does not even map level zero $(E^{1}_{\times 1})_{0}=E_{1}$ to level one $(E_{\times 1})_{1}=E_{2}$, let alone be continuous.

$\mathrm{Sc}$-subspaces II

Direct sum and ${\mathrm{sc}}$-complements of ${\mathrm{sc}}$-subspaces

Definition 2.3.14**.**

An ${\mathrm{sc}}$-subspace $F$ of a Banach scale $E$ is called sc-complemented if there is an ${\mathrm{sc}}$-subspace $G\subset E$ such that every Banach space direct sum of corresponding levels

[TABLE]

is equal to the ambient level $E_{m}$. Such $G$ is called an sc-complement of $F$. So the Banach space $F\oplus G$ carries the natural Banach scale structure (2.1.2). Such a pair $(F,G)$ or such direct sum $F\oplus G$ is called an sc-splitting of $E$.

Exercise 2.3.15**.**

Let $G$ be an ${\mathrm{sc}}$-complement of $F\subset E$. Check that the Banach space $F\oplus G$ together with the natural level structure (2.1.2) indeed satisfies the axioms of a Banach scale.

Exercise 2.3.16** (Sc-projections ${\mathrm{sc}}$-split).**

There is an ${\mathrm{sc}}$-splitting

[TABLE]

associated to any ${\mathrm{sc}}$-projection, i.e. any idempotent $P=P^{2}\in\mathcal{L}_{\mathrm{sc}}(E)$.

[Hint: $P_{m}=(P_{m})^{2}\in{\mathcal{L}}(E_{m})$ means $E_{m}=\ker P_{m}\oplus{\rm Fix}\,P_{m}$. Lemma 2.3.7.]

Proposition 2.3.17**.**

Finite dimensional ${\mathrm{sc}}$-subspaces are ${\mathrm{sc}}$-complemented.

Proof.

We recall the proof given in Hofer et al. (2017, Prop. 1.1). Suppose $F$ is a finite dimensional ${\mathrm{sc}}$-subspace of a Banach scale $E$. Then $F\subset E_{\infty}$ by Lemma 2.1.16 and $F$ generates the constant Banach scale with levels $F_{m}:=F\mathop{\cap}E_{m}=F$ by Exercise 2.1.9. Pick a basis $e_{1},\dots,e_{k}$ of $F\subset E_{\infty}$ and let $e_{1}^{*},\dots,e_{k}^{*}\in F^{*}$ be the dual basis. By the Hahn–Banach Theorem A.2.15 any $e_{i}^{*}$ extends to a continuous linear functional $\lambda_{i}$ on $E$. The linear operator $P\colon E\to E$ defined by $P(x):=\sum_{i=1}^{k}\lambda_{i}(x)e_{i}$ is continuous and satisfies $P\circ P=P$ by straightforward calculation. Note that the image of $P$ is $F$, that is $P(E)=F$, and that $F$ is contained in $E_{\infty}$, hence in every level $E_{m}$. This shows that $P$ is level preserving and admits level operators $P_{m}\colon E_{m}\to E_{m}$. By the continuous inclusion $E_{m}\hookrightarrow E$ the restrictions, still denoted by $\lambda_{i}$, are continuous linear functionals $\lambda_{i}\colon E_{m}\hookrightarrow E\to{\mathbb{R}}$ on every level $E_{m}$. By the same arguments as for $P$ every level operator $P_{m}$ is a continuous linear projection $P_{m}\colon E_{m}\to E_{m}$ with image $P(E_{m})=F$. Hence $P\in\mathcal{L}_{\mathrm{sc}}(E)$ is an ${\mathrm{sc}}$-projection.

Goal. Given the finite dimensional ${\mathrm{sc}}$-subspace $F\subset E$ (generating the constant Banach scale $F_{m}=F$), find a closed subspace $G\subset E$ such that

• a)

$G_{m}:=G\mathop{\cap}E_{m}$ are the levels of a Banach subscale ($G$ is an ${\mathrm{sc}}$-subspace);

• b)

$F_{m}\oplus G_{m}=E_{m}$ for every $m$.

Solution. The subspace of $E$ defined by $G:=({{\mathchoice{\rm 1\mskip-4.0mul}{\rm 1\mskip-4.0mul}{\rm 1\mskip-4.5mul}{\rm 1\mskip-5.0mul}}}-P)E$ is closed since $G=\ker P$ and $P$ is continuous. a) By Lemma 2.1.12 only (density) remains to be checked. To see that $G_{\infty}:=\bigcap_{m}G_{m}=G\mathop{\cap}E_{\infty}$ is dense in any level $G_{m}$ pick $g\in G_{m}\subset E_{m}$. By density of $E_{\infty}$ in $E_{m}$ choose a sequence $e_{\nu}\in E_{\infty}$ that converges in $E_{m}$ to $g\in E_{m}$. The sequence $g_{\nu}:=({{\mathchoice{\rm 1\mskip-4.0mul}{\rm 1\mskip-4.0mul}{\rm 1\mskip-4.5mul}{\rm 1\mskip-5.0mul}}}-P_{m})e_{\nu}$ lies in $G\mathop{\cap}E_{m}=:G_{m}$ and converges in $G_{m}$ to $g$: Indeed {{\color[rgb]{0,1,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,1,1}\pgfsys@color@cmyk@stroke{1}{0}{0}{0}\pgfsys@color@cmyk@fill{1}{0}{0}{0}e_{\nu}-g_{\nu}}=P_{m}e_{\nu}=P_{m}(e_{\nu}-g)}, since $g\in G_{\infty}\subset G_{m}=\ker P_{m}$, so together with $\mathopen{\|}P_{m}\mathclose{\|}\leq 1$ we get

[TABLE]

b) For $x\in F\mathop{\cap}G={\rm im\,}P\mathop{\cap}\ker P$ one has $x=Py$ for a $y\in E$, hence $0=Px=PPy=Py=x$. So the intersection of subspaces $F_{m}\mathop{\cap}G_{m}=\{0\}$ is trivial, too. It remains to show the equality ${\rm im\,}P_{m}+\ker P_{m}=E_{m}$, $\forall m$. ’$\subset$’ Obvious. ’$\supset$’ Pick $e\in E_{m}$ and set $f:=P_{m}e$ and $g:=e-P_{m}e$. ∎

Exercise 2.3.18**.**

Give an example of a finite dimensional subspace $F$ of a Banach scale that is not ${\mathrm{sc}}$-complemented. [Hint: Pick $f\in C^{0}(\SS^{1})\setminus C^{1}(\SS^{1})$.]

Quotient Banach scales

If you are not familiar with the quotient construction for Banach spaces, have a look at the neighborhood of Proposition A.2.7 and its proof for definitions and explanations. Understanding that proof helps to prove

Proposition 2.3.19** (Quotient Banach scales).**

Let $E$ be a Banach scale and $A$ an ${\mathrm{sc}}$-subspace. Then the quotient Banach space $E/A$ with levels

[TABLE]

and inclusions

[TABLE]

is a Banach scale.

By Proposition A.2.7 the norm on the coset space $E_{m}/A_{m}$ defined by

[TABLE]

is complete. It is called the quotient norm and measures the distance between the coset $x+A$ and the zero coset, the subspace $A$ itself.

Proof.

By the ${\mathrm{sc}}$-subspace assumption on $A$ every level $A_{m}:=A\mathop{\cap}E_{m}$ is a Banach subspace of $E_{m}$, hence the quotient spaces $E_{m}/A_{m}$ endowed with the norms $\mathopen{\|}\cdot\mathclose{\|}_{m}$ are (Banach levels). To prove that the natural inclusions (2.3.5) are compact linear operators pick a sequence $x_{\nu}+A_{m+1}$ in the unit ball of $E_{m+1}/A_{m+1}$. (Note that $x_{\nu}\in E_{m+1}$.) This means that the distance of each $x_{\nu}$ to the zero coset $A_{m+1}$ of $E_{m+1}/A_{m+1}$ is not larger than $1$. Hence for each $x_{\nu}$ there is a point $a_{\nu}$ in the zero coset $A_{m+1}$ at $E_{m+1}$ distance less than $2$, that is $\mathopen{|}x_{\nu}-a_{\nu}\mathclose{|}_{m+1}<2$. What we did is to choose for the given bounded sequence of cosets $x_{\nu}+A_{m+1}=x_{\nu}-a_{\nu}+A_{m+1}$ a sequence of new representatives $x_{\nu}-a_{\nu}$ which, most importantly, is bounded in $E_{m+1}$. By compactness of the inclusion $E_{m+1}\hookrightarrow E_{m}$ there is a subsequence, still denoted by $x_{\nu}-a_{\nu}$, which converges to some element $x\in E_{m}$. By continuity of the quotient projection $\pi_{m}\colon E_{m}\to E_{m}/A_{m}$, $x\mapsto x+A_{m}$, see Proposition A.2.7, we obtain that

[TABLE]

This proves the (compactness) axiom. The set of smooth points

[TABLE]

is dense in every level $E_{m}/A_{m}$, because the image of a dense subset $E_{\infty}\subset E_{m}$ under the continuous surjection $\pi_{m}\colon E_{m}\to E_{m}/A_{m}$ is dense in the target space by Lemma A.1.23. This proves the (density) axiom and Proposition 2.3.19. ∎

It seems that so far the literature misses out on the analogues for finite codimensional ${\mathrm{sc}}$-subspaces of Proposition 2.3.17 (existence of ${\mathrm{sc}}$-complement for finite dimensional ${\mathrm{sc}}$-subspaces) and Lemma 2.1.16 (characterization of finite dimensional ${\mathrm{sc}}$-subspaces). Let’s change this.

Proposition 2.3.20**.**

Finite codimension ${\mathrm{sc}}$-subspaces are ${\mathrm{sc}}$-complemented.

Of course, the asserted ${\mathrm{sc}}$-complement $C$ in Proposition 2.3.20 has as dimension the mentioned finite codimension. Hence $C$ carries the constant Banach scale structure and consists of smooth points only; see Lemma 2.1.16.

Proof.

Let $A$ be an ${\mathrm{sc}}$-subspace of a scale Banach space $E$ of finite codimension $r={\rm codim\,}A:=\dim E/A$. By closedness and finite codimension the subspace $A$ of $E$ has a topological complement $C$; cf. Brezis (2011, Prop. 11.6). Since $A$ is the kernel of the quotient projection $\pi\colon E=A\oplus C\twoheadrightarrow E/A$ defined in (A.2.8) we get $\dim C=\dim E/A=r$ for any topological complement.

Recall that a finite dimensional ${\mathrm{sc}}$-complement of $A$ is an ${\mathrm{sc}}$-subspace $C$, endowed with constant levels $C_{m}=C$, such that

[TABLE]

Constructing such $C$ inside the vector space $E_{\infty}$ of smooth points, see Lemma 2.1.16, is equivalent to $C$ being an ${\mathrm{sc}}$-subspace.

To define $C$ observe that the Banach scale $E$ and the ${\mathrm{sc}}$-subspace $A$ give rise to the quotient Banach scale in Proposition 2.3.19. Because the top level $(E/A)_{0}=E/A$ is of finite dimension $r$, all sublevels are finite dimensional and therefore the quotient Banach scale is actually constant. Note that

[TABLE]

because both sides are of the same finite dimension $r$ and there is the natural inclusion $E_{\infty}/A_{\infty}\to E_{m}/A_{m}$, $\varphi+A_{\infty}\mapsto\varphi+A_{m}$. Pick a basis of

[TABLE]

say $\varphi_{1}+A_{\infty},\dots,\varphi_{r}+A_{\infty}$. Observe that each $\varphi_{j}\in E_{\infty}$ and define

[TABLE]

We show that $C$ is a topological complement of $A_{m}$. To prove $A_{m}\mathop{\cap}C=\{0\}$, pick $c\in A_{m}\mathop{\cap}C$. The quotient projection $\pi_{m}\colon E_{m}\to E_{m}/A_{m}$, $e\mapsto e+A_{m}$, whose kernel is $A_{m}$ maps $c$ to the zero coset $0+A_{m}$. On the other hand, the only element of $C$ that gets mapped to the zero coset under $\pi_{m}$ is $c=0$. We prove that $A_{m}+C=E_{m}$: “$\subset$” Obvious since $A_{m}\subset E_{m}$ and $C\subset E_{\infty}\subset E_{m}$. “$\supset$” Pick $e\in E_{m}$ and express $\pi_{m}(e)=e+A_{m}\in E_{m}/A_{m}$ in terms of the basis $\{\varphi_{j}+A_{m}\}_{j=1}^{r}$, let $c^{1},\dots,c^{r}\in{\mathbb{R}}$ be the coefficients. Set $c:=\sum_{j=1}^{r}c^{j}\varphi_{j}\in C$. Then $\pi_{m}(c)=\pi_{m}(e)$, hence $a:=e-c$ lies in the kernel of $\pi_{m}$ which is $A_{m}$. Hence $e=a+c$ is of the desired form. ∎

A finite codimensional closed subspace of an ordinary Banach space $X$ not only admits a topological complement, but there is even one in each dense subspace $X_{\infty}$ of $X$; see e.g. Hofer et al. (2007, Le. 2.12) or Brezis (2011, Prop. 11.6). This enters the proof of

Lemma 2.3.21** (Finite codimensional ${\mathrm{sc}}$-subspaces).**

Suppose $E$ is a scale Banach space and $A$ is a linear subspace, then

[TABLE]

*whenever $A\subset E$ is of finite codimension $r$. 999 A finite codimension subspace in Banach space need not be closed; see e.g. Brezis (2011, Prop. 11.5). *

Proof.

An ${\mathrm{sc}}$-subspace is closed by definition. To prove the reverse implication, let $A$ be a closed subspace of $E$ of finite codimension, say $r$. By the result mentioned above the subspace $A$ of $E$ admits a topological complement $C$ contained in the dense subset $E_{\infty}$. In the proof of Proposition 2.3.20 we saw that topological complements satisfy $\dim C={\rm codim\,}A=:r$.

We need to show that the levels defined by $A_{m}:=A\mathop{\cap}E_{m}\subset E_{m}$ satisfy the three axioms of a Banach scale. As $A$ is closed in $E$, by Lemma 2.1.12 only the (density) axiom remains to be shown: density of $A_{\infty}$ in each $A_{m}$.

The inclusion $\underline{C\subset E_{\infty}}$ means that $C$ is a constant Banach scale by Lemma 2.1.16 and Exercise 2.1.9. Before proving density we show that

[TABLE]

is a direct sum of closed subspaces of the Banach space $E_{m}$: Firstly, closedness of $A_{m}$ we already know and $C$ is closed due to its finite dimension. Secondly, trivial intersection $A_{m}\mathop{\cap}C=\{0\}$ holds true since it even holds for the larger space $A\supset A_{m}$. Thirdly, we prove $A_{m}+C=E_{m}$. “$\subset$” Obvious. “$\supset$” Any $e\in E_{m}\subset E=A+C$ is of the form $e=a+c$ for some $a\in A$ and $c\in C$. But $a=e-c$ is also in $E_{m}$ (so we are done), because both $e$ and $c\in\underline{C\subset E_{\infty}}\subset E_{m}$ are and $E_{m}$ is a vector space.

We prove density of $A_{\infty}$ in $A_{m}$. Given $a\in A_{m}\subset E_{m}$, by density of $E_{\infty}$ in $E_{m}$ there is some in the $E_{m}$ norm convergent sequence $E_{\infty}\ni e_{\nu}\to a\in E_{m}$. On the other hand, by (2.3.6) there is the direct sum of Banach spaces $E_{m}=A_{m}\oplus C$, so $e_{\nu}$ is of the form $e_{\nu}=a_{\nu}+c_{\nu}$ with $(a_{\nu},c_{\nu})\in A_{m}\times C$. Clearly $a_{\nu}-a+c_{\nu}=e_{\nu}-a\to 0$ in $E_{m}$. But most importantly $a_{\nu}=e_{\nu}-c_{\nu}\in E_{\infty}$ since the linear space $E_{\infty}$ contains $e_{\nu}$ and $c_{\nu}\in\underline{C\subset E_{\infty}}$. So

[TABLE]

Since $A_{m}$ and $C$ are topological complements of one another, see (2.3.6), the norm in $E_{m}$ splits in the following sense. By Brezis (2011, Thm. 2.10) there is a constant $\mu\geq 0$ such that for any element $e$ of $E_{m}$ the norms of its parts in $A_{m}$ and in $C$ are bounded above by $\mu\mathopen{|}e\mathclose{|}_{m}$. For $e:=e_{\nu}-a=(a_{\nu}-a)+c_{\nu}\in A_{m}+C$ we get that

[TABLE]

Hence $A_{\infty}\ni a_{\nu}\to a\in A_{m}$ in the $E_{m}$ norm. This proves Lemma 2.3.21. ∎

Corollary 2.3.22** (Closed finite codimensional subspaces ${\mathrm{sc}}$-split $E$).**

Given a scale Banach space $E$ and a finite codimension $r$ subspace $A$, then

[TABLE]

The ${\mathrm{sc}}$-splitting $E=A\oplus C$ has levels $E_{m}=(A\mathop{\cap}E_{m})\oplus C$ and $\dim C=r$.

Proof.

“$\Rightarrow$” Lemma 2.3.21 and Proposition 2.3.20. “$\Leftarrow$” An ${\mathrm{sc}}$-subspace is closed by definition. ∎

Exercise 2.3.23** (Intersection and sum of ${\mathrm{sc}}$-subspaces).**

a) If $A,B\subset E$ are finite dimensional ${\mathrm{sc}}$-subspaces, so are $A\mathop{\cap}B$ and $A+B$.

b) If $A,B\subset E$ are finite codimensional ${\mathrm{sc}}$-subspaces, so are $A\mathop{\cap}B$ and $A+B$.

[Hints: a) Lemma 2.1.16. b) By Lemma 2.3.21 it suffices to show for $A\mathop{\cap}B$ and for $A+B$ closedness101010 Brezis (2011, Prop. 11.5): A subspace containing a closed one of finite codimension is closed.

and finite codimension.111111 ${\rm codim\,}(A\mathop{\cap}B)\leq{\rm codim\,}A+{\rm codim\,}B$ and ${\rm codim\,}(A+B)\leq\min\{{\rm codim\,}A,{\rm codim\,}B\}$.

2.3.2 Scale Fredholm operators

Definition 2.3.24** (${\mathrm{sc}}$-Fredholm operators).**

An sc-Fredholm operator is an ${\mathrm{sc}}$-operator $T\colon E\to F$ that satisfies the following axioms, namely {labeling}(${\mathrm{sc}}$-isomorphism)

there are ${\mathrm{sc}}$-splittings $E=K\oplus X$, $F=Y\oplus C$ such that

$K$ is the kernel of $T$ and of finite dimension,

$Y$ is the image of $T$ and $C$ is of finite dimension,

the operator $T$ viewed as a map $T\colon X\to Y$ is an ${\mathrm{sc}}$-isomorphism.

The Fredholm index of $T$ is the integer

[TABLE]

By finite dimension the Banach subscales generated by $K$ and $C$ are constant. So trivially one gets the identities $K=K_{\infty}$ and $C=C_{\infty}$. Combined with the equally trivial inclusions $K_{\infty}\subset E_{\infty}$ and $C_{\infty}\subset F_{\infty}$ they provide the precious information that $K\subset E_{\infty}$ and $C\subset F_{\infty}$ consist of smooth points.

Proposition 2.3.25**.**

${\mathrm{sc}}$-Fredholm operators $T\colon E\to F$ are regularizing: If $T$ maps $e\in E$ to level $m$, then already $e$ was in level $m$; cf (2.3.10).

Proof.

Let $e\in E$ and $Te\in F_{m}$. But $F_{m}=T(X_{m})\oplus C$, so $Te=Tx+c$ for some $x\in X_{m}\subset E_{m}\subset E$ and $c\in C$. As $T(E)\mathop{\cap}C=Y\mathop{\cap}C=\{0\}$ and $e-x\in E$, the identity $T(e-x)=c$ shows that both sides are zero. So e-x\in\ker T=K\mathop{{\color[rgb]{1,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,1}\pgfsys@color@cmyk@stroke{0}{1}{0}{0}\pgfsys@color@cmyk@fill{0}{1}{0}{0}\text{\boldmath=}}}K_{\infty}\subset E_{\infty}\subset E_{m}. Therefore $e=(e-x)+x\in E_{m}$. ∎

Exercise 2.3.26** **(Intersection level $Y_{m}=Y\mathop{\cap}F_{m}$

is image of level operator $T_{m}$).

Consider an ${\mathrm{sc}}$-Fredholm operator $T\colon E=K\oplus X\to F=Y\oplus C$ where the ${\mathrm{sc}}$-subspace $Y:={\rm im\,}T$ is the image $T(E)=T(X)$. Recall that an ${\mathrm{sc}}$-subspace $Y$ generates a Banach subscale whose levels are given by intersection $Y\mathop{\cap}F_{m}$. Show that for ${\mathrm{sc}}$-Fredholm operators each intersection level is equal to the image of the corresponding level operator $T_{m}\colon E_{m}\to F_{m}$, i.e.

[TABLE]

[Hint: “\mathop{{\color[rgb]{1,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,1}\pgfsys@color@cmyk@stroke{0}{1}{0}{0}\pgfsys@color@cmyk@fill{0}{1}{0}{0}\text{\boldmath\subset}}}” Suppose $Te=:y\in F_{m}$ where $e\in E$.]

Exercise 2.3.27** (Isn’t the axiom (${\mathrm{sc}}$-isomorphism) superfluous?).**

In view of Exercise 2.3.26 the fourth axiom in Definition 2.3.24 seems to be a consequence of the previous three axioms. Is it?

Exercise 2.3.28**.**

The composition $T\circ T^{\prime}$ of two ${\mathrm{sc}}$-Fredholm operators is an ${\mathrm{sc}}$-Fredholm operator and $\mathop{\mathrm{index}}(T\circ T^{\prime})=\mathop{\mathrm{index}}T+\mathop{\mathrm{index}}T^{\prime}$.

Proposition 2.3.29** (Stability of ${\mathrm{sc}}$-Fredholm property).**

Consider an ${\mathrm{sc}}$-Fredholm operator $T\colon E\to F$ and an ${\mathrm{sc}}^{+}$-operator $S\colon E\to F$, then their sum $T+S$ is also an ${\mathrm{sc}}$-Fredholm operator of the same Fredholm index.

Proof.

The sum $T+S\colon E\to F$ is an ${\mathrm{sc}}$-operator. How about ${\mathrm{sc}}$-splittings?

Domain splitting. Both $T$ and $S$ provide level operators $E_{m}\to F_{m}$ that are Fredholm and compact, respectively. Hence the sum level operators $(T+S)_{m}=T_{m}+S_{m}\colon E_{m}\to F_{m}$ are Fredholm for each level $m$. Note that the kernel $K_{m}$ of $(T+S)_{m}$ contains $K_{m+1}$. To see the reverse inclusion pick $x\in K_{m}$. Then $Tx=-Sx\in F_{m\underline{+1}}$ by the ${\mathrm{sc}}^{+}$ nature of $S$. Thus $x\in E_{m+1}$ by the regularity Proposition 2.3.25. Hence $x\in K_{m+1}$. Thus $K_{m}=K_{m+1}$. So $K:=K_{0}=K_{m}=K_{\infty}$ is finite dimensional and $K=K_{\infty}\subset E_{\infty}$. Hence $K$ is an ${\mathrm{sc}}$-subspace by Lemma 2.1.16 and generates a constant Banach scale. By Proposition 2.3.17 the kernel scale $K$ admits an ${\mathrm{sc}}$-complement $X$ in $E$. Summarizing, we have

[TABLE]

for every $m\in{\mathbb{N}}_{0}$ and where $K$ does not depend on $m$.

Target splitting. Consider the image $Y:=(T+S)(E)=(T+S)(X)$ of the level zero Fredholm operator $T+S\colon E\to F$. But the image of a Fredholm operator is closed and of finite codimension, say $r$. Hence $Y$ is an ${\mathrm{sc}}$-subspace of $F$ by Lemma 2.3.21 and admits an $r$-dimensional ${\mathrm{sc}}$-complement $C\subset F_{\infty}$ by Corollary 2.3.22. Summarizing, we have

[TABLE]

for every $m\in{\mathbb{N}}_{0}$ and where $C$ does not depend on $m$.

${\mathrm{sc}}$-isomorphism. It is clear that $T$ as a map $T:X\to Y$ is bijective and level preserving with continuous level operators $T_{m}:X_{m}\to Y_{m}$, still injective. But why are these surjective? Exercise. Continuity of the inverse of $T_{m}$ then follows from the bounded inverse theorem.

Fredholm-index. Adding a compact operator, say $S:E\to F$, to a Fredholm operator, say $T:E\to F$, does not change the Fredholm index. This concludes the proof that $T+S$ is an ${\mathrm{sc}}$-Fredholm operator. ∎

Scale Fredholm operators –

naïve approach through level operators

Intuitively, if not naively, an ${\mathrm{sc}}$-Fredholm operator should be a level preserving linear operator $T\colon E\to F$ between Banach scales whose level operators $T_{m}\colon E_{m}\to F_{m}$ are Fredholm operators: Each $T_{m}$ is linear and continuous, has a finite dimensional kernel $K_{m}:=\ker T_{m}$, a closed image $Y_{m}:={\rm im\,}T_{m}$, and a finite dimensional cokernel ${\rm coker\,}T_{m}:=F_{m}/Y_{m}$. One calls the integer

[TABLE]

the Fredholm index of $T_{m}$. (As we’ll find out, one more condition to come.)

$\bullet$ Firstly, note that the kernels already form a nested sequence $K:=K_{0}\supset K_{1}\supset\dots$ of (by continuity of $T_{m}$) closed subspaces $K_{m}\subset E_{m}$. Note that $K_{m}=K\mathop{\cap}E_{m}$ since $\ker T|_{E_{m}}=\ker T\mathop{\cap}E_{m}$. For a Banach scale it only misses the (density) axiom saying that $K_{\infty}$ is dense in every level $K_{m}$.

Before adding a density requirement to the intuitive definition of an ${\mathrm{sc}}$-Fredholm operator let us investigate its consequences and see if a simpler condition could do the same job. If density holds, then $K$ is an ${\mathrm{sc}}$-subspace and, by finite dimension, generates the constant Banach scale (see Lemma 2.1.16), still denoted by $K$ and called the kernel Banach scale.

Therefore we add to the intuitive definition of ${\mathrm{sc}}$-Fredholm the requirement

[TABLE]

in symbols $K:=\ker T=\ker T_{m}\subset E_{m}$ $\forall m$.121212 Constant dimension $\dim\ker T_{m}=\dim\ker T$ suffices by the inclusions $K_{m-1}\supset K_{m}$.

By Proposition 2.3.17 the kernel ${\mathrm{sc}}$-subspace $K\subset E_{\infty}$ admits an ${\mathrm{sc}}$-complement in $E$, say $X$.

$\bullet$ Secondly, the images $Y_{m}:={\rm im\,}T_{m}=T(X_{m})\supset T(X_{m+1})$ form a nested sequence $Y:=Y_{0}\supset Y_{1}\supset\dots$ of closed subspaces $Y_{m}\subset F_{m}$ of finite codimensions $r_{m}$. What is missing that the image scale ${\rm im\,}T$ with levels ${\rm im\,}T_{m}$ is a Banach scale is I) (density) again, just as in case of the kernel scale. However, this time there is one more thing missing that was automatic for the kernel scale. Namely, we would like to have that II) the image scale is in fact generated by its top level $Y={\rm im\,}T$, that is we wish that

[TABLE]

Suppose I) and II) hold. Namely, the image scale with levels $Y_{m}:={\rm im\,}T_{m}$ is a Banach scale and arises by intersection with its top level $Y={\rm im\,}T$. In other words, the closed finite codimensional subspace $Y={\rm im\,}T$ is an ${\mathrm{sc}}$-subspace and the generated Banach subscale has intersection levels $Y\mathop{\cap}F_{m}={\rm im\,}T_{m}$ which are equal to the images of the level operators. Let $r=\dim F/Y$ be the codimension of $Y$. Then $Y$ admits by Proposition 2.3.20 an $r$ dimensional ${\mathrm{sc}}$-complement $C$ which necessarily generates the constant Banach scale $C_{m}=C$. Lemma 2.1.16 tells that $C\subset F_{\infty}$.

By Corollary 2.3.22 and Lemma 2.3.21 a sufficient condition that $Y={\rm im\,}T$ is an ${\mathrm{sc}}$-subspace, thus generating a Banach subscale, is the following which we add as a requirement to the intuitive definition of ${\mathrm{sc}}$-Fredholm:

[TABLE]

$\bullet$ Thirdly, to enforce that the intersection levels of the Banach scale generated by $Y={\rm im\,}T$ coincide with the images of the level operators, i.e.

[TABLE]

we add to the intuitive definition of ${\mathrm{sc}}$-Fredholm the requirement

[TABLE]

of level regularity. So $e\in E$ and $Te\in F_{m}$ together imply $e\in E_{m}$. The next exercise shows that (2.3.10) implies all three conditions (2.3.7–2.3.9).

Exercise 2.3.30**.**

To the naïve notion of ${\mathrm{sc}}$-Fredholm add (2.3.10) to prove

• a)

Constancy of kernel scale (2.3.7) holds true. (Thus $K$ is an ${\mathrm{sc}}$-subspace of finite dimension and therefore $K$ admits an ${\mathrm{sc}}$-complement $X$.)

• b)

Equality of scales (2.3.9) holds true. (That is the image scale $Y$ with levels $T(X_{m})$ equals the intersection scale with levels $Y\mathop{\cap}F_{m}$.)

• c)

The image scale is a Banach subscale of $F$ generated by its top level $Y={\rm im\,}T$. (That is $Y$ is an ${\mathrm{sc}}$-subspace. So (2.3.8) is satisfied by Proposition 2.3.20 and $C\subset F_{\infty}$ by Lemma 2.1.16.)

• d)

Each level operator as a map $T_{m}\colon X_{m}\to Y_{m}$ is an isomorphism.

[Hints: a) trivial. b) “$\subset$” easy, “$\supset$” trivial. c) It only remains to show density of $Y_{\infty}$ in $Y$. By a) $X$ generates a Banach subscale, so $X_{\infty}$ is dense in $X$. Show that $Y_{\infty}=T(X_{\infty})$, then apply Lemma A.1.23. d) Equality (2.3.9).]

Definition 2.3.31** (${\mathrm{sc}}$-Fredholm operator – via level operators).**

An sc-Fredholm operator is a level preserving linear operator $T\colon E\to F$ between Banach scales all of whose level operators $T_{m}\colon E_{m}\to F_{m}$ are Fredholm and which satisfies the level regularity condition (2.3.10).

Exercise 2.3.32**.**

Show that Definitions 2.3.24 and 2.3.32 are equivalent.

2.4 Scale differentiability

Motivated by properties of the shift map, see our discussion in the introduction around (1.0.1), the notion of scale differentiability was introduced by Hofer, Wysocki, and Zehnder (2007); see also Hofer et al. (2010, 2017).

Scale continuous maps – class ${\mathrm{sc}}^{0}$

An open subset $U$ of an ${\mathrm{sc}}$-Banach space $E$ induces via level-wise intersection a nested sequence $U^{\cap E}$ of open subsets $U_{m}=U\cap E_{m}$ of the corresponding Banach spaces $E_{m}$; cf. Lemma 2.1.12.

Definition 2.4.1**.**

A partial quadrant in a Banach scale $E$ is a closed convex subset $C$ of $E$ such that there is an ${\mathrm{sc}}$-isomorphism $T\colon E\to{\mathbb{R}}^{n}\oplus W$, for some $n$ and some ${\mathrm{sc}}$-Banach space $W$, satisfying $T(C)=[0,\infty)^{n}\oplus W$. Note that $C$ necessarily contains the origin [math] of $E$.

An ${\mathrm{sc}}$-triple $(U,C,E)$ consists of a Banach scale $E$, a partial quadrant $C\subset E$, and a relatively open subset $U\subset E$. Observe that both $U$ and $C$ inherit from $E$ nested sequences of subsets whose levels are the closed subsets $C_{m}:=C\cap E_{m}\subset E_{m}$ and the relatively open subsets $U_{m}:=U\cap C_{m}\subset C_{m}$.

The notion of partial quadrant is introduced to describe boundaries and corners. At first reading think of $C=E$, so $U$ is an open subset of $E$.

Definition 2.4.2** (Scale continuity).**

Let $(U,C,E)$ and $(V,D,F)$ be ${\mathrm{sc}}$-triples. A map $f\colon U\to V$ is called scale continuous or of class ${\mathrm{sc}}^{0}$ if

• (i)

$f$ islevel!preserving level preserving, that is $f(U_{m})\subset V_{m}$ for every $m$, and

• (ii)

each restriction viewed as a map $f_{m}:=f|_{U_{m}}\colon U_{m}\to V_{m}$ to level $m$ is continuous. The maps $f_{m}$ are called level maps.

Let us abbreviate terminology as follows.

Convention 2.4.3**.**

If we say “suppose $f\colon U\to V$ is of class ${{\mathrm{sc}}}^{k}$” it means that $f$ is an ${\mathrm{sc}}^{k}$ map between ${\mathrm{sc}}$-triples $(U,C,E)$ and $(V,D,F)$ – suppose at first reading between $(U,E,E)$ and $(V,F,F)$ ;-)

Given Banach scales $E$ and $F$, an operator $T\colon E\to F$ can have the property of being ${\mathrm{sc}}$-linear between the Banach scales $E$ and $F$, i.e. $T\in\mathcal{L}_{\mathrm{sc}}(E,F)$, or it can be continuous and linear in the usual sense between the Banach spaces $E$ and $F$, i.e. $T\in{\mathcal{L}}(E,F)$. In the latter case, for extra emphasis, we often write $E_{0}$ and $F_{0}$, instead of $E$ and $F$, and $T\in\boldmath{\mathcal{L}}(E_{0},F_{0})$.

Definition 2.4.4** (Diagonal maps of height $\ell$).**

Let $f\colon U\to V$ be an ${\mathrm{sc}}^{0}$-map. Pick $\ell\in{\mathbb{N}}$. View a level map $f_{m+\ell}$ as a map into the higher level $V_{m}$

[TABLE]

to obtain a continuous map $f_{m+\ell}^{m}=f|\colon U_{m+\ell}\to V_{m}$ given by restriction of $f$ and called a diagonal map of height $\ell$. For simplicity one usually writes $f\colon U_{m+\ell}\to V_{m}$ and calls it an induced map. The collection of all diagonal maps of $f$ of height $\ell$ is denoted by

[TABLE]

with level maps $(f^{-\ell})_{m}=f_{m+\ell}^{m}$. It is of class ${\mathrm{sc}}^{0}$, called the induced ${\mathrm{sc}}$-map of height $\ell$. If we just say diagonal map we mean one of height $1$.

Continuously scale differentiable maps – class ${\mathrm{sc}}^{1}$

To define scale differentiability let us introduce the notion of tangent bundle. The tangent bundle of a Banach scale $E$ is defined as the Banach scale

[TABLE]

If $A\subset E$ is a subset we denote by $A^{k}\subset E^{k}$, as in Definition 2.1.6, the shifted scale of subsets whose levels are given by $(A^{k})_{m}=A_{k+m}$ where $m\in{\mathbb{N}}_{0}$.

Definition 2.4.5**.**

The tangent bundle of an ${\mathrm{sc}}$-triple $(U,C,E)$ is the ${\mathrm{sc}}$-triple $T(U,C,E):=(TU,TC,TE)$ where131313 The symbol $U^{1}\oplus E^{0}$ actually denotes the subset $U^{1}\times E^{0}$ of the ${\mathrm{sc}}$-Banach space $E^{1}\oplus E^{0}$ and is just meant to remind us that the ambient Banach space is a direct sum.

[TABLE]

Note that the levels, for instance of $TU$, are given by

[TABLE]

Definition 2.4.6** (Scale differentiability).**

Suppose $f\colon U\to V$ is of class ${\mathrm{sc}}^{0}$. Then $f$ is called continuously scale differentiable or of class ${\mathrm{sc}}^{1}$ if for every point $x$ in the first sublevel $U_{1}\subset U$ there is a bounded linear operator

[TABLE]

between the top level Banach spaces, called the ${\mathrm{sc}}$-derivative of $f$ at $x$ or the ${\mathrm{sc}}$-linearization, such that the following three conditions hold. {labeling}(ptw diff’able)

The upmost diagonal map $f\colon U_{1}\to V_{0}$ is pointwise differentiable in the usual sense, notation $df(x)\in{\mathcal{L}}(E_{1},F_{0})$; see Definition A.2.22.

The ${\mathrm{sc}}$-derivative $Df(x)$ extends $df(x)$ from $E_{1}$ to $E_{0}$, i.e. the diagram

[TABLE]

commutes.141414 So $df(x)\colon E_{1}\to F_{0}$ is compact. This implies $f\in C^{1}(U_{1},V_{0})$; see Lemma 2.5.2 (ii).

Motivated by the diagram let us call $df(x)$ a diagonal derivative if the level index between domain and target drops by 1.

The tangent map $Tf\colon TU\to TV$ defined by

[TABLE]

for $(x,\xi)\in U^{1}\oplus E^{0}=TU$ is of class ${\mathrm{sc}}^{0}$.

Remark 2.4.7** (A continuity property of $Df$).**

Suppose $f\in{\mathrm{sc}}^{1}(U,V)$. By ${\mathrm{sc}}^{0}$ there are continuous level maps $f_{m}=F|\colon U_{m}\to V_{m}$, whereas the axiom ($Tf$ is $\text{sc}^{0}$) requires continuous level maps

[TABLE]

In particular, for each $m\in{\mathbb{N}}_{0}$ the second component map

[TABLE]

still denoted by $Df$, is continuous whenever $f\in{\mathrm{sc}}^{1}(U,V)$. It is linear in $\xi$.

Remark 2.4.8** (Continuity in compact-open, but not in norm, topology).**

The compact-open and the norm topologies are reviewed in great detail in Appendix A.1. Continuity of the map $\Phi=Df$ in (2.4.14) means that

[TABLE]

is continuous whenever the target carries the compact-open topology. Let’s refer to this as horizontal continuity in the compact-open topology, because both E_{{\color[rgb]{0,1,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,1,1}\pgfsys@color@cmyk@stroke{1}{0}{0}{0}\pgfsys@color@cmyk@fill{1}{0}{0}{0}m}} and F_{{\color[rgb]{0,1,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,1,1}\pgfsys@color@cmyk@stroke{1}{0}{0}{0}\pgfsys@color@cmyk@fill{1}{0}{0}{0}m}} are of the same level $m$. It is crucial that the domain has better regularity $m+1$, see Lemma 2.4.12.

In general, continuity is not true in the norm topology, that is with respect to {\mathcal{L}}(E_{{\color[rgb]{0,1,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,1,1}\pgfsys@color@cmyk@stroke{1}{0}{0}{0}\pgfsys@color@cmyk@fill{1}{0}{0}{0}m}},F_{{\color[rgb]{0,1,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,1,1}\pgfsys@color@cmyk@stroke{1}{0}{0}{0}\pgfsys@color@cmyk@fill{1}{0}{0}{0}m}}). The map which prompted the discovery of scale calculus, the shift map (1.0.1), provides a counterexample to continuity of

[TABLE]

for details see e.g. Frauenfelder and Weber (2018, §2).

Things improve drastically if instead of $E_{m}$ one starts at better regularity E_{m{\color[rgb]{.75,.5,.25}\definecolor[named]{pgfstrokecolor}{rgb}{.75,.5,.25}+1}}, see Lemma 2.4.12. Now the linear map Df(x):E_{m{\color[rgb]{.75,.5,.25}\definecolor[named]{pgfstrokecolor}{rgb}{.75,.5,.25}+1}}\to F_{m} changes level, we say “is diagonal”, and one has $Df=df$ and norm continuity, that is

[TABLE]

referred to as diagonal continuity in the norm topology.

Remark 2.4.9** (Uniqueness of extension).**

Since $E_{1}$ is dense in the Banach space $E_{0}$ the scale derivative $Df(x)$ is uniquely determined by the requirement (2.4.12) to restrict along $E_{1}$ to $df(x)$. However, observe that the mere requirement that $f\colon U_{1}\to F_{0}$ is pointwise differentiable does not guarantee that a bounded extension of $df(x)\in{\mathcal{L}}(E_{1},F_{0})$ from $E_{1}$ to $E_{0}$ exists. Here the B.L.T. Theorem A.2.8 does not help, because the completion of $E_{1}$ is $E_{1}$ itself.. Existence of such an extension is part of the definition of ${\mathrm{sc}^{1}}$.

Exercise 2.4.10**.**

Show that for constant Banach scales $E$ and $F$, in other words, for finite dimensional normed spaces equipped with the constant scale structure, a map $f\colon U\to V$ is of class ${\mathrm{sc}}^{1}$ iff it is of class $C^{1}$.

Exercise 2.4.11**.**

What changes in Exercise 2.4.10 if $E$ or $F$ are constant?

Scale derivative $Df(x)$ induces only

some level operators

Lemma 2.4.12** (Level preservation and continuity properties of $Df(x)$).**

Let $f\colon U\to V$ be of class ${\mathrm{sc}}^{1}$ and $m\in{\mathbb{N}}_{0}$. Then the following is true for every point $x\in U$ of regularity $m+1$, that is $x\in U_{m+1}$.

• (a)

Existence of level operators down to one level above $x$:* That the ${\mathrm{sc}}$-derivative $Df(x)\in{\mathcal{L}}(E_{0},F_{0})$ is level preserving is guaranteed only for levels $0,\dots,m$.*

• (b)

Continuity of these level operators:* The** induced** level operators are bounded linear operators, in symbols*

[TABLE]

• (c)

Horizontal continuity in compact-open topology:* By continuity of the map $\Phi\colon U_{m+1}\oplus E_{m}\to F_{m}$ in (2.4.14), still denoted by $D_{m}f$ or even $Df$, it holds that Df\in C^{0}(U_{m+1},\mathcal{L}_{\mathrm{c}}(E_{{\color[rgb]{0,1,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,1,1}\pgfsys@color@cmyk@stroke{1}{0}{0}{0}\pgfsys@color@cmyk@fill{1}{0}{0}{0}m}},F_{{\color[rgb]{0,1,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,1,1}\pgfsys@color@cmyk@stroke{1}{0}{0}{0}\pgfsys@color@cmyk@fill{1}{0}{0}{0}m}})). By** linearity** of $Df(x)$ this simply means that along any convergent sequence $x_{\nu}\to x$ in $U_{k+1}$ the scale derivative applied to any individual $\xi\in E_{m}$ converges, that is*

[TABLE]

• (d)

Diagonal continuity in norm topology:* The ${\mathrm{sc}}$-derivative as a map*

[TABLE]

is continuous. Actually Df=df\colon U_{m+1}\to{\mathcal{L}}(E_{m{\color[rgb]{.75,.5,.25}\definecolor[named]{pgfstrokecolor}{rgb}{.75,.5,.25}+1}},F_{m}); see (2.5.22).

Corollary 2.4.13**.**

At smooth points ${\mathrm{sc}}$-derivatives are ${\mathrm{sc}}$-operators, that is

[TABLE]

Proof of Lemma 2.4.12.

Let $f\colon U\to V$ be of class ${\mathrm{sc}}^{1}$. Pick $x\in U_{m+1}$ and $\ell\in\{0,\dots,m\}$. Hence $x\in U_{\ell+1}$ and so $(x,\xi)\in U_{\ell+1}\oplus E_{\ell}=(TU)_{\ell}$ for $\xi\in E_{\ell}$. The axiom ($Tf$ is $\text{sc}^{0}$) means by definition of ${\mathrm{sc}}^{0}$ that every level map $(Tf)_{\ell}$, see (2.4.13), is continuous. In particular, for fixed $x\in U_{m+1}\subset U_{\ell+1}$ the map between second components $E_{\ell}\to F_{\ell}$, $\xi\mapsto Df(x)\xi$, is continuous. This proves (a–b). Since $\Phi$ in (2.4.14) is continuous so is $\Phi(\cdot,\xi)\colon U_{m+1}\to F_{m}$ for each fixed $\xi\in E_{m}$. This proves (c). Part (d) holds true by Proposition A.2.13 c) for the above map $\Phi$ and the compact operator $S:=I_{m+1}\colon E_{m+1}\hookrightarrow E_{m}$. ∎

Characterization of ${\mathrm{sc}}^{1}$ in terms of the

scale derivative $Df(x)$

The next lemma and proof are taken from Frauenfelder and Weber (2018).

Lemma 2.4.14** **(Characterization of ${\mathrm{sc}}^{1}$ in terms of the

${\mathrm{sc}}$-derivative).

---

Let $f\colon U\to V$ be ${\mathrm{sc}}^{0}$. Then $f$ is ${\mathrm{sc}}^{1}$ iff the following conditions hold: {labeling}(level operators)*

(i) The restriction $f\colon U_{1}\to F_{0}$, that is the top diagonal map, is pointwise differentiable in the usual sense.

(ii) Its derivative $df(x)\in{\mathcal{L}}(E_{1},F_{0})$ at any $x\in U_{1}$ has a continuous extension $Df(x)\colon E_{0}\to F_{0}$.

(iii) The continuous extension $Df(x)\colon E_{0}\to F_{0}$ restricts, for all levels $m\in{\mathbb{N}}_{0}$ and base points $x\in U_{m+1}$, to continuous linear operators (called level operators)

[TABLE]

such that the corresponding maps

[TABLE]

are continuous.

Proof.

’$\Rightarrow$’ Suppose $f$ is ${\mathrm{sc}}^{1}$. Then statements (i) and (ii) are obvious and in statement (iii) the restriction assertion holds by Lemma 2.4.12 part (b), the continuity assertion by part (c).

’$\Leftarrow$’ Suppose $f$ is ${\mathrm{sc}}^{0}$ and satisfies (i–iii). It remains to show that the tangent map is ${\mathrm{sc}}^{0}$, namely, a) level preserving and b) admitting continuous level maps. a) To see that $Tf$ maps $(TU)_{m}$ to $(TV)_{m}$ for every $m\in{\mathbb{N}}_{0}$, pick $(x,\xi)\in(TU)_{m}=U_{m+1}\oplus E_{m}$. Since $f$ is ${\mathrm{sc}}^{0}$ we have that $f(x)\in V_{m+1}$. By (iii) we have that $Df(x)\xi\in F_{m}$. Hence

[TABLE]

b) To see that $TF$ as a map $Tf|_{(TU)_{m}}\colon(TU)_{m}\to(TV)_{m}$ is continuous assume $(x_{\nu},\xi_{\nu})\in(TU)_{m}=U_{m+1}\oplus E_{m}$ is a sequence which converges to $(x,\xi)\in(TU)_{m}$. Because $f$ is ${\mathrm{sc}}^{0}$, it follows that

[TABLE]

Continuity of $Df$ provided by (iii) guarantees that

[TABLE]

Therefore

[TABLE]

This proves continuity b) and hence the lemma holds. ∎

Higher scale differentiability – class ${\mathrm{sc}}^{k}$

For $k\geq 2$ one defines higher continuous scale differentiability ${\mathrm{sc}}^{k}$ recursively as follows. In the definition of ${\mathrm{sc}}^{1}$ one requires a map $f\colon U\to V$ between open subsets of Banach scales $E$ and $F$ to be ${\mathrm{sc}}^{0}$ and then defines a tangent map $F:=Tf\colon TU\to TV$, again between open subsets of Banach scales $TE$ and $TF$, which among other things is required to be ${\mathrm{sc}}^{0}$, too. If the map $F$ itself is of class ${\mathrm{sc}}^{1}$, that is if among other things $TF=TTf\colon TTU\to TTV$ is of class ${\mathrm{sc}}^{0}$, one says that $f$ is of class ${\mathrm{sc}}^{2}$, and so on.

Definition 2.4.15** (Higher scale differentiability).**

An ${\mathrm{sc}}^{1}$-map $f\colon U\to V$ is of class ${\mathrm{sc}}^{k}$ if and only if its tangent map $Tf\colon TU\to TV$ is ${\mathrm{sc}}^{k-1}$. It is called sc-smooth, or of class ${\mathrm{sc}}^{\infty}$, if it is of class ${\mathrm{sc}}^{k}$ for every $k\in{\mathbb{N}}$.

An ${\mathrm{sc}}^{k}$-map has iterated tangent maps as follows. Recursively one defines the iterated tangent bundle as

[TABLE]

Let us consider the example $T^{2}U$. Recall that for an open subset $U\subset E$ of a Banach scale we set $TU:=U^{1}\oplus E^{0}$. Now consider the open subset $TU$ of the Banach scale $TE:=E^{1}\oplus E^{0}$ to obtain that

[TABLE]

For $f$ of class ${\mathrm{sc}}^{k}$ define its iterated tangent map $T^{k}f\colon T^{k}U\to T^{k}V$ recursively as

[TABLE]

For example

[TABLE]

is (as shown in the proof of Lemma 2.4.16 below) given by

[TABLE]

Here $D^{2}f$ is the ${\mathrm{sc}}$-Hessian of $f$ which we introduce next. The following lemma and proof are taken from Frauenfelder and Weber (2018).

Lemma 2.4.16** **(Characterization of ${\mathrm{sc}}^{2}$ in terms of

the ${\mathrm{sc}}$-derivative).

Let $f\colon U\to V$ be ${\mathrm{sc}}^{1}$. Then $f$ is ${\mathrm{sc}}^{2}$ iff the following conditions hold:*

• (a)

*The restriction $f\colon U_{2}\to V_{0}$, that is the top diagonal map of height two, is pointwise twice differentiable in the usual sense. *

• (b)

Its second derivative $d^{2}f(x)\in{\mathcal{L}}(E_{2}\oplus E_{2},F_{0})$ at any $x\in U_{2}$ has a continuous extension $D^{2}f(x)\colon E_{1}\oplus E_{1}\to F_{0}$, the sc-Hessian of $f$ at $x$.

• (c)

The continuous extension $D^{2}f(x)\colon E_{1}\oplus E_{1}\to F_{0}$ restricts, for all $m\in{\mathbb{N}}_{0}$ and $x\in U_{m+2}$, to continuous bilinear maps

[TABLE]

such that the corresponding maps

[TABLE]

are continuous.

Proof.

’$\Leftarrow$’ Suppose $f\colon U\to F$ is $\mathrm{sc}^{1}$ and satisfies the three conditions (a-c) of the Lemma. We need to show that $f$ is $\mathrm{sc}^{2}$ (meaning by definition that $Tf\in\mathrm{sc}^{1}$). Since $f$ is $\mathrm{sc}^{1}$ we have a well defined tangent map

[TABLE]

of class $\mathrm{sc}^{0}$. Suppose that

[TABLE]

Hypotheses (a) and (b) guarantee that the linear map

[TABLE]

defined for $(\hat{x},\hat{\xi})\in E_{1}\oplus E_{0}=(TE)_{0}$ by

[TABLE]

is well defined and bounded. To see that this map is the ${\mathrm{sc}}$-derivative of $Tf$, see (2.4.11), we need to check the three axioms in the definition of scale differentiability for $Tf$. Concerning the first two axioms we need to investigate differentiability of the ’diagonal map’, i.e. the restriction of $Tf\colon(TU)_{0}\to(TF)_{0}$ to $(TU)_{1}$. It suffices to show that

[TABLE]

Since we already know that the first component $f$ of $Tf$ is $\mathrm{sc}^{1}$ it suffices to check the second component and show that

[TABLE]

We estimate

[TABLE]

Because $D^{2}f\colon U_{2}\oplus E_{1}\oplus E_{1}\to F_{0}$ is continuous by hypothesis (b) there exists an open neighborhood $V$ of $x$ in $U_{2}$ and $\delta>0$ such that for every $y\in V$ and every $v$ and $w$ in $B_{\delta}$, namely the $\delta$-ball around the origin of $E_{1}$, it holds

[TABLE]

By bilinearity of $D^{2}f(y)$ for any $v,w\in E_{1}$ we get the estimate

[TABLE]

at each $y\in V$. We can assume without loss of generality that $V$ is convex. We rewrite the first term in (2.4) as follows

[TABLE]

From uniform boundedness (2.4.19) we conclude that

[TABLE]

where $c\geq 1$ is a bound for the linear inclusion $E_{2}\hookrightarrow E_{1}$, so $\mathopen{\|}\frac{\hat{x}}{\mathopen{\|}\hat{x}\mathclose{\|}_{2}}\mathclose{\|}_{1}\leq c$. Hence in view of (2.4) in order to show (2.4.17) we are left with showing

[TABLE]

Fix a constant $\kappa\geq 1/\delta^{2}$ where $\delta$ is the constant in (2.4.19). Now choose $\epsilon>0$. By taking advantage of the fact that $E_{2}$ is dense in $E_{1}$ we can choose

[TABLE]

Choose $W\subset V$ a convex open neighborhood of $x$ with the property that for every $x+\hat{x}\in W$ it holds that

[TABLE]

Suppose that $x+\hat{x}\in W$. We are now ready to estimate

[TABLE]

To obtain the first inequality we wrote each of the three terms $\xi$ in line one in the form $\xi=\xi^{\prime}+(\xi-\xi^{\prime})$, we used that $df=Df$ for diagonal restrictions of $f$, and we used formula (2.4.20) for $\hat{\xi}=\xi-\xi^{\prime}$. The second inequality uses, in particular, the estimate (2.4.19) on both $D^{2}f$ terms. This proves (2.4.21) and therefore the first two axioms of scale differentiability of $Tf$.

It remains to prove axiom three, namely that the tangent map of $Tf$, i.e.

[TABLE]

is $\mathrm{sc}^{0}$: the map $T^{2}f$ must be level preserving and the corresponding level maps

[TABLE]

given by formula (2.4.16) must be continuous for all $m\in{\mathbb{N}}_{0}$. For the $Tf$ part both assertions are true, because $Tf\in{\mathrm{sc}^{1}}$. Concerning the $D(Tf)$ part there are three terms to be checked. Since $Tf\in{\mathrm{sc}^{1}}$ part (iii) of Lemma 2.4.14 applies and asserts that term one exists as a map $Df\colon U_{m+2}\oplus E_{m+1}\to F_{m+1}$ and is continuous, similarly for the map $Df\circ(\iota,{\rm Id})\colon U_{m+2}\oplus E_{m}\to U_{m+1}\oplus E_{m}\to F_{m}$ in term three. Concerning term two use hypothesis (c) to see that $D^{2}f\colon U_{m+2}\oplus E_{m+1}\oplus E_{m+1}\to F_{m}$ is well defined and continuous. This finishes the proof of the implication that under the assumptions (a-c) of the Lemma $f$ is $\mathrm{sc}^{2}$.

’$\Rightarrow$’ For the other implication, namely that if $f$ is $\mathrm{sc}^{2}$ it satisfies the conditions (a-c) of the Lemma, we point out that by a result of Hofer, Wysocki, and Zehnder Hofer et al. (2010, Prop. 2.3) it follows that $f$ is actually of class $C^{2}$ as a map $f\colon U_{m+2}\to F_{m}$ for every $m\in\mathbb{N}_{0}$. This in particular implies properties (a) and (b). Property (c) is straightforward; cf. proof of Lemma 2.4.14 (iii) based on Lemma 2.4.12 parts (b) and (c). This concludes the proof of Lemma 2.4.16. ∎

Exercise 2.4.17** (Symmetry of scale Hessian).**

Show that the scale Hessian $\mathrm{Hess}_{x}f:=D^{2}f(x)\colon E_{1}\oplus E_{1}\to F_{0}$ is symmetric, that is $\mathrm{Hess}_{x}f(\xi,\eta)=\mathrm{Hess}_{x}f(\eta,\xi)$ for all $\xi,\eta\in E_{1}$.

[Hint: The usual second derivative $d^{2}f(x)\colon E_{2}\oplus E_{2}\to F_{0}$ is symmetric and $E_{2}$ is a dense subset of the Banach space $E_{1}$.]

Applying the arguments in the proof of Lemma 2.4.16 inductively – Lemma 2.4.14 playing the role of the induction hypothesis – we obtain

Lemma 2.4.18** **(Characterizing ${\mathrm{sc}}^{k}$ by higher

${\mathrm{sc}}$-derivatives $D^{k}f(x)$).

Let $k\in{\mathbb{N}}$ and $f\colon U\to V$ be ${\mathrm{sc}}^{k-1}$. Then $f$ is ${\mathrm{sc}}^{k}$ iff the following conditions hold:

• (i)

The restriction $f\colon U_{k}\to V_{0}$, that is the top diagonal map of height $k$, is pointwise $k$ times differentiable in the usual sense.

• (ii)

Its $k^{\rm th}$ derivative $d^{k}f(x)\in{\mathcal{L}}(E_{k}\oplus\dots\oplus E_{k},F_{0})$ at any $x\in U_{k}$ has a continuous extension

[TABLE]

• (iii)

The continuous extension $D^{k}f(x)\colon E_{k-1}\oplus\dots\oplus E_{k-1}\to F_{0}$ restricts, for all $m\in{\mathbb{N}}_{0}$ and $x\in U_{m+k}$, to continuous $k$-fold multilinear maps

[TABLE]

such that the corresponding maps

[TABLE]

are continuous.

2.5 Differentiability – Scale vs Fréchet

First we investigate how the new class ${\mathrm{sc}}^{1}$ of continuously scale differentiable maps $f\colon U\to V$ relates to $C^{1}$ continuous differentiability in the usual Fréchet sense of all diagonal maps $f\colon U_{m+1}\to V_{m}$ of height $1$. Then we investigate how the class ${\mathrm{sc}}^{k}$ of higher scale differentiable maps $f\colon U\to V$ relates to $C^{\ell}$ differentiability of all diagonal maps $f\colon U_{m+\ell}\to V_{m}$ of height $\ell\in\{0,\dots,k\}$, hence up to height of at most $k$. For further details see Hofer et al. (2010).

Maps of class ${\mathrm{sc}}^{1}$

Convention 2.5.1** (Topologies).**

Given Banach spaces $E_{0}$ and $F_{0}$, then ${\mathcal{L}}(E_{0},F_{0})$ denotes the vector space of bounded linear maps $T\colon E_{0}\to F_{0}$ equipped with the (complete) operator norm (Section A.2.2). By $\mathcal{L}_{\mathrm{c}}(E_{0},F_{0})$ we denote the same vector space equipped with the compact-open topology.

Lemma 2.5.2** (Continuity properties of $Df$ and the diagonal differential $df$).**

Let $f\colon U\to V$ be of class ${\mathrm{sc}}^{1}$. Then the following is true.

• (i)

The map $U_{1}\oplus E_{0}\to F_{0}$, $(x,\xi)\mapsto Df(x)\xi$, is continuous.

• (ii)

The usual differential $df\colon U_{1}\to{\mathcal{L}}(E_{1},F_{0})$ of the diagonal map $f\colon U_{1}\to V_{0}$ is continuous, in symbols $f\in C^{1}(U_{1},V_{0})$.

• (iii)

Every diagonal map $f\colon U_{m+1}\to V_{m}$ is of class $C^{1}$. In other words, its differential, the so-called* diagonal differential*

[TABLE]

is a continuous map.

• (iv)

At $x\in U_{m+1}$ the diagonal derivative $df(x)\colon E_{m+1}\to F_{m}$ in (iii) extends to $E_{m}$ and the extension is the restriction $Df(x)|_{E_{m}}\in{\mathcal{L}}(E_{m},F_{m})$ of the ${\mathrm{sc}}$-derivative (2.4.11); cf. Lemma 2.4.12 (b). That is, the diagram

[TABLE]

commutes. As a map $Df:U_{m+1}\oplus E_{m}\to F_{m}$ the level $m$ scale derivative is continuous; cf. (2.4.15).

Of course, the lemma could be stated more economically, but we enlist the assertions in their order of proof.

Proof.

We follow essentially Cieliebak (2018). (i) By assumption $f$ is ${\mathrm{sc}}^{1}$, so the induced map $f\colon U_{1}\to V_{0}$ is pointwise differentiable and for every $x\in U_{1}$ the usual derivative $df(x)\in{\mathcal{L}}(E_{1},F_{0})$ extends from $E_{1}$ to a map $Df(x)\in{\mathcal{L}}(E_{0},F_{0})$. Moreover, by axiom ($Tf$ is $\text{sc}^{0}$) the map

[TABLE]

is continuous, cf. (2.4.13), which is assertion (i).

(ii) As the inclusion $S:=I_{1}\colon E_{1}\hookrightarrow E_{0}$ is compact, continuity of the map

[TABLE]

holds by Proposition A.2.13 c). We used that $Df(x)=df(x)$ along $E_{1}$.

(iii+iv) For $m=0$ the assertions are true by (i) and (ii) and (ii) will be a key input for the present proof, see Step 1 below, that

• a)

as a map $f\colon U_{m+1}\to V_{m}$ is of class $C^{1}$, thereby proving (iii), and

• b)

its derivative $df(x)=Df(x)|_{E_{m+1}}\colon E_{m+1}\to F_{m}$ is the ${\mathrm{sc}}$-derivative $Df(x)\colon E_{0}\to V_{0}$ applied to the elements of $E_{m+1}$ or, equivalently, the restriction to the dense subset $E_{m+1}$ of the level operator $D_{m}f(x)=Df(x)|_{E_{m}}\in{\mathcal{L}}(E_{m},F_{m})$ which exists by Lemma 2.4.12 (b).

By density $E_{m+1}\subset E_{m}$ part b) shows that the continuous extension of $df(x)\colon E_{m+1}\to F_{m}$ to $E_{m}$ is the level operator $D_{m}f(x)$. The yet missing continuity assertion in (iv) holds true by (2.4.14). Step 2 below will prove a) and b) which then completes the proof of (iii+iv). Step 1 is just a preliminary.

Step 1. Given $x\in U_{1}$, let $\xi\in E_{1}$ be sufficiently small such that the image of the map $\gamma\colon[0,1]\to U_{1}\subset E_{1}$, $t\mapsto x+t\xi$, is contained in $U_{1}$. Then

[TABLE]

Proof of Step 1. As $f\in C^{1}(U_{1},V_{0})$ by (ii), identity one is the integral form of the mean value theorem; see e.g. Lang (1993, XIII Thm. 4.2). Identity two holds since $\frac{d}{dt}f(x+t\xi)=df(x+t\xi)\xi=Df(x+t\xi)\xi$. Equality two is by (2.4.12) – by definition of ${\mathrm{sc}}^{1}$ the scale derivative restricted to $E_{1}$ is $df(x)$.

In the proof of Step 2 we will use Step 1 for the elements of the subset $U_{m+1}\subset U_{1}$ and for small $\xi\in E_{m+1}\subset E_{1}$. For such $x$ and $\xi$ the term $f(x+\xi)-f(x)$ even lies in $F_{m+1}$, since $f$ is level preserving (it is of class ${\mathrm{sc}}^{0}$ by assumption). However, we shall only estimate the $F_{m}$ norm, as this gives us the opportunity to bring in compactness of the inclusion $S:=I_{m+1}\colon E_{m+1}\hookrightarrow E_{m}$ on the domain side of $f$.

Step 2. $\forall m$ the map $f\colon U_{m+1}\to V_{m}$ is of class $C^{1}$ with derivative {\color[rgb]{.75,.5,.25}\definecolor[named]{pgfstrokecolor}{rgb}{.75,.5,.25}df=Df}.

Proof of Step 2. Pick $x\in U_{m+1}$ and a non-zero short vector $\xi\in E_{m+1}$ to get

[TABLE]

Here the equality holds by Step 1. Concerning inequality one note that the path $[0,1]\to F_{m}$, $t\mapsto Df(x+t\xi)\xi-Df(x)\xi$, is continuous by Lemma 2.4.12 (c) since $x+t\xi\in U_{m+1}$ and $\xi\in E_{m+1}\subset E_{m}$. So the map is in ${\rm L}^{1}([0,1],F_{m})$, hence the norm of the integral is less or equal than the integral along the norm; see e.g. Lang (1993, VI §4 (4)). Inequality two holds by definition of the operator norm.

We prove convergence to zero. This will follow from continuity of the map $Df\colon U_{m+1}\to{\mathcal{L}}(E_{m+1},F_{m})$, see Lemma 2.4.12 (d). However, due to infinite dimension it is not just some compactness argument: Let $\xi_{\nu}\to 0$ be any in $E_{m+1}$ convergent sequence. Then the family of bounded linear operators

[TABLE]

generates, for each element $\zeta\in E_{m+1}$, a bounded orbit

[TABLE]

Indeed by continuity of the map $Df\colon U_{m+1}\to{\mathcal{L}}(E_{m+1},F_{m})$, as guaranteed by Lemma 2.4.12 (d), and convergence $\xi_{\nu}\to 0$ there is a radius $R=R(\zeta)$ such that the whole sequence of elements $Df(x+\xi_{\nu})\zeta$ of $F_{m}$ lies in the ball $B_{R}\subset F_{m}$ of radius $R$ and centered at $Df(x)\zeta$. But by convexity of $B_{R}$ all segments from the center $Df(x)\zeta$ to $Df(x+\xi_{\nu})\zeta$ also lie in $B_{R}$. The Banach–Steinhaus Theorem A.2.12 then provides a uniform upper bound $c_{\mathcal{F}}$ for the operator norms of all members of ${\mathcal{F}}$. Now the constant function $g\equiv 2c_{\mathcal{F}}\colon[0,1]\to[0,\infty)$ is integrable and dominates ($g\geq\mathopen{|}F_{\nu}\mathclose{|}$) each function

[TABLE]

The pointwise limit $F_{\nu}(t)\to 0$, as $\nu\to\infty$, is the constant function [math] on $[0,1]$, again by continuity of $Df$ and by continuity of the norm function. Thus the dominated convergence theorem applies, see e.g. Lang (1993, VI Thm. 5.8), and yields $\lim_{\nu}\int F_{\nu}=\int\lim_{\nu}F_{\nu}=\int 0=0$. This proves convergence to zero.

It remains to prove continuity of the map

[TABLE]

Continuity holds by Proposition A.2.13 c) for the by (2.4.13), cf. Lemma 2.4.14, continuous map $\Phi\colon U_{m+1}\oplus E_{m}\to F_{m}$, $(x,\xi)\mapsto Df(x)\xi$, and the compact inclusion $S:=I_{m+1}\colon E_{m+1}\to E_{m}$. As any $\xi\in E_{m+1}$ lies in $E_{1}$, one has

[TABLE]

since the diagram (2.4.12) commutes. This proves Lemma 2.5.2. ∎

Lemma 2.5.3** **(Characterization of ${\mathrm{sc}}^{1}$ via

diagonal maps being of class $C^{1}$).

An ${\mathrm{sc}}^{0}$-map $f\colon U\to V$ is of class ${\mathrm{sc}}^{1}$ iff

• (i)

all diagonal maps $f\colon U_{m+1}\to V_{m}$ are of class $C^{1}$ and for each of them

• (ii)

the derivative $df(x)\in{\mathcal{L}}(E_{m+1},F_{m})$, at any $x\in U_{m+1}$, extends to a continuous linear operator on $E_{m}$, notation $D_{m}f(x):E_{m}\to F_{m}$, and

• (iii)

the extension as a map $D_{m}f:U_{m+1}\oplus E_{m}\to F_{m}$, $(x,\xi)\mapsto D_{m}f(x)\xi$, is continuous; cf. (2.4.15).

Proof.

’$\Rightarrow$’ Lemma 2.5.2. ’$\Leftarrow$’ By (i-ii) for $m=0$ the first two axioms of ${\mathrm{sc}}^{1}$ are satisfied. By (ii-iii) for all $m$ the axiom ($Tf$ is $\text{sc}^{0}$) is also satisfied. ∎

Remark 2.5.4**.**

For any map $f\colon U\to V$ of class ${\mathrm{sc}}^{1}$ the induced map $f\colon E_{\infty}\supset U_{\infty}\to V_{\infty}\subset F_{\infty}$ between Fréchet spaces is of class $C^{1}$; cf. Cieliebak (2018, Probl. 5.5).

Maps of class ${\mathrm{sc}}^{k}$

It is an immediate consequence of Lemma 2.5.3, together with the identity $(TU)^{1}=T(U^{1})$, that for an ${\mathrm{sc}}^{k}$ map $f\colon U\to V$ one can lift both indices equally and still have an ${\mathrm{sc}}^{k}$ map, say $f\colon U^{\ell}\to V^{\ell}$.

Lemma 2.5.5** (Lifting indices, Hofer et al. (2010, Prop. 2.2)).**

If $f\colon U\to V$ is an ${\mathrm{sc}}^{k}$-map, then the induced map $f\colon U^{1}\to V^{1}$ is also of class ${\mathrm{sc}}^{k}$.

Proof.

Induction over $k\in{\mathbb{N}}$. Case $k=1$: This holds true by Lemma 2.5.3 which characterizes ${\mathrm{sc}}^{1}$ by some conditions on all 151515 saying “all diagonal maps $f\colon U_{m+1}\to V_{m}$” refers to the set $\{f\colon U_{m+1}\to V_{m}\}_{m\in{\mathbb{N}}_{0}}$

diagonal maps $f\colon U_{m+1}\to V_{m}$ and their extensions $D_{m}f(x)$. Replacing $U,V$ by $U^{1},V^{1}$ means to simply forgetting the two maps for $m=0$.

Induction step $k\Rightarrow k+1$: Let $f\colon U\to V$ be ${\mathrm{sc}}^{k+1}$. By definition this means that $f$ is ${\mathrm{sc}}^{1}$ and $Tf$ is ${\mathrm{sc}}^{k}$. So by induction hypothesis applied to $Tf\in{\mathrm{sc}}^{k}(TU,TV)$ that same map just between shifted spaces, namely $Tf\colon(TU)^{1}\to(TV)^{1}$, is as well of class ${\mathrm{sc}}^{k}$. But $(TU)^{1}=T(U^{1})$, hence $Tf\colon T(U^{1})\to T(V^{1})$ is ${\mathrm{sc}}^{k}$. Note that $f\colon U^{1}\to V^{1}$ is also of class ${\mathrm{sc}}^{1}$ as a consequence of the case $k=1$ applied to $f\in{\mathrm{sc}}^{1}(U,V)$. But an ${\mathrm{sc}}^{1}$ map, say $f\colon U^{1}\to V^{1}$, whose tangent map is ${\mathrm{sc}}^{k}$ is of class ${\mathrm{sc}}^{k+1}$ by Definition 2.4.15. ∎

Lemma 2.5.6** (Necessary and sufficient conditions for ${\mathrm{sc}}^{k}$-smoothness).**

Let $U,V$ be relatively open subsets of partial quadrants in ${\mathrm{sc}}$-Banach spaces $E,F$. {labeling}(Necessary)**

If $f\colon U\to V$ is ${\mathrm{sc}}^{k}$, then all diagonal maps $f\colon U_{m+\ell}\to V_{m}$ of height $\ell$ are of class $C^{\ell}$ for all heights from [math] up to $k$.

Assume that a map $f\colon U\to V$ induces for every level $m\in{\mathbb{N}}_{0}$ and every height $\ell$ between [math] and $k$ a diagonal map $f\colon U_{m+\ell}\to V_{m}$ which, moreover, is of class $C^{\ell+1}$. Such a map $f\colon U\to V$ is of class ${\mathrm{sc}}^{k+1}$.

Sketch of proof.

Necessary. Suppose $f\in{\mathrm{sc}}^{k}(U,V)$. Firstly, it suffices to prove the case $\ell=k$, because an ${\mathrm{sc}}^{k}$ map is also an ${\mathrm{sc}}^{\ell}$ map for $\ell\in\{0,\dots,k\}$. Secondly, it suffices to prove the case $m=0$, namely the

Claim. The map $f\colon U_{k}\to V_{0}$ is of class $C^{k}$.

Given $m\in{\mathbb{N}}_{0}$, the claim implies that $f\colon U_{m+k}\to V_{m}$ is of class $C^{k}$ and we are done. Indeed by Lemma 2.5.5 the map $f\colon U^{m}\to V^{m}$ is also of class ${\mathrm{sc}}^{k}$ and for this map the claim asserts that $f\colon(U^{m})_{k}\to(V^{m})_{0}$ is of class $C^{k}$.

One proves the claim by induction over $k$. For $k=0$ the map $f\colon U_{0}\to V_{0}$ is $C^{0}$ as $f\in{\mathrm{sc}}^{0}$, for $k=1$ the map $f\colon U_{1}\to V_{0}$ is $C^{1}$ by Lemma 2.5.2 (ii).

The induction step $k\Rightarrow k+1$ is very similar in character to the proof of Lemma 2.5.2 (iii+iv) just more technical as one is looking at $k$-fold derivatives, thus $k$-multilinear maps. For details see Hofer et al. (2010, Prop. 2.3).

Sufficient. Proof by induction over $k$. Case $k=0$. By assumption there is for each $m\in{\mathbb{N}}_{0}$ a $C^{1}$ level map $f_{m}:=f|_{U_{m}}\colon U_{m}\to V_{m}$. Together with the restriction $i_{m+1}$ of the linear, hence smooth, embedding $I_{m+1}\colon E_{m+1}\hookrightarrow E_{m}$ one has a commutative diagram

[TABLE]

of $C^{1}$ maps, in particular, all diagonal maps $f\colon U_{m+1}\to V_{m}$ are $C^{1}$. By the chain rule one gets the identity

[TABLE]

for $x\in U_{m+1}$. The identity also shows that $df_{m}(x)\in{\mathcal{L}}(E_{m},F_{m})$ extends $df(x)$ from $E_{m+1}$ to $E_{m}$. Thus (i) and (ii) in Lemma 2.5.3 are satisfied and it remains to check (iii). But this follows by pre-composing the first variable of the (by the $C^{1}$-assumption on $f_{m}$) continuous map $U_{m}\oplus E_{m}\to F_{m}$, $(x,\xi)\mapsto df_{m}(x)\xi$, with the continuous embedding $U_{m+1}\hookrightarrow U_{m}$. The step $k\Rightarrow k+1$ is very technical, see Hofer et al. (2010, Prop. 2.4). ∎

2.6 Chain rule

A key element of calculus, the chain rule, is also available in ${\mathrm{sc}}$-calculus. This is rather surprising given the fact that the ${\mathrm{sc}}$-derivative arises by differentiating the diagonal map $f\colon U_{1}\to V_{0}$ thereby loosing one level, so for a composition one would expect the loss of two levels. However, using (compactness) of the embeddings $E_{m+1}\hookrightarrow E_{m}$ one can avoid loosing two levels.

Theorem 2.6.1** (Chain rule Hofer et al. (2007, Thm. 2.16)).**

Suppose $f\colon U\to V$ and $g\colon V\to W$ are ${\mathrm{sc}}^{1}$-maps. Then the composition $g\circ f\colon U\to W$ is also ${\mathrm{sc}}^{1}$ and

[TABLE]

Equivalently, in terms of ${\mathrm{sc}}$-derivatives it holds that

[TABLE]

Proof.

The main principles and tools of the proof have been detailed and referenced in the slightly simpler setting of proving Step 2 in the proof of Lemma 2.5.2 (iii+iv). Fix $x\in U_{1}$. Because $V_{1}$ is an open neighborhood of $f(x)$ in the cone $D_{1}\subset F_{1}$ and because the level map $f\colon U_{1}\to V_{1}$ is continuous, there is a radius $\delta>0$ open ball $B_{\delta}$ in $E_{1}$ centered at [math] such that $x+B_{\delta}$ is contained in $U_{1}$ and such that the map

[TABLE]

takes values in $V_{1}$ for all $t\in[0,1]$ and $\xi\in B_{\delta}$. Because $g$ is ${\mathrm{sc}}^{1}$, as a map $g\colon V_{1}\to W_{0}$ it is of class $C^{1}$ by Lemma 2.5.3 (i). Apply the mean value theorem, observing that ${\partial}_{t}\phi(t,h)=f(x+\xi)-f(x)$, and add zero to obtain

[TABLE]

Divide by $\mathopen{|}\xi\mathclose{|}_{E_{1}}$, so the first integral becomes

[TABLE]

Since $f$ is ${\mathrm{sc}}^{1}$ the restriction of $Df|_{x}\colon E_{0}\to F_{0}$ to $E_{1}$ is $df|_{x}$ whenever $x\in U_{1}$, see (2.4.12), hence $h(\xi)\to 0$ as $\mathopen{|}\xi\mathclose{|}_{E_{1}}\to 0$ by Definition A.2.22 of the Fréchet derivative $df|_{x}:=df(x)$. Now $\phi\colon[0,1]\times B_{\delta}\to V_{1}$ is continuous and $\phi(t,\xi)\to f(x)$, as $\mathopen{|}\xi\mathclose{|}_{E_{1}}\to 0$, uniformly in $t\in[0,1]$. Since $g$ is ${\mathrm{sc}}^{1}$ Lemma 2.4.14 (iii) guarantees that the map $V_{1}\oplus F_{0}\to G_{0}$, $(y,\eta)\mapsto Dg|_{y}\eta$, is continuous. Thus $Dg|_{\phi(t,\xi)}h(\xi)\to 0$, as $\mathopen{|}\xi\mathclose{|}_{E_{1}}\to 0$, uniformly in $t\in[0,1]$. So the integral (2.6.24) vanishes in the limit as $\mathopen{|}\xi\mathclose{|}_{E_{1}}\to 0$.

The second integral divided by $\mathopen{|}\xi\mathclose{|}_{E_{1}}$ becomes

[TABLE]

By (compactness) of the inclusion $E_{1}\hookrightarrow E_{0}$ and continuity of the ${\mathrm{sc}}$-derivative $Df|_{x}\colon E_{0}\to F_{0}$ the set of all $Df|_{x}\xi/\mathopen{|}\xi\mathclose{|}_{E_{1}}$ with $0\not=\xi\in B_{\delta}$ has compact closure in $F_{0}$.161616 images of compact sets under continuous maps are compact

Since the map $V_{1}\oplus F_{0}\to G_{0}$, $(y,\eta)\mapsto Dg|_{y}\eta$, is continuous by Lemma 2.4.14 (iii) – due to $g$ being ${\mathrm{sc}}^{1}$ – it follows as above that the integrand in (2.6.25) converges in $G_{0}$ to [math] uniformly in $t\in[0,1]$, so the integral (2.6.25) converges in $G_{0}$ to [math], both as $\mathopen{|}\xi\mathclose{|}_{E_{1}}\to 0$.

This shows that the ${\mathrm{sc}}^{0}$ map given by the composition $g\circ f\colon U\to W$ satisfies the first two axioms in Definition 2.4.6 of ${\mathrm{sc}}^{1}$. Indeed as a map $g\circ f\colon U_{1}\to V_{1}\to W_{0}$ is pointwise differentiable and at $x\in U_{1}$ the derivative $d(g\circ f)|_{x}\colon E_{1}\to G_{0}$ has a continuous extension, namely the composition of bounded linear operators $Dg|_{f(x)}Df|_{x}\colon E_{0}\to G_{0}$. So by definition this composition is the ${\mathrm{sc}}$-derivative $D(g\circ f)|_{x}$ associated to $g\circ f$. Thus $T(g\circ f)=Tg\circ Tf\colon TU\to TW$. Because both $Tf$ and $Tg$ are ${\mathrm{sc}}^{0}$, so is $T(g\circ f)$. Thus $g\circ f$ satisfies axiom three in Definition 2.4.6. So $g\circ f$ is ${\mathrm{sc}}^{1}$. ∎

2.7 Boundary recognition

Let $C$ be a partial quadrant in an ${\mathrm{sc}}$-Banach space $E$. Pick a linear ${\mathrm{sc}}$-isomorphism $T\colon E\to{\mathbb{R}}^{n}\oplus W$ with $T(C)=[0,\infty)^{n}\oplus W$. For $x\in C$ write $Tx=(a_{1},\dots,a_{n},w)\in[0,\infty)^{n}\oplus W$ and define its degeneracy index by

[TABLE]

A point $x\in C$ satisfying $d_{C}(x)=0$ is an interior point of $C$, a boundary point if $d_{C}(x)=1$, and a corner point if $d_{C}(x)\geq 2$. See Figure 2.2.

Exercise 2.7.1**.**

The degeneracy index $d_{C}$ does not depend on the choice of linear ${\mathrm{sc}}$-isomorphism $T\colon E\to{\mathbb{R}}^{n}\oplus W$.

Theorem 2.7.2** (Invariance under ${\mathrm{sc}}^{1}$-diffeomorphisms).**

Let $(U,C,E)$ and $(V,D,F)$ be partial quadrants. Let $f\colon U\to V$ be an ${\mathrm{sc}}^{1}$-diffeomorphism, that is an ${\mathrm{sc}}^{1}$-map with an ${\mathrm{sc}}^{1}$-inverse, then for every $x\in U$ one gets equality

[TABLE]

Proof.

Hofer et al. (2007, Thm. 1.19) ∎

2.8 Sc-manifolds

The new notion of differentiability of maps between the new linear spaces – ${\mathrm{sc}}^{k}$ differentiability of maps between ${\mathrm{sc}}$-Banach spaces – allows to carry over the new calculus to topological spaces modeled locally on ${\mathrm{sc}}$-Banach spaces. This results in a new class of manifolds, called ${\mathrm{sc}}^{k}$-manifolds. Their construction parallels the definition of $C^{k}$ Banach manifolds; see Section A.2.4.

To complement Section A.2.4 (case $C^{k}$) we spell out here the smooth case (case ${\mathrm{sc}}^{\infty}$). Suppose $X$ is a topological space. An ${\mathrm{sc}}$-chart $(V,\phi,(U,C,E))$ for $X$ consists of an ${\mathrm{sc}}$-triple $(U,C,E)$ and a homeomorphism $\phi\colon X\supset V\to U\subset C$ between open subsets. Two ${\mathrm{sc}}$-charts are called ${\mathrm{sc}}$-smoothly compatible if the transition map (cf. Figure 2.3)

[TABLE]

is

an ${\mathrm{sc}}$-smooth diffeomorphism (invertible ${\mathrm{sc}}$-smooth map with ${\mathrm{sc}}$-smooth inverse). An ${\mathrm{sc}}$-smooth atlas for $X$ is a collection ${\mathcal{A}}$ of pairwise ${\mathrm{sc}}$-smooth compatible Banach ${\mathrm{sc}}$-charts for $X$ such that the chart domains form a cover $\{V_{i}\}_{i}$ of $X$. Two atlases are called equivalent if their union forms an atlas.

Exercise 2.8.1**.**

Let $X$ be a topological space endowed with an ${\mathrm{sc}}$-smooth atlas ${\mathcal{A}}$. a) Is it true that $X$ is connected iff it is path connected? b) Show that if $X$ is connected, then all model ${\mathrm{sc}}$-Banach spaces $\widetilde{E}$ appearing in the charts of ${\mathcal{A}}$ are (linearly) ${\mathrm{sc}}$-isomorphic to one and the same ${\mathrm{sc}}$-Banach space, say $E$. In this case one says that $(X,{\mathcal{A}})$ is modeled on $E$.

[Hint: b) Given a transition map $\psi\colon E\supset U\to\widetilde{U}\subset\widetilde{E}$ between two ${\mathrm{sc}}$-charts, observe that $U_{\infty}$ is a dense subset of $U$ and that ${\mathrm{sc}}$-derivatives taken at smooth points are ${\mathrm{sc}}$-operators by Corollary 2.4.13.]

Definition 2.8.2**.**

An ${\mathrm{sc}}$-manifold is a paracompact Hausdorff space $X$, see Definition A.1.21, endowed with an equivalence class of ${\mathrm{sc}}$-smooth atlases. If all model spaces are ${\mathrm{sc}}$-Hilbert spaces one speaks of an Hilbert ${\mathrm{sc}}$-manifold.

Definition 2.8.3** (Sc-smooth maps between ${\mathrm{sc}}$-manifolds).**

a) A continuous map $f\colon X\to Y$ between ${\mathrm{sc}}$-manifolds is called ${\mathrm{sc}}$-smooth if for all ${\mathrm{sc}}$-charts $\phi\colon X\supset V\to C\subset E$ and $\psi\colon Y\subset W\to D\subset F$ the chart representative

[TABLE]

is of class ${\mathrm{sc}}^{\infty}$ as a map from an open subset of the partial quadrant $C$ in the ${\mathrm{sc}}$-Banach space $E$

into the ${\mathrm{sc}}$-Banach space $F$. See Figure 2.4.

b) An ${\mathrm{sc}}$-diffeomorphism between ${\mathrm{sc}}$-manifolds is an invertible ${\mathrm{sc}}$-smooth map whose inverse is ${\mathrm{sc}}$-smooth.

Detecting boundaries and corners

Suppose $X$ is an ${\mathrm{sc}}$-manifold. To define the degeneracy index of a point $x\in X$, pick an ${\mathrm{sc}}$-chart $\phi\colon X\supset V\to C\subset E$ about $x$ and set

[TABLE]

By Theorem 2.7.2 the definition does not depend171717 For M-polyfolds the definition might depend on the choice of chart. The way out will be to take the minimum over all charts.

on the choice of ${\mathrm{sc}}$-chart. One calls a point $x$ of degeneracy index $d_{X}(x)=k$ an interior point if $k=0$, a boundary point if $k=1$, and a corner point of complexity $k$ in case $k\geq 2$. This is illustrated by Figure 2.2 for $X=C=[0,\infty)^{2}$.

Levels of ${\mathrm{sc}}$-manifolds are

topological Banach manifolds

A point $x$ of an ${\mathrm{sc}}$-manifold $X$ is said to be on level $m$ if $\phi(x)\in E_{m}$ lies on level $m$ for some (thus every) ${\mathrm{sc}}$-chart $\phi\colon X\supset V\to C\subset E$ about $x$. Indeed the definition does not depend on the choice of chart, even for topological ${\mathrm{sc}}$-manifolds (those of class ${\mathrm{sc}}^{0}$), since any transition map is of class ${\mathrm{sc}}^{0}$, hence level preserving (with continuous level maps). Level $m$ of the ${\mathrm{sc}}$-manifold is the set

[TABLE]

By levelwise continuity of transition maps each level $X_{m}$ of an ${\mathrm{sc}}$-manifold is a topological Banach manifold (in general not $C^{1}$).

To summarize, an ${\mathrm{sc}}$-manifold $X$ decomposes into a nested sequence of topological Banach manifolds

[TABLE]

whose intersection $X_{\infty}$ carries the structure of a smooth Fréchet manifold with boundaries and corners; cf. Cieliebak (2018, §5.3).

Furthermore, each level $X_{k}$ of an ${\mathrm{sc}}$-manifold $X$ inherits the structure of an ${\mathrm{sc}}$-manifold denoted by $X^{k}$ and called the shifted ${\mathrm{sc}}$-manifold $X^{k}$. By definition level $m$ of $X^{k}$ is level $X_{k+m}$ of $X$.

Levels of strong ${\mathrm{sc}}$-manifolds

are smooth Banach manifolds

Suppose $(U,C,E)$ and $(V,D,F)$ are ${\mathrm{sc}}$-triples. The notion of scale differentiability ${\mathrm{sc}}^{1}$ is based on usual $C^{1}$ differentiability of all diagonal maps of height one. A natural way to strengthen this is to ask all level maps (height zero) to be $C^{1}$ (or $C^{k}$). Given $k\in{\mathbb{N}}$ or $k=\infty$, an ${\mathrm{sc}}^{0}$ map $f\colon U\to V$ between ${\mathrm{sc}}$-triples is called strongly ${\mathrm{sc}}^{k}$ or of class ${\mathrm{ssc}}^{k}$ if all level maps $f_{m}\colon U_{m}\to V_{m}$ are of class $C^{k}$. This means that on each level one works with the usual calculus on Banach spaces. Now one calls a paracompact Hausdorff space $X$ an ${\mathrm{ssc}}^{k}$-manifold if all transition maps are of class ${\mathrm{ssc}}^{k}$, that is if they are level-wise $C^{k}$.

Important classes of function spaces fit into the framework of strong scale differentiability, for instance loop spaces of finite dimensional manifolds.

Example 2.8.4** (Loop spaces are ${\mathrm{ssc}}^{\infty}$-manifolds).**

Let $M$ be a manifold of finite dimension. Then the loop space

[TABLE]

that consists of all absolutely continuous maps $x\colon{\mathbb{R}}\to M$ of period one, that is $x(t+1)=x(t)$ for every $t$, is a strongly ${\mathrm{sc}}$-smooth manifold.

Example 2.8.5**.**

The previous example generalizes to $X:=W^{k,p}(N,M)$ where $N$ can be any compact manifold-with-boundary of finite dimension $n$ and where the numbers $k\in{\mathbb{N}}$ and $p\in[1,\infty)$ must satisfy the condition $k>n/p$ (assuring continuity of the functions that are the elements of $X$).

Tangent bundle of ${\mathrm{sc}}$-manifolds

Let $X$ be an ${\mathrm{sc}}$-manifold. For an ${\mathrm{sc}}$-chart $(V,\phi,(U,C,E))$ we shall use the short notation $(V,\phi)$ with the understanding that $U=\phi(V)$ is an open subset of a partial quadrant $C$ in an ${\mathrm{sc}}$-Banach space $E$. Recall that $X^{1}$ denotes the ${\mathrm{sc}}$-manifold that arises from $X$ by forgetting level zero. Let $V^{1}\subset X^{1}$ denote the corresponding scale of levelwise open subsets generated by $V_{1}:=V\mathop{\cap}X_{1}$. Now consider tuples $(V,\phi,x,\xi)$ where $(V,\phi)$ is an ${\mathrm{sc}}$-chart of $X$, the point $x\in V^{1}$ lies on level one, and $\xi\in E^{0}$ is a vector in level zero of the ${\mathrm{sc}}$-Banach space $E$. Two tuples are called equivalent if the two points $x,\widetilde{x}\in X^{1}$ are equal and the two vectors correspond to one another through the ${\mathrm{sc}}$-derivative, in symbols

[TABLE]

An equivalence class $[V,\phi,x,\xi]$ is called a tangent vector to the ${\mathrm{sc}}$-manifold at a point $x$ on level one. There is a canonical projection defined on $TX$, the set of all tangent vectors at all points of $X_{1}$, namely

[TABLE]

Exercise 2.8.6** (Tangent bundle as ${\mathrm{sc}}$-manifold).**

Naturally endow the set $TX$ with the structure of an ${\mathrm{sc}}$-manifold such that the projection $p\colon TX\to X^{1}$ becomes ${\mathrm{sc}}$-smooth as a map between ${\mathrm{sc}}$-manifolds.

[Hint: See Remark A.2.24. Each chart $\phi\colon X\supset V\to E$ of $X$ gives a bijection

[TABLE]

onto the open subset $U^{1}\oplus E^{0}$ of $E^{1}\oplus E^{0}$ where $U=\phi(V)$.]

Chapter 3 Sc-retracts – local models

Let us indoctrinate you right away to the intuition behind the key players $O$ and maps between them. Think of an ${\mathrm{sc}}$-retract $O$ as a compressed open set – the image of some idempotent map $r=r\circ r\colon U\to U$, called a projection or retraction. Vice versa, think of the open set $U$, likewise $r$, as a decompression of $O$. The great variety of possible properties of such $O$ – there can be corners and even jumping dimension – are desirable in applications, because solution spaces to PDEs often exhibit such behavior. In contrast, to do analysis it is desirable that domains of maps are open, so difference quotients, hence derivatives, can be defined. Idempotents $r=r\circ r\colon U\to U$ combine and provide both of these, somehow contradictory, properties. One uses such $O$ as geometric model space and when it comes to analysis one just decompresses $O=r(U)$ and uses the open set $U$ as domain. For instance, to define differentiability of a function $f\colon O\to{\mathbb{R}}$ one decompresses the domain $O$ and calls $f$ differentiable if the pre-composition $f\circ r\colon U\to{\mathbb{R}}$ is. In such context we often call $f\circ r$ or $r$ itself a decompression of $f$. A second highly useful property of images $O=r(U)$ of projections $r\colon U\to U$ is that any such is precisely the fixed point set ${\rm Fix}\,r=O$ of $r$.

In Chapter 3 our main source is again Hofer et al. (2017), together with Cieliebak (2018) and Fabert et al. (2016). Concerning terminology our convention is and was to assign the adjective ${\mathrm{sc}}$-smooth (or the equivalent symbol ${\mathrm{sc}}^{\infty}$) to maps that are $k=\infty$ many times continuously scale differentiable. In case of sets, e.g. ${\mathrm{sc}}$-manifolds or ${\mathrm{sc}}$-retracts, the “${\mathrm{sc}}$” itself already indicates ${\mathrm{sc}}$-smooth.

Outline of Chapter 3. In the present chapter M-polyfolds111 M-polyfolds are defined analogous to manifolds, just based on ${\mathrm{sc}}$-differentiability and more general model spaces. In contrast, polyfolds correspond classically to orbifolds.

are constructed based on the new notion of scale differentiability and locally modeled on rather general topological spaces $O$ which might have corners, even jumping dimension along components, but they will still be accessible to the new weaker form of calculus – ${\mathrm{sc}}$-calculus. The class of spaces are ${\mathrm{sc}}$-retracts, generalizing smooth retracts in Banach manifolds. Section 3.1 “Cartan’s last theorem” deals with smooth retracts and is the motivation for the generalizations in the following sections. Section 3.2 “Sc-smooth retractions and their images $O$” provides the local model spaces $O$ for M-polyfolds. A key step is to extend ${\mathrm{sc}}$-calculus from ${\mathrm{sc}}$-Banach spaces $E$ to ${\mathrm{sc}}$-smooth retracts $O$. Section 3.3 “M-polyfolds and their tangent bundles” defines M-polyfolds, in analogy to manifolds, by patching together local models and asking transition maps to be ${\mathrm{sc}}$-smooth (in the sense of the extended ${\mathrm{sc}}$-calculus). Section 3.4 “Strong bundles over M-polyfolds” provides the environment to implement ${\mathrm{sc}}$-Fredholm sections $f$. The need for ${\mathrm{sc}}^{+}$-sections requires fibers be shiftable in scale by $+1$ leading to double scale structures. In practice $f$ arises as a differential operator of order $\ell$ leading to asymmetry in base and fiber levels.

Detailed summary of Chapter 3

Section 3.1 “Cartan’s last theorem” recalls and proves the surprising result that the image $O=r(U)$ of a smooth idempotent map $r=r\circ r\colon U\to U$ on a Banach manifold, called a smooth retraction, is a smooth submanifold.

Section 3.2 “Sc-smooth retractions and their images $O$” is at the heart of the whole theory. It introduces the local model spaces for M-polyfolds, called ${\mathrm{sc}}$-retracts and denoted by $(O,C,E)$, or simply $O$. These are images $O$ of ${\mathrm{sc}}$-smooth idempotents $r=r^{2}\colon U\to U$, called ${\mathrm{sc}}$-retractions, defined on ${\mathrm{sc}}$-triples $(U,C,E)$. It is useful to observe that image and fixed point set of $r$ coincide and to think of $r$ as a projection onto its fixed point set, in symbols

[TABLE]

While the domain $U$ is a (relatively) open subset of a partial quadrant $C$ in an ${\mathrm{sc}}$-Banach space $E$, its image $O=r(U)$ is a projected or compressed version of $U$. Motivated by continuous retractions one might expect that the compressed set, the ${\mathrm{sc}}$-retract $O=r(U)$ has non-smooth properties, e.g. jumping dimension or having corners, as illustrated by Figure 3.1. In contrast, the images of in the usual sense smooth retractions on Banach manifolds are smooth Banach submanifolds by Theorem 3.1.1.

How can one do analysis and define a derivative on a possibly non-open set $O=r(U)$? The key idea is to decompress $O$ and use the open subset $U$ of $E$ as domain. (We assume $C=E$ for illustration). Let us call $U$, likewise $r$, a decompression of $O$. Of course, if one defines a property of $O$ using a decompression one needs to check independence of the chosen decompression of $O$. For instance, one defines ${\mathrm{sc}}$-smoothness of a map between ${\mathrm{sc}}$-retracts

[TABLE]

if some, hence by Lemma 3.2.4 any, decompression

[TABLE]

of $f$ is an ${\mathrm{sc}}$-smooth map in the ordinary sense; see Definition 2.4.15. Such $f$ is called an ${\mathrm{sc}}$-smooth retract map – the future M-polyfold transition maps. Given an ${\mathrm{sc}}$-retract $(O,C,E)$, the tangent map of a decompression $r=r\circ r\colon U\to U$ of $O=r(U)$ is an ${\mathrm{sc}}$-smooth retraction itself

[TABLE]

Hence the image

[TABLE]

is an ${\mathrm{sc}}$-retract $(TO,TC,TE)$ in the tangent ${\mathrm{sc}}$-triple $(TU,TC,TE)$. Here $TO$ is independent of the choice of the decompression $r$ of $O$ by Lemma 3.2.6. The tangent bundle of the ${\mathrm{sc}}$-retract $O$ is the natural surjection

[TABLE]

It is an ${\mathrm{sc}}$-smooth map between ${\mathrm{sc}}$-retracts. The tangent space at $x\in O^{1}$

[TABLE]

is a Banach subspace, even an ${\mathrm{sc}}$-subspace for $x\in O_{\infty}$, by Corollary 2.4.13.

The tangent map of an ${\mathrm{sc}}$-smooth retract map $f\colon O\to O^{\prime}$ is defined as the restriction to $TO$ of the tangent map

[TABLE]

of some, by Lemma 3.2.10 any, decompression $f\circ r$. Here $f\circ r(x)=f(x)$ since $O^{1}\subset O={\rm Fix}\,r$ and $D(f\circ r)|_{x}=Df(x)$ on $T_{x}O={\rm Fix}\,Dr(x)\subset E$. Section 3.2 on ${\mathrm{sc}}$-retracts is rounded off by the chain rule for compositions of ${\mathrm{sc}}$-smooth retract maps.

Section 3.3 “M-polyfolds and their tangent bundles” defines M-polyfolds in analogy to Banach manifolds just using the rather general class of ${\mathrm{sc}}$-retracts as local models and requiring only scale smoothness of the transition maps. In particular, to define an M-polyfold $X$ one starts with a paracompact Hausdorff space $X$. E.g. ${\mathrm{sc}}$-manifolds are M-polyfolds ($r=id$ and $O=U$) and so are open subsets of M-polyfolds. Sc-smoothness of maps

[TABLE]

between M-polyfolds is defined in terms of local coordinate representatives of $f$ which are required to be ${\mathrm{sc}}$-smooth retract maps. An M-polyfold $X$ inherits a set scale structure from the local model spaces. Let $X_{m}$, called level $m$ of $X$, consist of all points of $X$ which are mapped in some, hence any, coordinate chart into level $m$ of model space. Each level $X_{m}$ is a topological Banach manifold and inherits the structure of an M-polyfold denoted by $X^{m}$.

To construct the tangent bundle $p\colon TX\to X^{1}$ one first defines $TX$ as a set and then a natural map $p$, using the local coordinate charts $\phi\colon V\to O$ of $X$ to define bijections denoted by $T\phi\colon TX\supset TV\to TO$. Given an atlas ${\mathcal{A}}$ of $X$, these bijections induce the collection

[TABLE]

of subsets of $TX$. It forms a basis of a paracompact Hausdorff topology. Endowing $X$ with that topology the bijections $T\phi$ become homeomorphisms and one gets a natural M-polyfold atlas $T{\mathcal{A}}$ for $TX$.

Sub-M-polyfolds. A subset $A\subset X$ of an M-polyfold is a sub-M-polyfold if around any point $a\in A$ there is an open neighborhood $V\subset X$ and an ${\mathrm{sc}}$-smooth retraction $r=r^{2}\colon V\to V$ such that $A\mathop{\cap}V=r(V)={\rm Fix}\,r$. Such $r$ is called a local generator for the sub-M-polyfold $A$. Viewed as a map $r\colon V\to A$ a local generator is ${\mathrm{sc}}$-smooth and $T_{a}r(T_{a}X)=T_{a}A$ at any point $a\in A\mathop{\cap}V$. At smooth points the tangent space $T_{a}A$ is ${\mathrm{sc}}$-complemented in $T_{a}X$.

Boundaries and corners – tameness. Recall from (2.7.26) that the degeneracy index $k=d_{C}(p)$ of a point $p$ of a partial quadrant $C$ tells whether $p$ is an interior point ($k=0$), a boundary point ($k=1$), or a corner point of complexity $k\geq 2$. Unfortunately, for points $x$ of M-polyfolds $X$ the degeneracy index $d_{X}(x):=d_{C}(\phi(x))$ defined in terms of an M-polyfold chart $\phi$ may depend on the chart; see Figure 3.4. Thus one introduces a new class, the so-called tame M-polyfolds, for which there is no dependence on $\phi$.

Section 3.4 “Strong bundles over M-polyfolds” provides the environment to implement partial differential operators whose zero sets will represent the moduli spaces which are under investigation in many different geometric analytic situations. Often moduli spaces, hence zero sets, are of finite dimension and are modeled on the kernels of surjective Fredholm operators. To achieve surjectivity in a given geometric PDE scenario one usually perturbs some already present, but inessential, quantity. These perturbations should be related to bounded operators, so the overall Fredholm property is preserved.

Recall from Proposition 2.3.29 that the ${\mathrm{sc}}$-Fredholm property of a linear map $T\colon E\to F$ is preserved under addition of ${\mathrm{sc}}^{+}$-operators $S\colon E\to F$. The latter operators are characterized by the property of improving their output regularity by one level, that is $S(E_{m})\subset F_{m+1}$. As a consequence all level operators $S_{m}\colon E_{m}\to F_{m+1}\hookrightarrow F_{m}$ are compact.

Motivation. Replacing now the linear domain $E$ by an M-polyfold $X$ as domain of a partial differential operator $f$ of order, say $\ell$, the task at hand222 freely borrowed from one of my favorite authors

is to construct vector bundles $P\colon Y\to X$ with fibers modeled on an ${\mathrm{sc}}$-Banach space $F$, so that the differential operator becomes a section $f\colon X\to Y$. Concerning the implementation of Fredholm properties one has to allow for fiber level shifts by $+1$, that is all fibers $Y_{x}:=P^{-1}(0)$ should be identifiable with the ${\mathrm{sc}}$-Banach space $F^{0}=F$, as well as with the shifted one $F^{1}$; cf. Remark 2.3.12. In practice, the level indices $m$ correspond to the degree of differentiability of the level elements. So the domain of $f$ should be $X^{\ell+m}$ in which case $f$ takes values in level $m$, sometimes even $m+1$. Then one can exploit composition with compact embeddings up to level [math]; see Remark 3.4.2. This motivates the following asymmetric double scale structure which must be subsequently reduced to two versions of individual scales, in order to be accessible to scale calculus (there is no double scale calculus).

Trivial-strong-bundle retracts $K$ – the local models. Let $E,F$ be Banach scales and $U\subset E$ be open. The non-symmetric product $U\triangleright F$ is the subset $U\times F$ of the Banach space $E\oplus F$ endowed with the double scale, also called double filtration, defined by

[TABLE]

Projection onto the first component

[TABLE]

is called the trivial-strong-bundle projection. However, for ${\mathrm{sc}}$-calculus one needs one scale structure, not a double scale. Consider the ${\mathrm{sc}}$-manifolds

[TABLE]

For $i\in\{0,1\}$ projection on component one is an ${\mathrm{sc}}$-smooth map

[TABLE]

between ${\mathrm{sc}}$-manifolds called a trivial strong sc-bundle. A trivial-strong-bundle retraction is an idempotent **trivial-strong-bundle map333 i.e. double scale preserving and with $\rho_{u}\xi:=\rho(u,\xi)$ being linear in $\xi$ **

[TABLE]

The first component $r$ of $R$ is necessarily an ${\mathrm{sc}}$-smooth retraction on $U$, called associated base retraction. Its image, the ${\mathrm{sc}}$-retract $O=r(U)$, is called the associated base retract. A **trivial-strong-bundle retract444 ’strong’ indicates ’doubly scaled’ and the retraction acts on a ’trivial bundle’ ** $(K,C\triangleright F,E\triangleright F)$ is the image

[TABLE]

of a trivial-strong-bundle retraction $R=R\circ R$ on $U\triangleright F$ where $O=r(U)$ is the associated base retract. One likewise calls the natural surjection

[TABLE]

a trivial-strong-bundle retract. Call $K:=R(U\triangleright F)$ tame if $R$ is tame. As a subset of the doubly scaled space $U\triangleright F$ there is an induced double scale

[TABLE]

for $m\in{\mathbb{N}}_{0}$ and $k\in\{0,\dots m+1\}$. The spaces

[TABLE]

with levels $K^{[i]}_{m}=K_{m,m+i}$ are ${\mathrm{sc}}$-retracts, hence M-polyfolds. The surjections

[TABLE]

are ${\mathrm{sc}}$-smooth maps between ${\mathrm{sc}}$-retracts.

A section of a trivial-strong-bundle retract $p\colon K\to O$ is a map $s\colon O\to K$ that satisfies $p\circ s={\rm id}_{O}$. If $s$ is ${\mathrm{sc}}$-smooth as an ${\mathrm{sc}}$-retract map

[TABLE]

it is called an ${\mathrm{sc}}$-section (case $i=0$) or an ${\mathrm{sc}}^{+}$-section (case $i=1$). The map {\text{\boldmaths}}^{[i]}\colon O\to F^{i} is called the principal part of the section.

Strong bundles. A strong bundle over an M-polyfold $X$ is a continuous surjection $P\colon Y\to X$ defined on a paracompact Hausdorff space $Y$ such that each pre-image $Y_{x}:=P^{-1}(x)$ is a Banachable space, together with an equivalence class of strong bundle atlases.

As usual, one patches together local model bundles which in our case are the trivial-strong-bundle retracts $K=R(U\triangleright F)\to O$ outlined above. A strong bundle atlas for $P\colon Y\to X$ consists of suitably compatible strong bundle charts

[TABLE]

Such tuple consists of

• •

a trivial-strong-bundle retract $K$, that is $p\colon K=R(U\triangleright F)\to O$ where $O=r(U)$ is the associated base retract;

• •

a homeomorphism $\varphi\colon X\supset V\to O$ between an open subset of the base M-polyfold $X$ of $Y$ and the base retract $O$ of $K$;

• •

a homeomorphism $\Phi\colon P^{-1}(V)\to K$ which covers $\varphi$ in the sense that the diagram

[TABLE]

commutes. Consequently, for every point $v\in V$ the restriction of $\Phi$ to $P^{-1}(v)$ takes values in $p^{-1}(\varphi(v))$. It is also required that $\Phi$ as a map

[TABLE]

is a continuous linear bijection between the Banach/able space fibers.

A strong bundle atlas ${\mathcal{A}}^{Y}_{X}$ for $P\colon Y\to X$ provides a double scale structure on $X$ induced by local charts. As earlier, one extracts two individual scale structures and obtains two induced ${\mathrm{sc}}$-bundle atlases ${\mathcal{A}}^{Y^{[0]}}_{X}$ and ${\mathcal{A}}^{Y^{[1]}}_{X}$ for ${\mathrm{sc}}$-bundles555 The definition of ${\mathrm{sc}}$-bundles is indicated around (3.4.7).

[TABLE]

A section of a strong bundle $P\colon Y\to X$ is a map $s\colon X\to Y$ that satisfies $P\circ s={\rm id}_{X}$. If $s$ is ${\mathrm{sc}}$-smooth as a map between M-polyfolds

[TABLE]

then $s$ is called in case $i=0$ an ${\mathrm{sc}}$-section of $P\colon Y\to X$ and in case $i=1$ an ${\mathrm{sc}}^{+}$-section of $P\colon Y\to X$.

3.1 Cartan’s last theorem

In the realm of continuous linear operators $R$ on a Banach space $E$ an idempotent $R=R^{2}$ is called a projection. Note that the image ${\rm im\,}R={\rm Fix}\,R$ is equal to the fixed point set of $R$. (Both inclusions are immediate, only ’$\subset$’ uses idempotency.) But the image of a linear operator is a linear subspace and the fixed point set of a continuous map is a closed subset. So the image of a projection is a closed linear subspace which, furthermore, is complemented by the (again due to continuity) closed linear subspace $\ker R$. To summarize

[TABLE]

More generally, given a topological space $X$, a continuous idempotent map $r=r\circ r\colon X\to X$ is called a retraction on $X$ and the closed subset

[TABLE]

is called a retract of $X$.

Theorem 3.1.1** (Cartan (1986)).**

The image of a smooth retraction $r\colon X\to X$ on a Banach manifold is a topologically closed smooth submanifold of $X$.

Proof.

We follow Cieliebak (2018). Closedness of the set ${\rm im\,}r={\rm Fix}\,r$ holds by continuity of $r$. To be a submanifold is a local property. Pick $x\in{\rm Fix}\,r$ and a Banach chart $(V,\phi,E)$ about $x$ with $\phi(x)=0$; cf. Section A.2.4. It suffices to show that ${\rm Fix}\,r$ is locally near $x$ the image under a diffeomorphism, say $\alpha^{-1}$, of an open subset of a linear subspace, say ${\rm Fix}\,[R\colon E\to E]$ for some $R=R^{2}\in{\mathcal{L}}(E)$, of the Banach space $E$. This takes three steps.

Step 1. (Localize) The retraction $r$ on $X$ descends to a smooth retraction on an open subset $U\subset E$ of the local model Banach space, still denoted by

[TABLE]

The derivative $R:=dr(0)=R^{2}\in{\mathcal{L}}(E)$ is a projection and the maps

[TABLE]

take on the same value $\alpha(0)={{\mathchoice{\rm 1\mskip-4.0mul}{\rm 1\mskip-4.0mul}{\rm 1\mskip-4.5mul}{\rm 1\mskip-5.0mul}}}-R=\beta(0)$ at the origin.

Proof of Step 1. Observe that $V\mathop{\cap}r^{-1}(V)$ is not only an open neighborhood of the fixed point $r(x)=x\in V$, but it is also invariant under $r$: Indeed

[TABLE]

where both inclusions are immediate, only the second one uses $r\circ r=r$. Hence the local representative $\phi\circ r\circ\phi$, hereafter still denoted by $r$, is a smooth retraction on $U$ and it maps $0=\phi(x)$ to itself. The latter fixed point property enters the identity $R=dr|_{0}=d(r\circ r)|0=dr_{r(0)}\circ dr|_{0}=R\circ R$.

Step 2. (Local diffeomorphism) The map $\alpha$ conjugates $r$ and $R$

[TABLE]

and it holds that $\alpha(0)=0$ and $d\alpha(0)={{\mathchoice{\rm 1\mskip-4.0mul}{\rm 1\mskip-4.0mul}{\rm 1\mskip-4.5mul}{\rm 1\mskip-5.0mul}}}$.

Proof of Step 2. The retraction properties of $r$ and $R$ imply the identities

[TABLE]

and

[TABLE]

These two identities imply, respectively, the identities

[TABLE]

and

[TABLE]

Thus $\alpha\circ r=R\circ\alpha$. Hence $\alpha(0)=\alpha(r(0)=R(\alpha(0))=R({{\mathchoice{\rm 1\mskip-4.0mul}{\rm 1\mskip-4.0mul}{\rm 1\mskip-4.5mul}{\rm 1\mskip-5.0mul}}}-R)=0$ and

[TABLE]

Step 3. (Conjugation to linearization) There is an open subset $W\subset U$ of $E$ such that $\alpha\colon W\to E$ is a diffeomorphism onto its image $\alpha(W)$ and $r(W)\subset W$. Moreover, the linear retraction $R=dr(0)\colon E\to E$ restricts to a smooth retraction on $W$ and coincides with the composition

[TABLE]

Proof of Step 3. Since $d\alpha(0)={{\mathchoice{\rm 1\mskip-4.0mul}{\rm 1\mskip-4.0mul}{\rm 1\mskip-4.5mul}{\rm 1\mskip-5.0mul}}}$ is invertible there is by the inverse function theorem an open neighborhood $W^{\prime}\subset U$ of the fixed point $0\in E$ of $\alpha$ and $r$ such that the restriction $\alpha\colon W^{\prime}\to E$ is a diffeomorphism onto its image. To obtain, in addition, invariance under $r$ replace $W^{\prime}$ by $W:=W^{\prime}\mathop{\cap}r^{-1}(W^{\prime})$. To see this repeat the arguments that led to (3.1.1).

Step 4. (Diffeomorphism to open set in Banach space) Step 3 shows

[TABLE]

Step 4 proves Theorem 3.1.1: Indeed $\alpha(W)$ is an open neighborhood in $E$ of the fixed point [math] of $r$ and ${\rm Fix}\,R$ is a (closed) linear subspace of $E$. So the intersection is an open neighborhood of [math] in the Banach space ${\rm Fix}\,R$. But that intersection is diffeomorphic, under $\alpha^{-1}$, to the part of ${\rm Fix}\,r$ in the open set $W$. ∎

3.2 Sc-smooth retractions and their images $O$

In this section the local model spaces for M-polyfolds are constructed and the maps between them are endowed with an adequate notion of ${\mathrm{sc}}$-smoothness, namely, ${\mathrm{sc}}$-smoothness when viewed as maps between decompressed domains. The model spaces are images $O$ of ${\mathrm{sc}}$-smooth retractions $r=r^{2}\colon U\to U$ on ${\mathrm{sc}}$-triples $(U,C,E)$. It is useful to observe that image and fixed point set of $r$ coincide and to think of $r$ as a projection onto its image

[TABLE]

Sc-retracts and sc-smoothness of maps between them

Definition 3.2.1** (Sc-retracts $O$).**

An sc-smooth retraction on an ${\mathrm{sc}}$-triple $(U,C,E)$ is an ${\mathrm{sc}}$-smooth idempotent map $r=r\circ r\colon U\to U$. Note that

[TABLE]

${\rm Fix}\,r\subset U$ is (relatively) closed by continuity of $r$. An sc-retract $O\subset C\subset E$ in a partial quadrant $C$ in a Banach scale $E$ is the image (fixed point set)

[TABLE]

of some ${\mathrm{sc}}$-smooth retraction $r$ whose domain $U\subset C$ is (relatively) open. Usually we abbreviate the notation $(O,C,E)$ of an ${\mathrm{sc}}$-retract by simply writing $O$. As pointed out in Hofer et al. (2017, before Prop. 2.3), the ambient partial quadrant $C\subset E$ matters, because it is possible that $O$ is an ${\mathrm{sc}}$-retract with respect to some non-trivial $C$, but not for $C=E\colon$ Think of $(O,E,E)$ as local models for M-polyfolds in regions without boundary and $(O,C,E)$ as such near boundaries with corners; cf. Hofer et al. (2010, after Def. 1.13).

Lemma 3.2.2**.**

If $r\colon U\to U$ is an ${\mathrm{sc}}$-smooth retraction, then all level maps are continuous retractions

[TABLE]

and $O_{m}:=O\cap E_{m}$ is equal to the image $r(U_{m})$. In terms of shifted scales

[TABLE]

Proof.

To be shown is the equality of sets $O_{m}=r(U_{m})$. ’$\subset$’ Pick $x\in O\mathop{\cap}E_{m}\subset U\mathop{\cap}E_{m}$, then $x=r(x)\in r(U_{m})$. ’$\supset$’ Pick $x\in U_{m}$, then $r(x)\in r(U)\mathop{\cap}E_{m}=O\mathop{\cap}E_{m}$ since $U_{m}\subset U$ and $r$ is level preserving, respectively. ∎

Whereas the image of a smooth retraction on a Banach manifold is a smooth submanifold by Cartan’s last theorem, Theorem 3.1.1, an ${\mathrm{sc}}$-retract can be connected and nevertheless have pieces of various dimensions; see Figure 3.1.

How can one ever do analysis on such spaces? Let’s see:

Decompression. To start with, given a map $f\colon O\to O^{\prime}$ between ${\mathrm{sc}}$-retracts, one can “decompress” or “unpack” the, possibly “cornered”, domain $O$ of the map $f$ into an open set by pre-composing with an ${\mathrm{sc}}$-smooth retraction $r\colon U\to U$ whose image is $O=r(U)$. Indeed the map $f\circ r\colon U\to O^{\prime}\subset U^{\prime}$ has the same image as $f$, but lies within the reach of ${\mathrm{sc}}$-calculus since domain and target are (relatively) open subsets of partial quadrants $C$ and $C^{\prime}$ in ${\mathrm{sc}}$-Banach spaces.

Definition 3.2.3** **(Sc-smooth maps among ${\mathrm{sc}}$-retracts –

decompress domain).

A map $f\colon O\to O^{\prime}$ between ${\mathrm{sc}}$-retracts is called an ${\mathrm{sc}}$-smooth retract map if the composition $f\circ r\colon U\to U^{\prime}$ is ${\mathrm{sc}}$-smooth666 Sc-smoothness of $f\circ r\colon U\to U^{\prime}$ implies continuity of $f\colon O\to O^{\prime}$.

for some, thus by Lemma 3.2.4 for every, ${\mathrm{sc}}$-smooth retraction $r$ whose image is $O=r(U)$. Let us refer to such pre-composition process as decompressing (the domain of) $f$. Sc-smooth retract maps $O\to O^{\prime}$ are continuous.

Lemma 3.2.4**.**

Given ${\mathrm{sc}}$-smooth retractions with equal image $O=r(U)=s(V)$ and a map $f\colon O\to U^{\prime}$, then if one of the maps

[TABLE]

is ${\mathrm{sc}}$-smooth, so is the other one.

Proof.

By assumption ${\rm im\,}r=O={\rm Fix}\,s$ and ${\rm im\,}s=O={\rm Fix}\,r$, hence

[TABLE]

By hypothesis $s$ is ${\mathrm{sc}}$-smooth. If also $f\circ r$ is ${\mathrm{sc}}$-smooth, so is by the chain rule their composition $(f\circ r)\circ s$. But $f\circ(r\circ s)=f\circ s$. ∎

Tangent bundle of sc-retracts and tangent map

of sc-retract maps

Lemma 3.2.5** (Tangent map of retraction is itself a retraction).**

Let $r\colon U\to U$ be an ${\mathrm{sc}}$-smooth retraction on an ${\mathrm{sc}}$-triple $(U,C,E)$. Then its tangent map

[TABLE]

is an ${\mathrm{sc}}$-smooth retraction on the tangent ${\mathrm{sc}}$-triple

[TABLE]

Proof.

The tangent map of an ${\mathrm{sc}}$-smooth map is ${\mathrm{sc}}$-smooth by the iterative definition of ${\mathrm{sc}}^{k}$-smoothness; see Definition 2.4.15. It remains to show that

[TABLE]

’$\supset$’ A fixed point $x=f(x)$ of a map lies in its image. ’$\subset$’ An element of $Tr(TU)$ is of the form (y,{\color[rgb]{.75,.5,.25}\definecolor[named]{pgfstrokecolor}{rgb}{.75,.5,.25}\eta})=(r(x),{\color[rgb]{.75,.5,.25}\definecolor[named]{pgfstrokecolor}{rgb}{.75,.5,.25}Dr|_{x}\,\xi}) for some $(x,\xi)\in U_{1}\oplus E^{0}$. Hence $r(y)=r\circ r(x)=r(x)=y$ and

[TABLE]

where we used the chain rule (2.6.23). Hence $Tr(y,\eta)=(y,\eta)\in{\rm Fix}\,Tr$. ∎

Lemma 3.2.6** (Tangent maps of two decompressions have same image).**

Assume an ${\mathrm{sc}}$-smooth retract $(O,C,E)$ is the image of two ${\mathrm{sc}}$-smooth retractions

[TABLE]

Then both tangent maps have equal image $Tr(TU)=Ts(TV)\subset(O^{1}\oplus E^{0})$.

Proof.

We need to show ${\rm Fix}\,Tr={\rm Fix}\,Ts$. ’$\subset$’ Pick $(x,\xi)\in{\rm Fix}\,Tr=Tr(TU)\subset TU\subset E^{1}\oplus E^{0}$. Then {\color[rgb]{.75,.5,.25}\definecolor[named]{pgfstrokecolor}{rgb}{.75,.5,.25}\xi\in{\rm Fix}\,Dr|_{x}}\subset E and with (3.2.2) for $r$ and $s$ we conclude

[TABLE]

So $(x,\xi)\in V^{1}\oplus E^{0}=TV$ lies in the domain of $Ts$ and we get that

[TABLE]

Here we used twice the identity $s\circ r=r$ which holds since $r(U)=O={\rm Fix}\,s$ by hypothesis. ’$\supset$’ Same argument. ∎

Definition 3.2.7** (Tangent of ${\mathrm{sc}}$-retract).**

If $O=r(U)$ is an ${\mathrm{sc}}$-retract, then

[TABLE]

is an ${\mathrm{sc}}$-retract, too. Notation $T(O,C,E):=(TO,TC,TE)$. The definition of $TO$ does not depend on the choice of $(r,U)$ by Lemma 3.2.6.

Lemma 3.2.8** (Tangent bundle of ${\mathrm{sc}}$-retract $O$).**

The natural projection

[TABLE]

is an open surjective ${\mathrm{sc}}$-smooth retract map, cf. Definition 3.2.3, called tangent bundle of the ${\mathrm{sc}}$-retract $O$. The pre-image of a point, denoted by

[TABLE]

is a Banach subspace of $E$, an ${\mathrm{sc}}$-subspace whenever $x\in O_{\infty}$.

Proof.

Let $O=r(U)$. Then $TO:=Tr(TU)$ where $TU:=U^{1}\oplus E^{0}$. Hence the first component of $Tr$ is the map $r$ on the domain $U^{1}$, see (3.2.3). But $r(U^{1})=O^{1}$ by Lemma 3.2.2 which proves surjectivity of $p$. The decompression

[TABLE]

of $p$ is constant in $\xi$, and in $x$ it is the map $r\colon U^{1}\to U^{1}$ which is ${\mathrm{sc}}$-smooth by Lemma 2.5.5, since $r\colon U\to U$ is ${\mathrm{sc}}$-smooth by assumption. The pre-image

[TABLE]

is the fixed point set of a linear operator on the Banach space $E$ and therefore it is a linear subspace. It is a closed linear subspace, because the linear operator is continuous. For simplicity we shall simply write

[TABLE]

The ${\mathrm{sc}}$-derivative $Dr(x)\colon E\to E$ at any $x\in O_{\infty}$ restricts to a continuous linear operator on every level $E_{m}$ by Corollary 2.4.13. Hence $(p^{-1}(x))_{m}:=p^{-1}(x)\mathop{\cap}E_{m}$ are the levels of a Banach scale by Exercise 3.2.9. ∎

Exercise 3.2.9**.**

a) Show that $T_{x}O$ is an ${\mathrm{sc}}$-subspace of $E$ whenever $x\in O_{\infty}$. b) Show that the projection $p\colon TO\to O^{1}$ is an open map.

[Hint: a) Let $O=r(U)$. Show that $F_{m}:=(p^{-1}(x))_{m}:=p^{-1}(x)\mathop{\cap}E_{m}$ equals $F_{m}={\rm Fix}\,[Dr(x)\colon E_{m}\to E_{m}]$ and $F=F_{0}\supset F_{1}\supset\dots$ satisfies the three axioms (Banach levels), (compactness), and (density) of a Banach scale. b) First consider the case $C=E$, decompress $p$.]

Lemma 3.2.10**.**

Let $f\colon O\to O^{\prime}$ be an ${\mathrm{sc}}$-smooth retract map. If $r\colon U\to U$ and $s\colon V\to V$ are ${\mathrm{sc}}$-smooth retractions with image $O$, then the restrictions

[TABLE]

a) coincide and b) take values in $TO^{\prime}$ and c) are ${\mathrm{sc}}$-smooth retract maps.

Proof.

a) For (x,\xi)\in TO={\color[rgb]{.75,.5,.25}\definecolor[named]{pgfstrokecolor}{rgb}{.75,.5,.25}{\rm Fix}\,Ts}, as $r\circ s=s$ (${\rm Fix}\,r=O={\rm im\,}s$), we get

[TABLE]

b) Let $O^{\prime}=t(W)$, then it suffices to show ${\rm im\,}T(f\circ r)\subset{\rm Fix}\,Tt$. Observe that $t\circ f=f$ since ${\rm im\,}f\subset O^{\prime}={\rm Fix}\,t$. Hence $(x,\xi)\in TU$ provides a fixed point

[TABLE]

c) The decompression of $T(f\circ r)|_{TO}$ given by

[TABLE]

is ${\mathrm{sc}}$-smooth, because $f\circ r\colon U\to U^{\prime}$ is ${\mathrm{sc}}$-smooth due to the assumption that $f\colon O\to O^{\prime}$ is an ${\mathrm{sc}}$-smooth retract map, see Definition 3.2.3. ∎

Definition 3.2.11** **(Tangent of retract maps via domain

decompression).

The tangent map of an ${\mathrm{sc}}$-smooth retract map $f\colon O\to O^{\prime}$ is the restriction

[TABLE]

of the tangent map $T(f\circ r)\colon TU\to TU^{\prime}$ for a decompression $r$ of $O=r(U)$.

Some remarks are in order. Firstly, by Lemma 3.2.10 the definition of $Tf$ does not depend on the ${\mathrm{sc}}$-smooth retraction $r\colon U\to U$ with image $O$. Secondly, concerning component one $f\circ r(x)=f(x)$ since $x\in O^{1}\subset O={\rm Fix}\,r$. Thirdly, concerning component two

[TABLE]

since $T_{x}O$ is the fixed point set of $Dr(x)$.

Theorem 3.2.12** (Chain rule for ${\mathrm{sc}}$-smooth retract maps).**

Let $f\colon O\to O^{\prime}$ and $g\colon O^{\prime}\to O^{\prime\prime}$ be ${\mathrm{sc}}$-smooth retract maps. Then the composition $g\circ f\colon O\to O^{\prime\prime}$ is also a ${\mathrm{sc}}$-smooth retract map and the tangent maps satisfy

[TABLE]

Proof.

Sc-smoothness of the retract maps $f$ and $g$ by definition means ${\mathrm{sc}}$-smoothness of $f\circ r$ and $g\circ r^{\prime}$ where $r\colon U\to U$ and $r^{\prime}\colon U^{\prime}\to U^{\prime}$ are ${\mathrm{sc}}$-smooth retractions with images $O$ and $O^{\prime}$, respectively. The inclusion ${\rm im\,}f\subset O^{\prime}=Fix\,r^{\prime}$ provides the identity $f=r^{\prime}\circ f$. Hence $(g\circ f)\circ r=(g\circ r^{\prime})\circ(f\circ r)$ is a composition of two ${\mathrm{sc}}$-smooth maps, so it is ${\mathrm{sc}}$-smooth itself by the chain rule for ${\mathrm{sc}}$-smooth maps, Theorem 2.6.1. By definition of $Tf$, the chain rule, and $f=r^{\prime}\circ f$ we get $Tg\circ Tf=T(g\circ r^{\prime})\circ T(f\circ r)|_{TO}=T(g\circ f\circ r)|_{TO}=:T(g\circ f)$. ∎

In Section 3.3 the next exercise will be useful a) to show that open subsets of M-polyfolds are M-polyfolds and b) to construct sub-M-polyfold charts.

Exercise 3.2.13**.**

Given an ${\mathrm{sc}}$-retract $(O,C,E)$, prove the following.

• a)

Open subsets $O^{\prime}$ of the ${\mathrm{sc}}$-retract $O$ are ${\mathrm{sc}}$-retracts in $C$.

• b)

Suppose $V$ is an open subset of $O$ and $s=s\circ s\colon V\to V$ is an idempotent ${\mathrm{sc}}$-smooth retract map. The image of such $s$ is an ${\mathrm{sc}}$-retract $(o,C,E)$.

[Hints: Let $O={\rm im\,}[r\colon U\to U]$. a) How about $U^{\prime}:=r^{-1}(O^{\prime})$ and $r^{\prime}:=r|_{U^{\prime}}$ ? b) Let $o:={\rm im\,}s={\rm Fix}\,s$, then $o\subset s^{-1}(o)=V\subset O={\rm Fix}\,r$. How about $U^{\prime}:=r^{-1}(V)=r^{-1}(s^{-1}(o))\subset U$ and the ${\mathrm{sc}}$-smooth map $s\circ r\colon U^{\prime}\to U^{\prime}$ ?]

3.2.1 Special case: Splicings and splicing cores

Following Hofer et al. (2017, Def. 2.18), an ${\mathrm{sc}}$-smooth splicing on an ${\mathrm{sc}}$-Banach space $E$ consists of the following data. A relatively open neighborhood $V$ of [math] in a partial quadrant $[0,\infty)^{\ell}\times{\mathbb{R}}^{d-\ell}$ in ${\mathbb{R}}^{d}$ and a family $\{\pi_{v}\}_{v\in V}$ of ${\mathrm{sc}}$-projections $\pi_{v}=\pi_{v}\circ\pi_{v}\in\mathcal{L}_{\mathrm{sc}}(E)$ such that the map

[TABLE]

is ${\mathrm{sc}}$-smooth. Note that in this case each projection $\pi_{v}$ restricts to a continuous linear operator $\pi_{v}|_{E_{m}}\in{\mathcal{L}}(E_{m})$ on every level. But in the operator norm these operators do not, in general, depend continuously on $v$. The subset of ${\mathbb{R}}^{d}\oplus E$ composed of the images (fixed points) of each projection, i.e.

[TABLE]

is called the splicing core of the splicing.

Exercise 3.2.14** (Induced sc-smooth retraction).**

Given an ${\mathrm{sc}}$-smooth splicing $\{\pi_{v}\}_{v\in V\subset{\mathbb{R}}^{d}}$ on an ${\mathrm{sc}}$-Banach space $E$, consider the map given by

[TABLE]

Show that the map $r_{\pi}$ defines an ${\mathrm{sc}}$-smooth retraction on the ${\mathrm{sc}}$-triple $(V\oplus E,([0,\infty)^{\ell}\times{\mathbb{R}}^{d-\ell})\oplus E,{\mathbb{R}}^{d}\oplus E)$ and that its image is the splicing core $K^{\pi}$.

As remarked in Fabert et al. (2016, previous to Def. 5.6), this setup of splicing with finitely many “gluing” parameters covers the ${\mathrm{sc}}$-retractions relevant for Morse theory and holomorphic curve moduli spaces.

3.2.2 Splicing core with jumping finite dimension

Fix a smooth bump function $\beta\geq 0$ on ${\mathbb{R}}$ supported in $[-1,1]$ of unit $L^{2}$ norm. For $t>0$ consider the family $\beta_{t}(s):=\beta(s+e^{1/t})$ of left translates of $\beta$ by $e^{1/t}$ – huge left translations for $t$ near [math] and almost no translation for $t\sim\infty$. Fix a strictly increasing sequence of reals $\delta_{m}$ starting at $\delta_{0}=0$ and let $E=L^{2}({\mathbb{R}})$ be the ${\mathrm{sc}}$-Hilbert space whose levels are given by the weighted Sobolev spaces $E_{m}:=W^{m,2}_{\delta_{m}}({\mathbb{R}})$ introduced in Exercise 2.2.4. Consider the family

[TABLE]

of linear operators on $E$. Note that the image of $\pi_{t}$ is $\{0\}$ whenever $t\leq 0$, whereas for each $t>0$ the image of $\pi_{t}$ is ${\mathbb{R}}\beta_{t}$, hence one dimensional.

Exercise 3.2.15**.**

Check that each linear operator $\pi_{t}\colon E\to E$ is continuous and a projection, that is $\pi_{t}\circ\pi_{t}=\pi_{t}$.

Proposition 3.2.16**.**

The map $\pi\colon{\mathbb{R}}\oplus E\to E$, $(t,f)\mapsto\pi_{t}f$, is ${\mathrm{sc}}$-smooth.

Proof.

The result and details of the (hard) proof of ${\mathrm{sc}}$-smoothness are given in Hofer et al. (2010, Ex. 1.22 and Le. 1.23); see also Cieliebak (2018, Prop. 6.8). ∎

To summarize, the family $\{\pi_{t}\}_{t\in{\mathbb{R}}}$ of projections defines an ${\mathrm{sc}}$-smooth splicing on $E=L^{2}$. The corresponding splicing core $K^{\pi}\subset{\mathbb{R}}\oplus E$ is represented in Figure 3.1 as a subset of ${\mathbb{R}}^{2}$ homeomorphic to $K^{\pi}={\rm im\,}r_{\pi}=r_{\pi}({\mathbb{R}}\oplus E)$. Although connected, there are parts of dimension one and two.

3.3 M-polyfolds and their tangent bundles

M-polyfolds are defined analogous to ${\mathrm{sc}}$-manifolds, just use as local models instead of ${\mathrm{sc}}$-triples $(U,C,E)$ ${\mathrm{sc}}$-retracts $O={\rm im\,}[r=r^{2}\colon U\to U]$ in $C\subset E$.

Recall two standard methods to define manifolds. Method 1 starts with a topological space $X$, then one defines a collection of homeomorphisms to open sets in model Banach spaces, whose domains are open subsets of $X$ which together cover $X$. The collection must be suitably compatible on overlaps. Method 2 starts with only a set $S$, then one defines a collection of bijections between subsets of $S$ onto open subsets of local model Banach spaces, again the domains together must cover $S$. Now one uses the bijections to define a topology on the set $S$, essentially by declaring pre-images of open sets in model space to be open sets in $S$.

In practice one often employs Method 1 to define a manifold $X$. Then one employs Method 2 in order to define the tangent bundle $TX$. Namely, as a set called $TX$ of equivalence classes whose definition utilizes the manifold charts of $X$ and their tangent maps. The latter are used to define the required bijections that endow the set $TX$ of equivalence classes with a topology.

M-polyfolds and maps between them

Definition 3.3.1**.**

Let $X$ be a topological space. An M-polyfold chart $(V,\phi,(O,C,E))$, often abbreviated $(V,\phi,O)$, consists of

• •

an open set $V$ in $X$;

• •

an ${\mathrm{sc}}$-retract $O=r(U)={\rm Fix}\,r$ in a partial quadrant $C\subset E$;

• •

a homeomorphism $\phi\colon V\to O$ (open sets in ${\mathrm{sc}}$-retracts are ${\mathrm{sc}}$-retracts).

Two M-polyfold charts are ${\mathrm{sc}}$-smoothly compatible if the transition map

[TABLE]

and its inverse are both ${\mathrm{sc}}$-smooth retract maps777 Indeed open subsets of ${\mathrm{sc}}$-retracts are ${\mathrm{sc}}$-retracts by Exercise 3.2.13.

(i.e. ${\mathrm{sc}}$-smooth after domain decompression). An M-polyfold atlas for $X$ is a collection ${\mathcal{A}}$ of pairwise ${\mathrm{sc}}$-smoothly compatible M-polyfold charts $\phi\colon V\to O$ whose domains cover $X$. Two atlases are called equivalent if their union is again an M-polyfold atlas.

Definition 3.3.2**.**

An M-polyfold is a paracompact Hausdorff space $X$ endowed with an equivalence class of M-polyfold atlases.

Definition 3.3.3**.**

A map $f\colon X\to X^{\prime}$ between M-polyfolds is called an ${\mathrm{sc}}$-smooth M-polyfold map if every local M-polyfold chart representative

[TABLE]

of $f$ is an ${\mathrm{sc}}$-smooth retract map. An ${\mathrm{sc}}$-smooth diffeomorphism between M-polyfolds

is a bijective ${\mathrm{sc}}$-smooth map between M-polyfolds whose inverse is also ${\mathrm{sc}}$-smooth.

Exercise 3.3.4**.**

a) Sc-manifolds are M-polyfolds.

b) Open subsets of M-polyfolds are M-polyfolds.

c) Check that $X$ and $X^{\prime}$ in Figure 3.3 are M-polyfolds.

d) Use the ${\mathrm{sc}}$-retract $O={\rm im\,}r_{\pi}$ in Figure 3.1 to built further fun M-polyfolds.

e) It is an open problem, Hofer et al. (2017, Quest. 4.1), whether there is an ${\mathrm{sc}}$-smooth retract $(O,E,E)$ so that $O$ is homeomorphic to the letter T in ${\mathbb{R}}^{2}$.

Definition 3.3.5**.**

One defines level $m$ of an M-polyfold $X$ to be the set $X_{m}$ that consists of all points $x\in X$ which are mapped to level $m$ in some, hence any,888 transition maps are ${\mathrm{sc}}$-smooth, thus level preserving

M-polyfold chart.

Thus for an M-polyfold $X$ there is the nested sequence of levels

[TABLE]

Each level $X_{m}$ inherits the structure of an M-polyfold, notation $X^{m}$, see Hofer et al. (2017, p. 21) for charts, and each inclusion $X_{m+1}\hookrightarrow X_{m}$ is continuous (as a map between topological spaces), see Hofer et al. (2017, Le. 2.1).

Construction of the M-polyfold tangent bundle

The base M-polyfold. Let $X$ be an M-polyfold, in particular, a paracompact Hausdorff space, with M-polyfold atlas ${\mathcal{A}}=\{(V_{i},\phi_{i},(O_{i},C_{i},E_{i}))\}_{i\in I}$.

The tangent bundle as a set. By definition $TX$ is the set of equivalence classes of tuples $(x,V,\phi,(O,C,E),\xi)$, abbreviated $(x,\phi,\xi)$, that consist of

• •

a point $x\in X_{1}$ on level 1;

• •

an M-polyfold chart $\phi\colon X\supset V\to O=r(U)\subset C\subset E$ for $X$ about $x$;

• •

a tangent vector $\xi\in T_{\phi(x)}O={\rm Fix}\,[Dr|_{\phi(x)}\in{\mathcal{L}}(E)]$, see (3.2.4).

Two tuples $(x,V,\phi,O,\xi)\sim(x^{\prime},V^{\prime},\phi^{\prime},O^{\prime},\xi^{\prime})$ are said equivalent if

[TABLE]

The natural projection. There is a natural projection

[TABLE]

The pre-image of any point $x\in X^{1}$, denoted by

[TABLE]

and called the tangent space of $X$ at $x$, is a linear space over the reals:

[TABLE]

To represent the two input equivalence classes choose the same M-polyfold chart $\phi$ about $x$ for both of them (choose any two representatives and restrict to the intersection of their domains). This way $\xi$ and $\eta$ are both in the same vector space, here $T_{\phi(x)}O$, and so adding them makes sense.

The induced bijections. For every M-polyfold chart $\phi\colon V\to O$, say where $O={\rm im\,}r={\rm Fix}\,r$ for some ${\mathrm{sc}}$-smooth retraction $r$ on an ${\mathrm{sc}}$-triple $(U,C,E)$, the map named and defined by

[TABLE]

is a bijection. For a given level 1 point $x\in V_{1}:=V\mathop{\cap}X_{1}$ the map

[TABLE]

is a bijection (the identity) on the Banach subspace $T_{\phi(x)}O={\rm Fix}\,Dr|_{\phi(x)}$ of $E$; cf. (3.2.4). So $T_{x}X$ inherits the Banach space structure of $T_{\phi(x)}O$. At smooth points $T_{\phi(x)}O$ is an ${\mathrm{sc}}$-subspace of $E$ by Exercise 3.2.9, so the linear bijection $T_{x}\phi$ endows $T_{x}X$ with the structure of an ${\mathrm{sc}}$-Banach space.

The induced topology. Consider the collection ${\mathcal{B}}$ that consists of all subsets of $TX$ that are pre-images under $T\phi$ of all open subsets in the target space $TO$, for all M-polyfold charts $\phi\colon V\to O$ of $X$, in symbols

[TABLE]

Exercise 3.3.6**.**

Show that ${\mathcal{B}}$ is a basis for a topology; cf. Theorem A.1.14.

By definition the topology on $TX$ is the topology generated by the basis ${\mathcal{B}}$: The open sets in $TX$ are arbitrary unions of members of ${\mathcal{B}}$.

Proposition 3.3.7**.**

The topology on $TX$ is Hausdorff and paracompact.

Proof.

Hofer et al. (2017, § 2.6.3) ∎

Exercise 3.3.8**.**

The map $p\colon TX\to X^{1}$ in (3.3.5) is continuous and open.

The M-polyfold charts. For any M-polyfold chart $\phi\colon V\to O$ of $X$, where $O=r(U)$ say, the bijection $T\phi\colon TV\to TO$ defined by (3.3.6) is an M-polyfold chart for $TX$:

• •

$TV\subset TX$ is open by definition of ${\mathcal{B}}$;

• •

$TO=Tr(TU)$ is an ${\mathrm{sc}}$-retract by Definition 3.2.7;

• •

$T\phi\colon TV\to TO$ is a homeomorphism by definition of ${\mathcal{B}}$.

Furthermore, if $\phi,\phi^{\prime}\in{\mathcal{A}}$ are compatible for $X$, then $T\phi,T\phi^{\prime}$ are compatible for $TX$: We need to show that the map $(T\phi^{\prime})\circ(T\phi)^{-1}\colon TO\to TO^{\prime}$ given by

[TABLE]

is an ${\mathrm{sc}}$-smooth retract map. But $\phi^{\prime}\circ\phi^{-1}$ is an ${\mathrm{sc}}$-smooth retract map by the chain rule, Theorem 3.2.12, and so is the tangent map. This shows that an M-polyfold atlas ${\mathcal{A}}$ for $X$ induces an M-polyfold atlas for $TX$, namely

[TABLE]

Let us then summarize the previous constructions and findings in form of

Theorem 3.3.9**.**

Let $X$ be an M-polyfold. Then $TX$ is an M-polyfold and

[TABLE]

is an ${\mathrm{sc}}$-smooth map between M-polyfolds.

M-polyfold tangent maps

Definition 3.3.10**.**

The tangent map of an ${\mathrm{sc}}$-smooth M-polyfold map $f\colon X\to Y$ is the ${\mathrm{sc}}$-smooth M-polyfold map defined by

[TABLE]

where $\psi$ is any M-polyfold chart about $f(x)$.

Exercise 3.3.11**.**

Show that $Tf\colon TX\to TY$ is ${\mathrm{sc}}$-smooth as a map between M-polyfolds. Show that for $x\in X_{1}$ the map

[TABLE]

is a continuous linear operator and $T_{x}f$ is an ${\mathrm{sc}}$-operator whenever $x\in X_{\infty}$.

3.3.1 Sub-M-polyfolds

An M-polyfold $X$ is locally modeled on the images $O$ of ${\mathrm{sc}}$-smooth retractions $r=r^{2}\colon E\supset U\to U$ in an ${\mathrm{sc}}$-Banach space $E$. Thus it is natural to define a sub-M-polyfold of $X$ as a subset $A\subset X$ that is locally the image of an ${\mathrm{sc}}$-smooth retraction $r=r^{2}\colon V\to V$ acting on an open subset $V$ of $X$.

Definition 3.3.12**.**

A subset $A\subset X$ of an M-polyfold is called a sub-M-polyfold if around any point $a\in A$ there is an open neighborhood $V\subset X$ and an ${\mathrm{sc}}$-smooth retraction $r=r^{2}\colon V\to V$ such that $A\mathop{\cap}V=r(V)={\rm Fix}\,r$. Such $r$ is called a local generator for the sub-M-polyfold $A$.

Proposition 3.3.13**.**

Suppose $A\subset X$ is a sub-M-polyfold.

• (i)

A sub-M-polyfold $A$ inherits an M-polyfold structure from the ambient $X$.

• (ii)

The inclusion $\iota\colon A\hookrightarrow X$ is an ${\mathrm{sc}}$-smooth map between M-polyfolds and a homeomorphism onto its image.

• (iii)

A local generator $r$ for $A$, viewed as a map $r\colon V\to A$, is ${\mathrm{sc}}$-smooth and $T_{a}r(T_{a}X)=T_{a}A$ at any point $a\in A\mathop{\cap}V$.

• (iv)

At points $a\in A_{\infty}$ the tangent space $T_{a}A$ is ${\mathrm{sc}}$-complemented in $T_{a}X$.

Proof.

Hofer et al. (2017, Prop. 2.6). ∎

3.3.2 Boundary and corners – tameness

Unfortunately, on M-polyfolds the degeneracy index of a point, defined through an M-polyfold chart, might depend on the choice of chart, as this example shows: For a real parameter $a\geq 0$ define the orthogonal projection

[TABLE]

The image of the retraction $r_{a}$ is the half line $L_{a}=\{(x,ax)\mid x\geq 0\}$ in the quadrant $C$. On the M-polyfold $X=[0,\infty)$ we choose the global chart $\phi_{a}\colon X\to O=r_{a}(C)=L_{a}\subset C$ shown in Figure 3.4. In this chart the degeneracy index, see Section 2.7, of each point $x\in X$ (also depending on whether $a=0$ or $a>0$) is given by

[TABLE]

On the other hand, representing $X$ in the obvious global M-polyfold chart

[TABLE]

the degeneracy indices of points are the rather different, but

expected, values

[TABLE]

Of course, the discrepancy between $d_{C}$ and $d_{C^{\prime}}$ could be caused by incompatibility of charts. However, this is not the case, both transition retract maps are ${\mathrm{sc}}$-smoothly compatible. Indeed the decompression of $\psi:=\phi^{\prime}\circ{\phi_{a}}^{-1}$

[TABLE]

is even $C^{\infty}$ smooth and so is $\psi^{-1}\circ r^{\prime}\colon[0,\infty)\to O=L_{a}\subset C$, $x\mapsto(x,ax)$.

Definition 3.3.14** (Degeneracy index on M-polyfolds $X$).**

Given a point $x\in X$, just take the minimum

[TABLE]

over all M-polyfold charts $\phi\colon V\to O\subset C$ about the point $x$.

Degeneracy index stratification of quadrant – Tameness

To see what went wrong for the chart $\phi_{a}$ in the example above note that the quadrant C={\color[rgb]{.75,.5,.25}\definecolor[named]{pgfstrokecolor}{rgb}{.75,.5,.25}C_{0}}\mathop{\cup}{\color[rgb]{0,1,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,1,1}\pgfsys@color@cmyk@stroke{1}{0}{0}{0}\pgfsys@color@cmyk@fill{1}{0}{0}{0}C_{1}}\mathop{\cup}{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}C_{2}} decomposes into disjoint subsets $C_{i}:={d_{C}}^{-1}(i)$, the strata of the degeneracy index stratification. Now one identifies two problems:

• a)

The ${\mathrm{sc}}$-retraction $r_{a}$ does not preserve the degeneracy index strata.

• b)

The ${\mathrm{sc}}$-retract $r_{a}(C)=L_{a}$ is in a certain sense not transverse to the degeneracy index stratification of the quadrant $C$.

One avoids the problem by giving a name to retractions that do not have the defects a) and b) and then considers only such in theorems.

Definition 3.3.15**.**

An ${\mathrm{sc}}$-smooth retraction $r\colon U\to U$ on an ${\mathrm{sc}}$-triple $(U,C,E)$ is called tame if

• a)

the map $r$ preserves the $d_{C}$-stratification: $d_{C}(r(x))=d_{C}(x)$ $\forall x\in U$;

• b)

the image of $r$ is transverse to the $d_{C}$-stratification: For every smooth point $x$ in the image $r(U_{\infty})=O_{\infty}$ there must be an ${\mathrm{sc}}$-complement $A$ of the ${\mathrm{sc}}$-subspace $T_{x}O=Dr(x)E$ of $E$, cf. (3.2.4) and Exercise 3.2.9, with $A\subset E_{x}:=T_{x}C(x)$ where $C(x):={d_{C}}^{-1}(d_{C}(x))$ is the stratum of $x$.

If in b) above such $A$ exists, then one can choose $A=({{\mathchoice{\rm 1\mskip-4.0mul}{\rm 1\mskip-4.0mul}{\rm 1\mskip-4.5mul}{\rm 1\mskip-5.0mul}}}-Dr(x))E$ by Hofer et al. (2017, Prop. 2.9). So for tame ${\mathrm{sc}}$-smooth retractions $r\colon U\to U$ with image $O$ one has the ${\mathrm{sc}}$-splittings

[TABLE]

one for each smooth point of $O=r(U)$.

Remark 3.3.16** (Fixed origin).**

Consider the quadrant $C:=[0,\infty)^{n}$ in ${\mathbb{R}}^{n}$ and suppose $r\colon C\to C$ is a tame smooth retraction. Then the origin $0=r(0)$ is fixed by $r$. Indeed $x=0$ is the only point in $C$ with $d_{C}(x)=n$. Moreover, the image $r(C)$ is an open neighborhood of [math] in $C$; cf. Cieliebak (2018, Problem 6.5).

Definition 3.3.17**.**

An ${\mathrm{sc}}$-retract $(O,C,E)$ is called tame if $O=r(U)$ is the image of a tame ${\mathrm{sc}}$-smooth retraction $r$. An M-polyfold is called tame if $X$ admits an equivalent M-polyfold atlas modeled on tame ${\mathrm{sc}}$-smooth retracts.

For tame M-polyfolds $X$ the degeneracy index $d_{X}(x):=d_{C}(\phi(x))$ of a point $x\in X$ defined via an M-polyfold chart $\phi\colon V\to O\subset C$ about $x$ does not depend on the choice of the chart; see Hofer et al. (2017, Eq. (2.12)).

3.4 Strong bundles over M-polyfolds

We recall the notion of a vector bundle over a manifold and sketch how to generalize the base to M-polyfolds (bringing in scale and retracts) and accommodate Fredholm sections via ${\mathrm{sc}}^{+}$-sections (bringing in double scales).

Motivation and comparison of old and new concepts

The following overview is not meant to be, and is not, rigorous.

Classical vector bundles over manifolds – trivial bundles $U\times F\to U$.

A classical vector bundle over a manifold is locally modeled by trivial bundles $U\times F\to U$. Here $U$ is an open subset of a linear space $E$, the model space of the manifold, and $F$ is a linear space, the model of the fibers of the vector bundle. Any two local models must be related by a diffeomorphism

[TABLE]

called a vector bundle transition map, whose second component $(u,\xi)\mapsto{\mathcal{T}}(u,\xi)$ restricts at every point $u$ to a vector space isomorphism ${\mathcal{T}}_{u}:={\mathcal{T}}(u,\cdot)\colon F\to F$. So the building blocks for classical vector bundles are trivial bundles

[TABLE]

Sc-bundles over M-polyfolds – trivial-bundle sc-retracts $R(U\oplus F)\to r(U)$.

To define ${\mathrm{sc}}$-bundles over M-polyfolds one needs to generalize trivial bundles taking into account that now one deals with ${\mathrm{sc}}$-triples $(U,C,E)$ and ${\mathrm{sc}}$-Banach spaces $F$. A useful notation for trivial bundles is $U\oplus F\to U$ which indicates that the set $U\times F$ sits inside the ${\mathrm{sc}}$-direct sum $E\oplus F$ and thereby inherits the scale structure $(U\oplus F)_{m}=U_{m}\oplus F_{m}$.

Because local models for the base M-polyfold are ${\mathrm{sc}}$-retracts $O$, one should replace $U$ by its image $O=r(U)$ under an ${\mathrm{sc}}$-retraction. It is suggesting to replace the whole space $U\oplus F$ by its image under an ${\mathrm{sc}}$-retraction

[TABLE]

for which $F\to F$, $\xi\mapsto\rho_{u}\xi:=\rho(u,\xi)$ is linear, at any $u\in U$. Such retraction

• •

produces a local M-polyfold model $O=r(U)$ in the component $U$ and

• •

also respects the linear structure of the second component $F$.

A crucial observation is that along $O=r(U)={\rm Fix}\,r\subset U$ idempotency of $R$ implies that the linear map $\rho_{x}=(\rho_{x})^{2}\colon F\to F$ is also idempotent, hence a projection. Choose the identity retraction $R(u,\xi):=(u,\xi)$ and forget scale structures to recover trivial bundles $U\oplus F\to U$, hence classical vector bundles. The building blocks

[TABLE]

for ${\mathrm{sc}}$-bundles over M-polyfolds are called trivial-bundle sc-retracts. Projection onto the second component provides an ${\mathrm{sc}}$-smooth surjection

[TABLE]

onto an ${\mathrm{sc}}$-retract, the local model of an M-polyfold. Each pre-image $p^{-1}(x)$ is a closed linear subspace $\rho_{x}(F)={\rm Fix}\,\rho_{x}$ of $F$.

To summarize, the local models for ${\mathrm{sc}}$-bundles over M-polyfolds, called trivial-bundle sc-retracts, are ${\mathrm{sc}}$-retracts $K$ in $U\oplus F$ that are families

[TABLE]

of projection images ${\rm Fix}\,\rho_{x}$ in $F$ parametrized by local M-polyfold models $O$. To construct ${\mathrm{sc}}$-bundles over M-polyfolds one defines ${\mathrm{sc}}$-bundle charts as in Definition 3.4.12 disregarding double scales – just replace the double scale symbol $\triangleright$ by the ${\mathrm{sc}}$-direct sum $\oplus$. Compatibility of charts and ${\mathrm{sc}}$-bundle atlases are defined as usual.

Definition 3.4.1** (Sc-bundles over M-polyfolds).**

An sc-bundle over an M-polyfold $X$ is an ${\mathrm{sc}}$-smooth surjection $\pi\colon Y\to X$ between M-polyfolds endowed with an equivalence class of ${\mathrm{sc}}$-bundle atlases.

Accommodating Fredholm sections: double scale gives two scales $\mbox{}^{[0]}$ and $\mbox{}^{[1]}$.

The local model building blocks for strong bundles over M-polyfolds are trivial-strong-bundle retracts $K=R(U\triangleright F)$. These come with a double scale structure $K_{m,k}$ by definition and a natural projection $p:K\to O$ where the ${\mathrm{sc}}$-retract $O=r(U)$ is called the associated base retract and the ${\mathrm{sc}}$-retraction $r:U\to U$ is the first component of $R$. Reducing the double scale in two ways to a scale one obtains two ${\mathrm{sc}}$-bundles

[TABLE]

over the M-polyfold $O$. The sections of $p^{[1]}$ generalize ${\mathrm{sc}}^{+}$ operators.

3.4.1 Trivial-strong-bundle retracts - the local models

Throughout $F$ is an ${\mathrm{sc}}$-Banach space and $(U,C,E)$ an ${\mathrm{sc}}$-triple, that is $U$ is a relatively open subset of the partial quadrant $C$ in the ${\mathrm{sc}}$-Banach space $E$.

Remark 3.4.2** (Motivation for non-symmetric product and shift by $1$).**

At first sight the introduction of a double scale/filtration in Definition 3.4.3, even an asymmetric one, and its immediate reduction to a single scale in Definition 3.4.4, in two versions though, might be confusing and even appear superfluous given that the two versions inherit their scale structure as subsets of the simple and well known ${\mathrm{sc}}$-direct sums $U\oplus F$ and $U\oplus F^{1}$.

a) To perceive the need to shift the vector space part of $U\oplus F$ by one, recall stability of the ${\mathrm{sc}}$-Fredholm property under addition of ${\mathrm{sc}}^{+}$-operators; see Proposition 2.3.29.

b) In practice, when implementing a differential operator $f$ of order $\ell$ the level indices $m$ of $U$ indicate differentiability, simply speaking. So one needs to forget the first $\ell$ levels and choose $U_{\ell}$, or any sublevel of it, as domain for $f$. More precisely, one chooses the shifted scale $U^{\ell}$. Then $f\colon(U^{\ell})_{m}\to F_{m}$ and one can subsequently exploit composition with the compact embeddings $F_{m}\hookrightarrow F_{m-1}\dots\hookrightarrow F_{0}$.

Definition 3.4.3** (Non-symmetric product – double scale).**

The non-symmetric product $U\triangleright F$ is the subset $U\times F$ of the Banach space $E\oplus F$ endowed with the double scale, also called double filtration, defined by999 We use the symbol $U\triangleright F$, as opposed to $U\triangleleft F$, since the levels of $U$ are unlimited, any $m\in{\mathbb{N}}_{0}$ is allowed, whereas the ones of $F$ depend on $m$ and are restricted to $0,\dots,m+1$.

[TABLE]

Non-symmetric products $U\triangleright F$ serve as total spaces of strong trivial bundles. Projection onto the first component

[TABLE]

is called the trivial-strong-bundle projection. However, for ${\mathrm{sc}}$-calculus one needs one scale structure, not a double scale. To achieve this substitute $k$ by a useful function of $m$, say $k=m$ or $k=m+1$.

Definition 3.4.4** (Trivial strong bundles: Two relevant scale structures).**

Motivated by Hofer et al. (2017, §2.5) we denote the ${\mathrm{sc}}$-manifolds $U\oplus F$ and $U\oplus F^{1}$ by the symbols

[TABLE]

By definition of shifted scales the levels are101010 cf. Definition 2.1.6

[TABLE]

The projections onto the first component

[TABLE]

are ${\mathrm{sc}}$-smooth maps between ${\mathrm{sc}}$-manifolds called trivial strong bundles. We often write $p$ for simplicity and because the values do not depend on the choice of shift $i$ for the second component $F$. If the domain matters we shall write $p^{[i]}$.

Definition 3.4.5** (Morphisms of trivial strong bundles).**

a) A trivial-strong-bundle map $\Psi\colon U\triangleright F\to\tilde{U}\triangleright\tilde{F}$ is a map that preserves the double scale and is of the form

[TABLE]

where $\Gamma_{u}\xi:=\Gamma(u,\xi)$ is linear in $\xi$. Moreover, it is required that both induced maps between ${\mathrm{sc}}$-manifolds

[TABLE]

are ${\mathrm{sc}}$-smooth. b) A trivial-strong-bundle isomorphism is an invertible strong trivial bundle map whose inverse is also a strong trivial bundle map.

It is the previous definition where double scale preservation is required.

Exercise 3.4.6**.**

Check that the second component $\Gamma$ of a strong trivial bundle map $\Psi$ gives rise to an ${\mathrm{sc}}$-operator $\Gamma_{u}\in\mathcal{L}_{\mathrm{sc}}(F,\tilde{F})$ along the smooth points $u\in U_{\infty}$.

Definition 3.4.7** (Trivial-strong-bundle retraction).**

A trivial-strong-bundle retraction is an idempotent trivial-strong-bundle map

[TABLE]

The first component $r$ of $R$ is necessarily an ${\mathrm{sc}}$-smooth retraction on $U$, called the associated base retraction, whose image ${\mathrm{sc}}$-retract $O=r(U)$ is called the associated base retract. One calls $R$ tame in case the associated base retraction is tame.

Exercise 3.4.8**.**

Let $R(u,\xi)=\left(r(u),\rho_{u}\xi\right)$ be a strong trivial bundle retraction. Check that $r$ is an ${\mathrm{sc}}$-smooth retraction on $U$ and that $\rho_{x}\in{\mathcal{L}}(F)$ is a projection for $x\in{\rm Fix}\,r=r(U)=:O$, even an ${\mathrm{sc}}$-projection at smooth points, i.e. $x\in O_{\infty}$.

Definition 3.4.9** (Trivial-strong-bundle retracts $K$ – the local models).**

A **trivial-strong-bundle retract111111 ’strong’ indicates ’doubly scaled’ and the retraction acts on a ’trivial bundle’ ** $(K,C\triangleright F,E\triangleright F)$, or simply $K$, is given by the image

[TABLE]

of a trivial-strong-bundle retraction $R=R\circ R$ on $U\triangleright F$ where $O=r(U)$ is the associated base retract. One likewise calls the natural surjection

[TABLE]

trivial-strong-bundle retract. For simplicity we identify point pre-images

[TABLE]

with the Banach subspace $K_{x}:=\rho_{x}(F)$ of $F$, an ${\mathrm{sc}}$-subspace for smooth points, called the fiber of $K$ over $x$. Call $K=R(U\triangleright F)$ tame if $R$ is tame.

Being a subset of the doubly scaled space $U\triangleright F$ a trivial-strong-bundle retract $K$ inherits the double scale

[TABLE]

for $m\in{\mathbb{N}}_{0}$ and $k\in\{0,\dots m+1\}$. Note that the spaces121212 the symbol $R(U\triangleright F)^{[i]}$ abbreviates $R((U\triangleright F)^{[i]})$

[TABLE]

with levels $K^{[i]}_{m}=K_{m,m+i}$ are ${\mathrm{sc}}$-retracts, so M-polyfolds. The surjections

[TABLE]

are both ${\mathrm{sc}}$-smooth maps between ${\mathrm{sc}}$-retracts. Indeed by Definition 3.2.3 this requires that some, hence any, decompression, say

[TABLE]

be ${\mathrm{sc}}$-smooth. But the associated base retraction $r$ is ${\mathrm{sc}}$-smooth by assumption.

Definition 3.4.10** (Strong retract maps).**

A map ${\mathcal{F}}$ of the form

[TABLE]

between trivial-strong-bundle retracts is called a strong retract map if ${\mathcal{F}}$ is linear in the fibers, that is $\phi_{x}\colon K_{x}\to\tilde{K}_{f(x)}$ is linear, if ${\mathcal{F}}$ preserves the double filtrations, and if both induced maps between ${\mathrm{sc}}$-retracts

[TABLE]

are ${\mathrm{sc}}$-smooth (meaning ${\mathrm{sc}}$-smoothness after decompression).

Definition 3.4.11** **(${\mathrm{sc}}$- and ${\mathrm{sc}}^{+}$-sections of trivial-strong-bundle

retracts).

A section of a trivial-strong-bundle retract $p\colon K\to O$ is a map $s\colon O\to K$ that satisfies $p\circ s={\rm id}_{O}$. If $s$ is ${\mathrm{sc}}$-smooth as an ${\mathrm{sc}}$-retract map

[TABLE]

it is called in case $i=0$ an ${\mathrm{sc}}$-section and in case $i=1$ an ${\mathrm{sc}}^{+}$-section. The map {\text{\boldmaths}}^{[i]}\colon O\to F^{i} is called the principal part of the section.

Note that a section is ${\mathrm{sc}}$-smooth iff its principal part is. For simplicity we sometimes omit the superscript**[i]** if the level shift is clear from the context.

3.4.2 Strong bundles

Throughout $F$ is an ${\mathrm{sc}}$-Banach space and $(U,C,E)$ an ${\mathrm{sc}}$-triple, that is $U$ is a relatively open subset of the partial quadrant $C$ in the ${\mathrm{sc}}$-Banach space $E$.

Definition 3.4.12** (Strong bundle charts).**

Let $P\colon Y\to X$ be a continuous surjection from a paracompact Hausdorff space $Y$ onto an M-polyfold $X$ such that every pre-image $Y_{x}:=P^{-1}(x)$ has the structure of a Banachable space.131313 A Banachable space is an equivalence class that consists of all Banach spaces with pairwise equivalent norms.

A strong bundle chart for $P\colon Y\to X$ is a tuple

[TABLE]

that consists of

• •

a trivial-strong-bundle retract $p\colon K=R(U\triangleright F)\to O$ where $O=r(U)$ is the associated base retract;

• •

a homeomorphism $\varphi\colon V\to O$ between an open subset of the base M-polyfold $X$ of $Y$ and the base retract $O$ of $K$;

• •

a homeomorphism $\Phi\colon P^{-1}(V)\to K$ that covers $\varphi$, that is the diagram

[TABLE]

commutes. As a consequence, for every point $v\in V$ the restriction of $\Phi$ to $P^{-1}(v)$ takes values in $p^{-1}(\varphi(v))$. One also requires that $\Phi$ viewed as a map

[TABLE]

is a continuous linear bijection141414 making sense although the domain is just Banachable

between fibers.151515 strictly speaking, the target is $\{\varphi(v)\}\times\rho_{\varphi(v)}(F)$

Definition 3.4.13** (Strong bundle atlases).**

Two strong bundle charts are called compatible if, firstly, the transition map

[TABLE]

is a strong retract map, thus preserves the double scales, and, secondly, the two induced maps $\Psi^{[0]}$ and $\Psi^{[1]}$ between open subsets of ${\mathrm{sc}}$-retracts (cf. (3.4.8)), hence M-polyfolds, are ${\mathrm{sc}}$-smooth diffeomorphisms. A strong bundle atlas ${\mathcal{A}}^{Y}_{X}$ consists of pairwise compatible strong bundle charts covering $Y$. Two such atlases are called equivalent if their union is again a strong bundle atlas.

Definition 3.4.14** (Strong bundles over M-polyfolds).**

A strong bundle over an M-polyfold $X$ is a continuous surjection $P\colon Y\to X$ from a paracompact Hausdorff space equipped with an equivalence class of strong bundle atlases.

Exercise 3.4.15** (A strong bundle provides two M-polyfolds).**

Check that a strong bundle atlas ${\mathcal{A}}^{Y}_{X}$ for $Y\to X$ naturally provides two M-polyfold atlases ${\mathcal{A}}_{Y^{[0]}}$ and ${\mathcal{A}}_{Y^{[1]}}$ for M-polyfolds $Y^{[0]}$ and $Y^{[1]}$, respectively.

Exercise 3.4.16** (A strong bundle provides two ${\mathrm{sc}}$-bundles).**

A strong bundle atlas ${\mathcal{A}}^{Y}_{X}$ for $P\colon Y\to X$ naturally provides two induced ${\mathrm{sc}}$-bundle atlases ${\mathcal{A}}^{Y^{[0]}}_{X}$ and ${\mathcal{A}}^{Y^{[1]}}_{X}$ for ${\mathrm{sc}}$-bundles $P^{[0]}\colon Y^{[0]}\to X$ and $P^{[1]}\colon Y^{[1]}\to X$, respectively.

Induced double scale and section types

A strong bundle $P\colon Y\to X$ carries an asymmetric double scale structure $Y_{m,k}$, where $m\in{\mathbb{N}}_{0}$ and $k=0,\dots,m+1$, transmitted from the local models $K$ by the strong bundle charts. Here it enters that the transition maps $\Psi$ are strong retract maps, thus preserve the double scale of the local models $K$.

Definition 3.4.17** (${\mathrm{sc}}$- and ${\mathrm{sc}}^{+}$-sections of strong bundles).**

A section of a strong bundle $P\colon Y\to X$ is a map $s\colon X\to Y$ that satisfies $P\circ s={\rm id}_{X}$. If a section $s$ of $P\colon Y\to X$ is ${\mathrm{sc}}$-smooth as a map between M-polyfolds

[TABLE]

then $s$ is called an ${\mathrm{sc}}$-section (case $i=0$) or an ${\mathrm{sc}}^{+}$-section (case $i=1$).

Pull-back bundle

Suppose $f\colon Z\to X$ is an ${\mathrm{sc}}$-smooth map between M-polyfolds and $P\colon Y\to X$ is a strong bundle over $X$. The pull-back bundle $P_{f}\colon f^{*}Y\to X$ consists of the subset of $Z\times Y$ defined by

[TABLE]

and projection $P_{f}(z,y)=P_{1}(z,y)=z$ onto the first component. Together with projection onto the second component, denoted by $P_{2}$, the diagram

[TABLE]

commutes.

Exercise 3.4.18** (Induced strong bundle structure).**

Given an ${\mathrm{sc}}$-smooth map $f\colon Z\to X$ between M-polyfolds, show that a strong bundle structure on $P\colon Y\to X$ induces naturally a strong bundle structure on the pull-back bundle $f^{*}P\colon f^{*}Y\to Z$.

Appendix A Background from Topology and

Functional Analysis

A.1 Analysis on topological vector spaces

All vector spaces will be over the real numbers ${\mathbb{R}}$. Let us first repeat

Some basics about sets

The elements of a set $S$ are often called points. If a set $S$ contains only finitely many elements it is called finite. The number of elements of a finite set is denoted by $\mathopen{|}S\mathclose{|}$. The set with no element is called the empty set, denoted by $\emptyset$ or, in order to indicate the ambient universe $S$, by $\emptyset_{S}$. We avoid terminology like a set of sets, instead we shall speak of a family of sets or of a collection of sets. Let $2^{S}$ be the collection of all subsets of $S$. The empty set $\emptyset$ is a subset of any set $S$, in symbols $\emptyset\subset S$ or $\emptyset\in 2^{S}$. Our use of $\subset$ allows for equality, otherwise we write $\subsetneq$. For more basics on set theory and logic see e.g. Munkres (2000, Ch. I). See also Ch. I, in particular I.9 on axiomatics, in Dugundji (1966).

Definition A.1.1**.**

Let $S$ be a set. Given a family ${\mathcal{A}}\subset 2^{S}$ of subsets $A$ of $S$, union and intersection of the members of ${\mathcal{A}}$ are the subsets of $S$ defined by

[TABLE]

and

[TABLE]

Exercise A.1.2**.**

For ${\mathcal{A}}=\emptyset,\{\emptyset_{S}\}\subset 2^{S}$ show $\bigcup\{\emptyset_{S}\}=\emptyset_{S}=\bigcap\{\emptyset_{S}\}$, but

[TABLE]

[Hint: Final assertion – empty truth.]

Maps and exponential law

Suppose $A,B,C$ are sets. A map $f$ from $A$ to $B$, in symbols $f\colon A\to B$, is determined by a subset $G(f)\subset A\times B$ such that for each domain element $a\in A$ the set $\{b\in B\mid(a,b)\in G(f)\}$ has precisely 1 element. The unique $b\in B$ such that $(a,b)\in G(f)$ is denoted by $f(a)$ and called the image of $a$ under $f$. The set $A$ is the domain of $f$ and $B$ the codomain or the target. The subset $G(f)\subset A\times B$ is called the graph of $f$. A function is a map $f\colon A\to{\mathbb{R}}$ that takes values in the set of real numbers ${\mathbb{R}}$.

Let ${\rm Map}(A,B)$, or $B^{A}$, denote the set of all maps from $A$ to $B$. Motivated by the exponential notation the bijection

[TABLE]

is called the exponential map or the exponential law.

A.1.1 Topological spaces

For an elementary overview see e.g. Munkres (2000, Ch. II), for an exhaustive treatment Dugundji (1966), we also found extremely useful Müger (2016).

Definition A.1.3** (Topology).**

A topology on a set $M$ is a family ${\mathcal{T}}$ of subsets $U\subset M$, called the open!sets, such that the following axioms hold.

• (i)

Both the empty set $\emptyset$ and $M$ itself are open.

• (ii)

Arbitrary unions of open sets are open.

• (iii)

Finite intersections of open sets are open.

Such pair $(M,{\mathcal{T}})$ is called a topological space. The complements $U^{\rm C}:=X\setminus U$ of the open sets form the family of closed sets.

Exercise A.1.4**.**

The intersection of a collection of topologies is a topology.

A topology ${\mathcal{T}}$ on a set $M$ induces on any subset $A\subset M$ a topology ${\mathcal{T}}^{\cap A}$ which consists of the intersections of $A$ with all the members of the family ${\mathcal{T}}$ of subsets of $M$. The topology ${\mathcal{T}}^{\cap A}$ is called the subset topology or the induced topology on a subset $A$. A subspace is a subset of a topological space endowed with the subset topology.

Properties of topological spaces that are inherited by subspaces are called hereditary properties.

A topological space is called compact if every open cover admits a finite sub-cover. A subset $K$ of a topological space $(M,{\mathcal{T}})$ is called compact if the topology on $K$ induced by ${\mathcal{T}}$ is compact. A subset is called pre-compact if its closure is compact.

One often writes, instead of the pair $(M,{\mathcal{T}})$, simply $M$ and calls it a topological space. An open neighborhood of a subset $P\subset M$ is an open set $U$ that contains $P$, in symbols $P\subset U\in{\mathcal{T}}$. Any subset $A\subset M$ that contains an open neighborhood of $P$ is called a neighborhood of $P$. If $P=\{x\}$ is a point set we speak of a neighborhood of a point $x\in M$. It is convenient to write $U_{x}$ to indicate that a set $U$ contains the point $x$. With this convention “for any open neighborhood $U$ of $x$” becomes “for any open $U_{x}$”.

Basis of a given topology

Definition A.1.5** (Basis).**

Given a sub-collection ${\mathcal{C}}\subset{\mathcal{T}}$ of a topology, let

[TABLE]

be the collection of all unions of elements $C\subset M$ of ${\mathcal{C}}$. If a sub-collection ${\mathcal{B}}\subset{\mathcal{T}}$ satisfies ${\mathcal{T}}_{\mathcal{B}}={\mathcal{T}}$, i.e. if all open sets are unions of elements of ${\mathcal{B}}$, one calls ${\mathcal{B}}$ a basis of the topology ${\mathcal{T}}$ and says that the topology is generated by ${\mathcal{B}}$.

The elements of a basis ${\mathcal{B}}$ are called basic open sets. Any open set is a union of basic ones. Uniqueness of a basis fails as badly, as existence is trivial: Given ${\mathcal{T}}$, pick ${\mathcal{B}}:={\mathcal{T}}$. Often in practice, the smaller a basis, the better. So a criterion for being a basis is desirable.

Lemma A.1.6**.**

For a subset ${\mathcal{C}}\subset{\mathcal{T}}$ of a topology the following are equivalent:

• (i)

The collection ${\mathcal{C}}$ is a basis of ${\mathcal{T}}$, in symbols ${\mathcal{T}}_{\mathcal{C}}={\mathcal{T}}$.

• (ii)

The collection ${\mathcal{C}}$ is dominated by ${\mathcal{T}}$ in the following sense: Each point $x\in U\in{\mathcal{T}}$ of an open set also lies in a collection member $C\in{\mathcal{C}}$ that is contained in $U$, in symbols $x\in C\subset U$.

Proof.

See e.g. Dugundji (1966, III.2). ∎

Definition A.1.7** (Sub-basis).**

For a sub-collection ${\mathcal{S}}\subset{\mathcal{T}}$ of a topology, let

[TABLE]

be the collection of all finite intersections of elements of ${\mathcal{S}}$. If ${\mathcal{B}}_{\mathcal{S}}$ is a basis of ${\mathcal{T}}$, i.e. if all open sets are arbitrary unions of finite intersections of elements of ${\mathcal{S}}$, one calls ${\mathcal{S}}$ a sub-basis of the topology ${\mathcal{T}}$ and ${\mathcal{B}}_{\mathcal{S}}$ the basis generated by ${\mathcal{S}}$.

Definition A.1.8**.**

A topological space is called second countable if it admits a countable basis. This property is hereditary.

Definition A.1.9**.**

A subset of a topological space is called dense if it meets (has non-empty intersection with) every non-empty open set or, equivalently, if its closure is equal to the whole space. A topological space is called separable if it admits a dense sequence (countable subset). Separability is not hereditary.

Exercise A.1.10**.**

Show that second countability is hereditary, whereas separability is not, that second countable implies separable and that in metric spaces (endowed with the metric topology ${\mathcal{T}}_{d}$) the converse is true, too.

Definition A.1.11** (Local basis).**

Let $(M,{\mathcal{T}})$ be a topological space and $x\in M$. A collection ${\mathcal{B}}(x)$ of open neighborhoods $B_{x}$ of $x$ is called a local basis of the topology at $x$ if every open neighborhood $U_{x}$ of $x$ contains a member of ${\mathcal{B}}(x)$, in symbols $U_{x}\supset B_{x}\in{\mathcal{B}}(x)$.

Exercise A.1.12**.**

Let $(M,{\mathcal{T}})$ be a topological space.

• (i)

Given a basis ${\mathcal{B}}$ of ${\mathcal{T}}$, for every $x\in M$ the family ${\mathcal{U}}(x)=\{U\in{\mathcal{T}}\mid x\in U\}$ of all open neighborhoods of $x$ is a local basis of ${\mathcal{T}}$ at $x$.

• (ii)

Vice versa, given for every point $x$ of $M$ a local basis ${\mathcal{B}}(x)$ for ${\mathcal{T}}$ at $x$, show that their union ${\mathcal{B}}:=\bigcup_{x\in M}{\mathcal{B}}(x)=\{B\mid B\in{\mathcal{B}}(x),\;x\in M\}\subset{\mathcal{T}}\subset 2^{M}$ forms a basis of ${\mathcal{T}}$.

From sets to topologies

Starting with just a set $S$, let ${\mathcal{C}}$ be any collection of subsets of $S$. The definitions above still provide collections ${\mathcal{T}}_{\mathcal{C}},{\mathcal{B}}_{\mathcal{C}}\subset 2^{S}$. Note that always $\emptyset\in{\mathcal{T}}_{\mathcal{C}}$ and $M\in{\mathcal{B}}_{\mathcal{C}}$ (pick $\sigma:=\emptyset\subset{\mathcal{C}}$). It is a natural question to ask under what conditions on ${\mathcal{C}}$ the collections ${\mathcal{T}}_{\mathcal{C}}$ or ${\mathcal{T}}_{{\mathcal{B}}_{\mathcal{C}}}$ are topologies on $S$.

Exercise A.1.13** (Any collection is a sub-basis of some topology).**

Let $S$ be a set and ${\mathcal{S}}\subset 2^{S}$ any collection of subsets. Then ${\mathcal{T}}_{{\mathcal{B}}_{\mathcal{S}}}$ is a topology on $S$, the smallest topology that contains ${\mathcal{S}}$, and ${\mathcal{B}}_{\mathcal{S}}$ is a basis.

[Hints: Let ${\mathcal{T}}^{\mathcal{S}}$ be the intersection of all topologies ${\mathcal{T}}$ containing ${\mathcal{S}}$ (for example ${\mathcal{T}}=2^{S}$). Show ${\mathcal{T}}^{\mathcal{S}}={\mathcal{T}}_{{\mathcal{B}}_{\mathcal{S}}}$. See e.g. Dugundji (1966, III.3).]

While any collection of subsets of $S$ is a sub-basis of some topology on $S$, a sufficient condition for being a basis of some topology is the following.

Theorem A.1.14** (Being a basis of some topology).**

Given a set $S$, let ${\mathcal{B}}\subset 2^{S}$ be a collection of subsets $V$ of $S$ such that

• (i)

${\mathcal{B}}$* is a cover of $S$ (the union of all members of ${\mathcal{B}}$ is $S$) and*

• (ii)

every point $p\in V_{1}\mathop{\cap}V_{2}$ in an intersection of two ${\mathcal{B}}$ members simultaneously belongs to a ${\mathcal{B}}$ member $V_{3}\subset V_{1}\mathop{\cap}V_{2}$ contained in the intersection.

Under these conditions ${\mathcal{T}}_{\mathcal{B}}$ is a topology on $S$, the smallest topology containing ${\mathcal{B}}$, and ${\mathcal{B}}$ is a basis.

Proof.

See e.g. Dugundji (1966, III Thm. 3.2). ∎

Exercise A.1.15**.**

Let $S$ be a set. The three collections ${\mathcal{S}}=\emptyset$, $\{\emptyset_{S}\}$, $\{S\}\subset 2^{S}$ lead, respectively, to the three bases ${\mathcal{B}}_{\mathcal{S}}=\{S\}$, $\{\emptyset_{S},S\}$, $\{S\}\subset 2^{S}$ each of which generates the trivial, also called indiscrete, topology ${\mathcal{T}}_{{\mathcal{B}}_{\mathcal{S}}}=\{\emptyset_{S},S\}$.

Here is another method to topologize a set $S$ starting with a family of candidates for local bases, one candidate at each point $x$ of the set. It is a two step process. Firstly, at every point $x\in S$ we wish to specify a collection ${\mathcal{B}}(x)$ of subsets $V_{x}\subset S$ in such a way that, secondly, we can construct a unique topology ${\mathcal{T}}({\mathcal{B}})$ on $S$ for which the collection ${\mathcal{B}}(x)$ will be a local basis at $x$ and this is true for every $x\in S$. Since prior to step two there is no topology, hence no notion of local basis, we call ${\mathcal{B}}(x)$ a local pre-basis at $x$.

Definition A.1.16**.**

Let $S$ be a set and $x\in S$. Suppose ${\mathcal{B}}\subset 2^{S}$ is the union of a collection of non-empty families $\emptyset\not={\mathcal{B}}(x)$ of subsets of $S$, one family associated to each point $x$ of $S$, such that the following is satisfied at all points $x,y\in S$.

• (1)

Every member of ${\mathcal{B}}(x)$ contains $x$. ($\emptyset\notin{\mathcal{B}}(x)$)

• (2)

The intersection $V_{1}\mathop{\cap}V_{2}\supset V_{3}$ of any two members of ${\mathcal{B}}(x)$ contains a ${\mathcal{B}}(x)$-member $V_{3}$.111 Note that (2) makes sense since any intersection $V_{1}\mathop{\cap}V_{2}\ni x$ is non-empty.

(${\mathcal{B}}(x)$ downward directed)

• (3)

For any ${\mathcal{B}}(x)$-member $V_{x}$ each of its points $y$ belongs to a ${\mathcal{B}}(y)$-member $Y_{y}$ contained in $V_{x}$, i.e. any $V\in{\mathcal{B}}$ is a union of ${\mathcal{B}}$-members. (${\mathcal{B}}\subset{\mathcal{T}}({\mathcal{B}})$)

The family ${\mathcal{B}}(x)$ is called a local pre-basis at $x$, the union ${\mathcal{B}}:=\bigcup_{x\in M}{\mathcal{B}}(x)$ of all of them is called a pre-basis on the set $S$. The family of subsets

[TABLE]

is called the topology generated by the pre-basis ${\mathcal{B}}$ on the set $S$.

Exercise A.1.17**.**

a) Under conditions (1) and (2) show that ${\mathcal{T}}({\mathcal{B}})$ is a topology on $S$.222 While ${\mathcal{T}}({\mathcal{B}})$ under conditions (1) and (2) is already a topology, only in combination with (3) every member of ${\mathcal{B}}(x)$ will be an open set – a necessary condition for a local basis.

From now on suppose in addition condition (3). b) Show ${\mathcal{B}}\subset{\mathcal{T}}({\mathcal{B}})$. c) For each point $x\in S$ show that ${\mathcal{B}}(x)$ is a local basis of ${\mathcal{T}}({\mathcal{B}})$ at $x$. (Hence ${\mathcal{T}}_{\mathcal{B}}={\mathcal{T}}({\mathcal{B}})$ by Exercise A.1.12 (ii), i.e. ${\mathcal{B}}$ is a basis of ${\mathcal{T}}({\mathcal{B}})$.)

The conditions in Definition A.1.16 are related to the theory of filters; see e.g. Narici and Beckenstein (2011, §1.1.2). See also Narici and Beckenstein (2011, Thm. 2.3.1).

Convergence and continuity

Definition A.1.18** (Convergence).**

A subset sequence $(x_{n})\subset M$ in a topological space is said to converge to a point $z\in M$, in symbols $x_{n}\to z$, if any open neighborhood $U_{z}$ of $z$ contains all but finitely many of the sequence members.333 In symbols, there is $N\in{\mathbb{N}}$ such that $x_{n}\subset U_{z}$ whenever $n\geq N$.

Definition A.1.19** (Continuity).**

A map $f\colon M\to N$ between topological spaces is called continuous at a point $x$ if the pre-image of any open neighborhood $V_{f(x)}$ of the image point $f(x)$ contains an open neighborhood $U_{x}$ of $x$. A continuous map is one that is continuous at every point of its domain. Let $C(M,N)$ denote the set of continuous maps from $M$ to $N$.

Exercise A.1.20**.**

a) A map $f\colon M\to N$ between topological spaces is continuous at $x$ iff the pre-image of any open neighborhood $V_{f(x)}$ is open.

b) A map $f$ is continuous iff pre-images of open sets are open.

Hausdorffness and paracompactness

A cover of a topological space $(M,{\mathcal{T}})$ is a family of subsets of $M$ whose union is $M$. The members (elements) of such family are called the sets of the cover or simply the cover sets. A cover is called locally finite if every point of $M$ admits an open neighborhood which meets (intersects) only finitely many cover sets. A cover is called a refinement of another cover if every member of the former is a subset of some member of the latter. A cover ${\mathcal{U}}$ is called open if every cover set is open, in symbols ${\mathcal{U}}\subset{\mathcal{T}}$.

Definition A.1.21**.**

A topological space $M$ is called Hausdorff or $T_{2}$ whenever the topology separates points: Any two points admit disjoint open neighborhoods. Such a topology is called a Hausdorff topology. If the topology separates any two closed sets, then $M$ is called normal or $T_{4}$.

A topological space is called paracompact if every open cover ${\mathcal{U}}$ admits a locally finite open refinement ${\mathcal{V}}$.

Exercise A.1.22** (Hausdorff property).**

Show the following.

• a)

The Hausdorff property ($T_{2}$) is hereditary, normality ($T_{4}$) is not.444 However, closed subspaces of normal spaces are normal; cf. Müger (2016, Exc. 8.1.25).

• b)

In Hausdorff spaces points and, more generally, compact sets are closed. Thus normal implies Hausdorff. ($T_{4}$ $\Rightarrow$ $T_{2}$)

• c)

In Hausdorff spaces limits are unique:

[TABLE]

• d)

Metric spaces are normal ($T_{4}$). (With respect to the metric topology.)

[Hints: a) Counter-example $T_{4}$ Müger (2016, Cor. 8.1.47). b) Show the complement of a point is open. c) By contradiction $y\not=z$. d) Müger (2016, Le. 8.1.11).]

Whereas already Hausdorff by itself is useful to avoid pathological spaces like a real line with two origins, for a Hausdorff space paracompactness is equivalent to existence of a continuous partition of unity subordinate to any given open cover. For a concise presentation including proofs we recommend Cieliebak (2018, §2.2).

Surjections

Lemma A.1.23**.**

Let $M_{\infty}$ be a dense subset of a topological space $M$. Then the image of $M_{\infty}$ under any continuous surjection $f\colon M\twoheadrightarrow Y$ is a dense subset $f(M_{\infty})$ of the target topological space $Y$.

Proof.

Suppose by contradiction that there is a non-empty open set $V\subset Y$ disjoint to $f(M_{\infty})$. Then the pre-image

[TABLE]

is an open subset of $M$ by continuity of $f$ and non-empty as $f$ is surjective. But

[TABLE]

which contradicts density of $M_{\infty}$ in $M$. ∎

Compact-open topology

Let $C(M,N)$ be the set of continuous functions between topological spaces $M$ and $N$. Any pair given by a compact subset $K\subset M$ and an open subset $U\subset N$ determines a collection of continuous functions

[TABLE]

Let ${\mathcal{F}}=\{{\mathcal{F}}_{K,U}\}_{K,U}\subset 2^{C(M,N)}$ be the family of all such collections and denote by $\mathcal{T}_{\mathrm{c}}:={\mathcal{T}}_{{\mathcal{B}}_{\mathcal{F}}}$ the associated topology on the set $C(M,N)$; cf. Exercise A.1.13. It consists of arbitrary unions of finite intersections of elements of ${\mathcal{F}}$. One calls $\mathcal{T}_{\mathrm{c}}$ the compact-open topology on $C(M,N)$, cf. Narici and Beckenstein (2011, Ex. 2.6.9), notation

[TABLE]

Exercise A.1.24**.**

a) Show that $C_{\mathrm{c}}(M,N)$ is Hausdorff if the target $N$ is.

b) For metric spaces $(N,d)$ convergence in $\mathcal{T}_{\mathrm{c}}$ is equivalent to uniform convergence on compact sets: Show that $f_{n}\to f$ in $\mathcal{T}_{\mathrm{c}}$ if and only if

[TABLE]

for every compact subset $K\subset M$.

[Hints: a) Dugundji (1966, Ch. XII) or Müger (2016, Le. 7.9.1). b) Cf. Proposition A.1.59.]

Remark A.1.25** (Only sub-basis).**

In general, the collections ${\mathcal{F}}_{K,U}$ do not form a basis for the compact-open topology

[TABLE]

in symbols ${\mathcal{F}}\subsetneq{\mathcal{B}}_{\mathcal{F}}$, in general. Indeed it is not necessarily true that any non-empty intersection

[TABLE]

contains a non-empty family member ${\mathcal{F}}_{K,U}\in{\mathcal{F}}$ (let alone one that contains a given point; cf. Theorem A.1.14). Hence ${\mathcal{F}}$ cannot be a basis: Indeed if ${\mathcal{F}}$ was a basis, then the non-empty LHS was open, hence a union of members of ${\mathcal{F}}$ – at least one of which non-empty. We encountered two basis counter-examples on math.stackexchange.com:

Counter-example A. Let $M=N=\{a,b\}$ with the discrete topology ${\mathcal{T}}=2^{M}$ and let $K_{1}=U_{1}=\{a\}$ and $K_{2}=U_{2}=\{b\}$. Then ${\mathcal{F}}_{K_{1},U_{1}}\mathop{\cap}{\mathcal{F}}_{K_{2},U_{2}}=\{{\rm id}_{M}\}$ contains only one element, the identity map. The inclusion ${\mathcal{F}}_{K,U}\subset{\mathcal{F}}_{K_{1},U_{1}}\mathop{\cap}{\mathcal{F}}_{K_{2},U_{2}}$ implies $K\supset K_{1}\mathop{\cup}K_{2}=M\not=\emptyset$, hence $K=M$. Thus non-emptiness of ${\mathcal{F}}_{K,U}$ requires $U\not=\emptyset$. But ${\mathcal{F}}_{M,U}$ is not a subset of, equivalently equal to, the singleton $\{{\rm id}_{M}\}$ in any of the three possibilities $U=\{a\},\{b\},\{a,b\}$.

Counter-example B. $M=N={\mathbb{R}}$ with the standard topology. One can show that there are no subsets $K\subset{\mathbb{R}}$ compact and $U\subset{\mathbb{R}}$ open such that

[TABLE]

by constructing certain continuous functions subject to (non-linear) pointwise constraints.

A.1.2 Topological vector spaces

For topological vector spaces and, most importantly, topologies on the vector space of continuous linear maps between them we recommend the books by Rudin (1991), Schaefer and Wolff (1999, III §3), Narici and Beckenstein (2011, §2.6) (here the additive topological group is investigated first and scalar multiplication is superimposed only from Ch. 4 onward), and Treves (1967). There is a book of counter-examples by Khaleelulla (1982, CH. 2). The present section was originally inspired by the excellent Lecture Notes by Kai Cieliebak (2018).

Definition A.1.26**.**

A topological vector space (TVS) is a vector space $X$ endowed with a topology compatible with the vector space operations in the sense that both scalar multiplication ${\mathbb{R}}\times X\to X$ and addition $X\times X\to X$, are continuous maps. Also it is required that points are closed.555Many books on topological vector spaces do not require closedness of points.

Lemma A.1.27**.**

For a TVS $X$ (without using closedness of points) it holds:

• (i)

The closure of a linear subspace is again a linear subspace.

• (ii)

Given a vector $y\in X$ and a scalar $\alpha\in{\mathbb{R}}$, translation $y+\cdot\colon X\to X$ and dilation $\alpha\cdot\colon X\to X$ are linear homeomorphisms. Consequence:

Invariance under translation and dilation.* If $U$ is an open subset of $X$, then so are $x+U$ and $tU$ for all $x\in X$ and $t\in{\mathbb{R}}\setminus\{0\}$.*666 Consequently the open sets containing [math] determine all open sets, hence the topology. **

• (iii)

Any* open neighborhood $V$ of [math] contains an open neighborhood $U$ of [math] which is symmetric $(U=-U)$ and fits into $V$ “twice” $(U+U\subset V)$.*

• (iv)

Closed and compact subsets are separated in a strong sense.* For any closed set $C$ and any disjoint compact set $K$ there is an open neighborhood $U_{0}$ of [math] such that the open neighborhoods $C+U_{0}$ of $C$ and $K+U_{0}$ of $K$ are still disjoint,777 Disjointness remains true if one takes the closure of either $C+U_{0}$ or of $K+U_{0}$. * in symbols $(C+U_{0})\mathop{\cap}(K+U_{0})=\emptyset$.

Proof.

(i) Narici and Beckenstein (2011, Thm. 4.4.1). (ii) Narici and Beckenstein (2011, Thm. 4.3.1). (iii) By continuity of addition and as $0+0=0\in V$ there are open sets $W\ni 0$ and $\tilde{W}\ni 0$ with $W+\tilde{W}\subset V$. The open set $\tilde{U}:=W\mathop{\cap}\tilde{W}$ satisfies $\tilde{U}+\tilde{U}\subset V$. The open set $U:=\tilde{U}\mathop{\cap}-\tilde{U}$ is symmetric and $U+U\subset\tilde{U}+\tilde{U}\subset V$. (iv) Rudin (1991, Thm. 1.10). ∎

Because of the requirement that points of a TVS are closed, part (iv) of the previous lemma applies to $C=\{x\}$ and $K=\{y\}$ and yields disjoint open neighborhoods of any two points $x\not=y$ of $X$. This proves

Corollary A.1.28**.**

A TVS is Hausdorff.

Definition A.1.29**.**

(i) A subset $A$ of a TVS is called a bounded set if for each open neighborhood $U\subset X$ of [math] there is a constant $s>0$ such that $A\subset tU$ is contained in the rescaled neighborhood for all 888 If $A\subset tU$ for some $t$, isn’t the inclusion automatically true for all larger values of $t$?

parameters $t>s$.

(ii) A linear map $T\colon X\to Y$ between topological vector spaces is called bounded if it takes bounded sets to bounded sets and it is called compact if it takes bounded sets to pre-compact sets (compact closure).

Exercise A.1.30** (Bounded sets).**

Subsets of a bounded set are clearly bounded. If $A$ and $B$ are bounded sets, so are $A\mathop{\cup}B$, $A+B$ and $\alpha A$ whenever $\alpha\in{\mathbb{R}}$.

[Hint: If you get stuck consult Schaefer and Wolff (1999, I § 5.1).]

Lemma A.1.31**.**

In a TVS $X$ compact subsets are closed and bounded, whereas the reverse holds iff $\dim X<\infty$.

Proof.

Exercise A.1.22 b) and Rudin (1991, Thm. 1.15 b)). ∎

Spaces of linear maps as topological vector spaces

– ${\mathfrak{S}}$-topologies

Given topological vector spaces $X$ and $Y$, the set

[TABLE]

of all continuous linear operators $T\colon X\to Y$ is a vector space under addition of two operators $T,S\in{\mathcal{L}}(X,Y)$, defined by $(T+S)x:=Tx+Sx$, and scalar multiplication with real numbers $\alpha\in{\mathbb{R}}$, defined by $(\alpha T)x:=\alpha Tx$, both whenever $x\in X$.

We will review the standard abstract machinery that produces various topologies on ${\mathcal{L}}(X,Y)$ for which both operations are continuous, see e.g. Narici and Beckenstein (2011, §11.2) or Schaefer and Wolff (1999, III.3). For some of them points $T$ are closed, so the operator space ${\mathcal{L}}(X,Y)$ endowed with such topology is a TVS: An example is one of the most popular topologies, namely, the compact-open topology or c-topology on ${\mathcal{L}}(X,Y)$. Replacing the family of compact sets by any non-empty family of bounded sets closed under finite unions still guarantees that the generated topology is compatible with addition and scalar multiplication. Hausdorffness might be lost if the sets in the family are not any more compact, but it can be recovered by assumptions on $Y$, e.g. being normed.

Actually all one needs are topological spaces $M$ and $N$; cf. Exercise A.1.24. How one arrives at the c-topology by generalizing a natural construction which provides the point-open topology, or p-topology, is nicely explained in Müger (2016, § 7.9.1).

Exercise A.1.32**.**

Let $T\colon X\to Y$ be a linear map between topological vector spaces. (i) Show that $T$ is continuous iff it is continuous at [math], meaning that the pre-image of every open neighborhood of [math] is open. (ii) Show that continuity implies boundedness of $T$. (The reverse holds if the domain $X$ is a Fréchet space.)

Let $X$ and $Y$ be topological vector spaces. Let \text{\boldmath{\mathfrak{S}}}\subset 2^{X} be a non-empty family of subsets $A$ of $X$, closed under finite unions, that is

[TABLE]

Examples are the families

[TABLE]

Definition A.1.33** (Basic collections).**

For $A\in{\mathfrak{S}}\subset 2^{X}$ and any element $U$ of the family ${\mathcal{U}}_{0}$ of open neighborhoods of [math] in $Y$ consider the collection ${\mathcal{B}}_{A,U}$ of all continuous linear operators which map $A$ into $U$, in symbols

[TABLE]

Collections of the form ${\mathcal{B}}_{A,U}$ are called basic collections.

Lemma A.1.34**.**

*a) Any basic collection ${\mathcal{B}}_{A,U}\ni 0$ contains the zero operator.

b) Any intersection ${\mathcal{B}}_{12}:={\mathcal{B}}_{1}\mathop{\cap}{\mathcal{B}}_{2}$ of two basic collections contains one, i.e.*

[TABLE]

*for some $A_{3}\in{\mathfrak{S}}\subset 2^{X}$ and some open origin neighborhood $U_{3}\in{\mathcal{U}}_{0}\subset 2^{Y}$.

c) If $U+U\subset V$, then ${\mathcal{B}}_{A,U}+{\mathcal{B}}_{A,U}\subset{\mathcal{B}}_{A,V}$.

d) If $r\in{\mathbb{R}}\setminus\{0\}$, then $r{\mathcal{B}}_{A,U}={\mathcal{B}}_{r^{-1}A,U}={\mathcal{B}}_{A,rU}$.*

Proof.

Let $T,S\in{\mathcal{B}}_{A,U}$. a) Obvious. b) ${\mathcal{B}}_{A_{1}\mathop{\cup}A_{2},U_{1}\mathop{\cap}U_{2}}$. c) $\forall a\in A\colon$ $(T+S)a=Ta+Sa\in U+U\subset V$. d) $(rT)(r^{-1}A)\subset U$ and $(rT)(A)\subset rU$. ∎

We denote by ${\mathcal{B}}(0)$ the family of all basic collections,999 A collection of non-empty sets such that the intersection of any two of them contains another one is called a filter base. So ${\mathcal{B}}(0)$ is a filter base and so is each translate ${\mathcal{B}}(x)$.

in symbols

[TABLE]

The notation reminds us that each element ${\mathcal{B}}_{A,U}$ of ${\mathcal{B}}(0)$ contains the zero operator. For $T\in{\mathcal{L}}(X,Y)$ let ${\mathcal{B}}(T):=T+{\mathcal{B}}(0)$ be the translated family. We denote by

[TABLE]

the family of all translated basic collections.

Theorem A.1.35**.**

*Let $X$ and $Y$ be topological vector spaces. Let ${\mathcal{U}}_{0}\subset 2^{Y}$ be the collection of open sets containing the origin of $Y$. Suppose ${\mathfrak{S}}\subset 2^{X}$ is a non-empty family of bounded 101010 Boundedness leads to ${\mathcal{T}}_{\mathfrak{S}}$-continuity of “$+$” and scalar multiplication on ${\mathcal{L}}(X,Y)$.

subsets $A$ of $X$ which is closed under finite unions. Then the following is true (not using closedness of points in $X,Y$). {labeling}*(local basis)**

The family ${\mathcal{B}}(0)$ of all basic collections ${\mathcal{B}}_{A,U}\subset{\mathcal{L}}(X,Y)$ forms a local basis at [math] of a topology ${\mathcal{T}}_{{\mathfrak{S}}}$ on ${\mathcal{L}}(X,Y)$ for which addition and scalar multiplication are continuous; cf. Remark A.1.37.

*The family ${\mathcal{B}}={\mathcal{B}}_{\mathfrak{S}}^{{\mathcal{U}}_{0}}$ is a basis for a topology ${\mathcal{T}}_{\mathfrak{S}}$ on ${\mathcal{L}}(X,Y)$.

The topology ${\mathcal{T}}_{\mathfrak{S}}$ on ${\mathcal{L}}(X,Y)$, called ${\mathfrak{S}}$-topology, is*

• –

Hausdorff whenever the linear span of $\bigcup{\mathfrak{S}}$ is dense in $X$ and if $Y$ is Hausdorff;

• –

locally convex whenever $Y$ is.

Proof.

See e.g. Narici and Beckenstein (2011, Thm. 11.2.2). ∎

By Exercise A.1.57 any normed vector space $Y$ is a locally convex TVS, i.e. a TVS such that any neighborhood of [math] contains a convex111111 A subset $C$ of a vector space is called a convex set if $C$ contains every line segment $\{tx+(1-t)y\mid t\in[0,1]\}$ connecting two of its points $x,y\in C$.

one.

In contrast to the basis property of ${\mathcal{B}}$ in the linear setting, recall from Remark A.1.25 that in the general case of topological spaces even for the family of compact subsets the basic collections do not form a basis, only a sub-basis.

Corollary A.1.36**.**

Let $X$ and $Y$ be topological vector spaces. If ${\mathfrak{S}}$ covers $X$ (e.g. if ${\mathfrak{S}}={\mathfrak{S}}_{\rm p},{\mathfrak{S}}_{\rm c},{\mathfrak{S}}_{\rm b}$), then $\mathcal{L}_{\mathfrak{S}}(X,Y):=\left({\mathcal{L}}(X,Y),{\mathcal{T}}_{\mathfrak{S}}\right)$ is a TVS.

The following topologies associated to the indicated families ${\mathfrak{S}}$ are called

$\mathcal{L}_{\mathrm{p}}(X,Y):={\mathcal{L}}_{{\mathfrak{S}}_{\rm p}}(X,Y)$ point-open or p-topology

• -

$\mathcal{L}_{\mathrm{c}}(X,Y):={\mathcal{L}}_{{\mathfrak{S}}_{\rm c}}(X,Y)$ compact-open or c-topology

• -

$\mathcal{L}_{\mathrm{b}}(X,Y):={\mathcal{L}}_{{\mathfrak{S}}_{\rm b}}(X,Y)$ bounded-open or b-topology

Remark A.1.37** (Continuous vector operations).**

Suppose $X$ and $Y$ are topological vector spaces. Continuity of addition and scalar multiplication under a ${\mathfrak{S}}$-topology is equivalent to boundedness of every image $TA\subset Y$ where $T\in{\mathcal{L}}(X,Y)$ and $A\in{\mathfrak{S}}$; see e.g. Schaefer and Wolff (1999, III §3.1) or Bourbaki (1987, III §3 Prop. 1).

Exercise A.1.38** (Families of compact sets).**

For each of the three families ${\mathfrak{S}}={\mathfrak{S}}_{\rm p},{\mathfrak{S}}_{\rm c},{\mathfrak{S}}_{\rm b}$ show the particular assertion of Theorem A.1.35 that ${\mathcal{T}}_{\mathfrak{S}}$ is a topology on ${\mathcal{L}}(X,Y)$ and ${\mathcal{B}}={\mathcal{B}}_{\mathfrak{S}}^{{\mathcal{U}}_{0}}$ is a basis – in contrast to Remark A.1.25.

[Hints: Theorem A.1.14 or Exercise A.1.17. Lemma A.1.27 iv).]

Continuity properties

Proposition A.1.39**.**

Suppose $M$ is a topological space and $X,Y,Z$ are topological vector spaces. Then the following is true.

• a)

If the map $\varphi\colon M\times Y\to Z$ is continuous and, moreover, linear in the second variable, then the induced map

[TABLE]

is continuous, in symbols $\Phi\in C(M,\mathcal{L}_{\mathrm{c}}(Y,Z))$.

• b)

If $S\colon X\to Y$ is a compact linear operator, then the induced map

[TABLE]

is continuous.

• c)

For $\Phi$ and $S$ as in a) and b) the induced map

[TABLE]

is continuous. (Juxtaposition of linear maps means composition.)

For normed vector spaces $X$ and $Y$ both topological vector spaces $\mathcal{L}_{\mathrm{b}}(X,Y)$ and ${\mathcal{L}}(X,Y)$ with the operator norm topology coincide by Proposition A.1.59.

Operators similar to the one in (A.1.7) are well known in non-linear analysis under the name Nemitski operators associated to $\varphi$; see e.g. Ambrosetti and Prodi (1993, §1.2).

Proof of Proposition A.1.39.

a) is even true for topological spaces $M,Y,Z$ and continuous functions $\varphi\colon M\times Y\to Z$, not necessarily linear in the second variable; see e.g. Dugundji (1966, XII.3.1) or Müger (2016, Le. 7.9.5). Now the conclusion is that $\Phi$ is continuous as a map $M\to C_{\mathrm{c}}(Y,Z)$; cf. (A.1.2).

To prove this we must show that for all $p_{0}\in M$ and sub-basis elements ${\mathcal{F}}_{K,V}\subset C_{\mathrm{c}}(Y,Z)$ that contain $\Phi(p_{0})$ there is an open neighborhood $U_{p_{0}}$ of $p_{0}$ in $M$ whose image under $\Phi$ lies in ${\mathcal{F}}_{K,V}$, too. Equivalently, we have to show that $\varphi(p_{0}\times K)\subset V$ implies $\varphi(U_{p_{0}}\times K)\subset V$ for some open set $p_{0}\in U_{p_{0}}\subset M$. Continuity of $\varphi$ guarantees an open pre-image $\varphi^{-1}(V)\subset M\times Y$ which contains $p_{0}\times K$ by assumption. By compactness of $K$ the Slice Lemma, see e.g. Dugundji (1966, XI.2.6) or Müger (2016, Prop. 7.5.1), provides an open neighborhood $U_{p_{0}}$ of $p_{0}\in M$ such that the thickening $U_{p_{0}}\times K$ of $p_{0}\times K$ is still contained in $\varphi^{-1}(V)$.

b) Let’s show that the pre-image $\Phi^{-1}{\mathcal{B}}_{A,U}$ of any (open) basis element of the bounded-open topology $\mathcal{T}_{\mathrm{b}}(X,Z)$ is open in $\mathcal{L}_{\mathrm{c}}(Y,Z)$, i.e. contains some basis element ${\mathcal{B}}^{\prime}_{K,V}\in\mathcal{T}_{\mathrm{c}}(Y,Z)$ of the compact-open topology. Given $A\subset X$ bounded and $0\in U\subset Z$ open, note that $\Phi^{-1}{\mathcal{B}}_{A,U}={\mathcal{B}}^{\prime}_{S(A),U}\supset{\mathcal{B}}^{\prime}_{K,U}\in\mathcal{T}_{\mathrm{c}}(Y,Z)$ where by definition the compact set $K$ is the closure of the pre-compact set $S(A)\subset Y$.

c) The composition of continuous maps is continuous. But composing the continuous maps (A.1.5) and (A.1.6) is the map (A.1.7). ∎

Fréchet and Gâteaux derivative on TVS

Definition A.1.40** (Fréchet derivative on TVS).**

Suppose $f\colon X\supset U\to Y$ is a map between topological vector spaces defined on an open subset $U$.

In case $0\in U$ and $f(0)=0$ one says that $f$ has derivative zero at [math] if for each open neighborhood $W_{0}\subset Y$ of [math] there is an open neighborhood $V_{0}\subset X$ of [math] and a function $o\colon(-1,1)\to{\mathbb{R}}$ such that

[TABLE]

for every $t\in(-1,1)$.

In general, one calls $f$ differentiable at $x\in U$ if there is a continuous linear operator $D\colon X\to Y$ such that the map

[TABLE]

has derivative zero at [math]. In this case $df(x):=D\in{\mathcal{L}}(X,Y)$ is called the derivative of $f$ at $x$. If $f$ is differentiable at every point of $U$ one calls $f$ (Fréchet) differentiable on $U$. In this case the map

[TABLE]

into the vector space of continuous linear maps ${\mathcal{L}}(X,Y)$ is called the (Fréchet) differential of $f$.

By Corollary A.1.36 endowing ${\mathcal{L}}(X,Y)$ with the topology ${\mathcal{T}}_{\mathfrak{S}}$ associated to any of the families ${\mathfrak{S}}={\mathfrak{S}}_{\rm p},{\mathfrak{S}}_{\rm c},{\mathfrak{S}}_{\rm b}$ results in a TVS denoted by $\mathcal{L}_{\mathfrak{S}}(X,Y)$. Hence $df\colon U\to\mathcal{L}_{\mathfrak{S}}(X,Y)$ is a map between TVS and one defines iteratively the higher order differentials

[TABLE]

For normed vector spaces $X$ and $Y$ the bounded-open topology ${\mathcal{T}}_{{\mathfrak{S}}_{\rm p}}$ and the operator norm topology on ${\mathcal{L}}(X,Y)$ coincide by Proposition A.1.59 below.

We say that a map $f\colon X\supset U\to Y$ admits directional derivative at $x\in U$ in direction $\xi\in X$, if there are ${\varepsilon}>0$ and $\eta\in Y$ such that the map

[TABLE]

has derivative zero at [math]. In this case ${\partial}_{\xi}f(x):=\eta$ is called the derivative of $f$ at $x$ in direction $\xi$. If the map ${\partial}f(x)\colon X\to Y$, $\xi\mapsto{\partial}_{\xi}f(x)$, is defined for every $\xi\in X$ and is linear and continuous, then $f$ is said Gâteaux differentiable at $x$ with Gâteaux derivative ${\partial}f(x)\in{\mathcal{L}}(X,Y)$.

The (Fréchet) derivative on topological vector spaces enjoys some basic properties such as the chain rule and the fact that Fréchet differentiability implies continuity and Gâteaux differentiability. However, other fundamentals are not available, for instance the implicit function theorem and Cartan’s last theorem may fail on TVS; see examples in Cieliebak (2018, §4.2).

A.1.3 Metric spaces

Definition A.1.41**.**

A metric on a set $M$ is a function $d\colon M\times M\to[0,\infty)$ that satisfies the following three axioms whenever $x,y,z\in M$.

1. (i)

$d(x,y)=d(y,x)$ (Symmetry) 2. (ii)

$d(x,z)\leq d(x,y)+d(y,z)$ (Triangle inequality) 3. (iii)

$d(x,y)=0$ $\Leftrightarrow$ $x=y$ (Non-degeneracy)

Such pair $(M,d)$ is called a metric space.

The prototype example of a metric $d$ is the distance between two points in euclidean space. Hence a metric is also called a distance function. We often use the notation $M_{d}$ for a metric space, meaning that $M$ is a set endowed with the metric $d$.

Definition A.1.42**.**

A metric space $M_{d}$ comes naturally with the metric topology ${\mathcal{T}}_{d}$ whose basis ${\mathcal{B}}_{d}$ consists of the open balls $B^{d}_{x}({\varepsilon})$ of all radii ${\varepsilon}>0$ about all points $x$ of $M_{d}$. A metric space will be automatically endowed with the metric topology, unless mentioned otherwise.

Exercise A.1.43** (Metric topology).**

Check that the collection ${\mathcal{B}}_{d}$ of all open balls in $M_{d}$ indeed forms a basis for a topology, and not just a sub-basis.

As mentioned earlier, metric spaces are normal, thus Hausdorff. Moreover, second countability (countable basis) is equivalent to separability (dense sequence).

Exercise A.1.44** (Convergent sequence).**

Check that $x_{n}\to y\in M_{d}$, in the sense of Definition A.1.18, if and only if any ${\varepsilon}$-ball about $y$ contains all but finitely many sequence members $x_{n}$, in symbols

[TABLE]

Sequential convergence properties

Proposition A.1.45**.**

Let $Q$ be a compact topological space and $M_{d}$ a metric space. Then the compact-open topology $\mathcal{T}_{\mathrm{c}}$ on $C(Q,M_{d})$ coincides with the metric topology ${\mathcal{T}}_{d_{\infty}}$ associated to the supremum metric

[TABLE]

Proof.

Müger (2016, Prop. 7.9.2). (To show equality of two topologies one shows that the members of a basis, or of a sub-basis, of the first topology are open with respect to the second topology, and vice versa.) ∎

Exercise A.1.46**.**

If $Q$ is compact, then $d_{\infty}$ is a metric on $C(Q,M_{d})$.

[Hint: If stuck, consult e.g. Müger (2016, Prop. 2.1.25).]

Convergence $f_{\nu}\to g$ with respect to ${\mathcal{T}}_{d_{\infty}}={\mathcal{T}}_{d_{\infty}}(Q,M_{d})$, that is

[TABLE]

is called uniform convergence on the compact set $Q$.

Exercise A.1.47**.**

Let $N$ be a topological space. If $M_{d}$ is a metric space, the compact-open topology $\mathcal{T}_{\mathrm{c}}$ on $C(N,M_{d})$ coincides with the topology

[TABLE]

of uniform convergence on all compact subsets $K$ of $N$.

[Hint: If $N\supset Q$ compact, then $\mathcal{T}_{\mathrm{c}}(N,M_{d})\supset\mathcal{T}_{\mathrm{c}}(Q,M_{d})={\mathcal{T}}_{d_{\infty}}(Q,M_{d})$.]

Definition A.1.48** (Equicontinuous family).**

Let $N$ be a topological space and $M_{d}$ a metric space. A family ${\mathcal{F}}\subset{\rm Map}(N,M_{d})$ of maps, a-priori continuous or not, is called equicontinuous if for every $x\in N$ and every ${\varepsilon}>0$ there is an open neighborhood $U_{x}$ of $x$ such that for all neighborhood elements $x^{\prime}\in U_{x}$ and family members $f\in{\mathcal{F}}$ both values $f(x)$ and $f(x^{\prime})$ are ${\varepsilon}$-close, in symbols

[TABLE]

Exercise A.1.49**.**

The members of an equicontinuous family ${\mathcal{F}}$ are continuous.

Complete metric spaces –

Theorem of Baire and Arzelà–Ascoli

Definition A.1.50**.**

A sequence $(x_{n})$ in a metric space $M_{d}$ is called a Cauchy sequence if for every ${\varepsilon}>0$ there is a sequence member $x_{N}$ such that any two subsequent members are within distance ${\varepsilon}$ of one another, in symbols

[TABLE]

Exercise A.1.51**.**

Check that every convergent sequence in a metric space is a Cauchy sequence, but the converse is not true.

Definition A.1.52** (Complete metric space).**

A metric space in which every Cauchy sequence converges is called complete and so is the metric.

Exercise A.1.53**.**

Let $Q$ be a compact topological space. Then the metric space $\left(C(Q,M_{d}),d_{\infty}\right)$ is complete iff the target metric space $M_{d}$ is complete.

[Hint: If stuck, consult e.g. Müger (2016, Prop. 3.1.18 and Rmk. 5.2.12).]

Theorem A.1.54** (Baire’s Theorem).**

Let $M_{d}$ be a complete metric space and $(U_{n})$ a sequence of open and dense subsets. Then the intersection

[TABLE]

is dense in $M_{d}$.

Proof.

See e.g. Müger (2016, Thm. 3.3.1). ∎

Among the many applications of Baire’s Theorem are the open mapping theorem and the Banach–Steinhaus Theorem A.2.12, also called the principle of uniform boundedness.

Theorem A.1.55** (Arzelà–Ascoli Theorem).**

Let $Q$ be a compact topological space and $M_{d}$ a complete metric space. Then the following is true. A family

[TABLE]

of continuous maps is pre-compact (with respect to the supremum metric $d_{\infty}$) if and only if the family ${\mathcal{F}}$ is equicontinuous and the ${\mathcal{F}}$-orbit through each domain point $q\in Q$, namely each subset

[TABLE]

is pre-compact.

Proof.

See e.g. Müger (2016, Thm. 7.7.67). ∎

A.1.4 Normed vector spaces

Definition A.1.56**.**

A norm on a (real) vector space $X$ is a function $\mathopen{\|}\cdot\mathclose{\|}\colon X\to[0,\infty)$ that satisfies the following three axioms

1. (i)

$\mathopen{\|}\alpha x\mathclose{\|}=\mathopen{|}\alpha\mathclose{|}\mathopen{\|}x\mathclose{\|}$ (Homogeneity) 2. (ii)

$\mathopen{\|}x+y\mathclose{\|}\leq\mathopen{\|}x\mathclose{\|}+\mathopen{\|}y\mathclose{\|}$ (Triangle inequality) 3. (iii)

$\mathopen{\|}x\mathclose{\|}=0$ $\Leftrightarrow$ $x=0$ (Non-degeneracy)

for all $x,y\in X$ and $\alpha\in{\mathbb{R}}$. Such pair $(X,\mathopen{\|}\cdot\mathclose{\|})$ is called a normed vector space, often just denoted by $X$. If one drops the requirement $\mathopen{\|}x\mathclose{\|}=0$ $\Rightarrow$ $x=0$ in (iii), one obtains the definition of a semi-norm on $X$.

The prototype example of a norm $\mathopen{\|}\cdot\mathclose{\|}$ is the distance of a point in euclidean space from the origin.

Exercise A.1.57** (Normed $\Rightarrow$ metric and TVS with convex basis).**

Suppose $(X,\mathopen{\|}\cdot\mathclose{\|})$ is a normed vector space. Show the following.

a) The definition

[TABLE]

provides a (translation invariant: $d(x+z,y+z)=d(x,y)$) metric on $X$.

(So normed vector spaces are endowed with a natural topology, the metric topology ${\mathcal{T}}_{d_{\|}}$. Because metric topologies are Hausdorff, limits are unique.)

b) Both vector operations, addition and scalar multiplication, are continuous.

(So any normed vector space $X$ is a TVS.)

c) Open balls $B_{\varepsilon}(x)$ of radius ${\varepsilon}>0$ centered at $x\in X$ are convex sets. So the natural basis of the topology of a TVS $X$ given by all open balls consists of convex sets.

(So by Theorem A.1.35 the space ${\mathcal{L}}(X,Y)$ of continuous linear operators between normed vector spaces is a locally convex TVS under the point-open, compact-open, and bounded-open topologies; with respect to the latter it is even normed as we will see.)

[Hints: b) Addition: triangle inequality, scalar multiplication: homogeneity.]

Exercise A.1.58** (The normed vector space ${\mathcal{L}}(X,Y)$).**

Let $X$ and $Y$ be normed vector spaces. Recall Definition A.1.29 on boundedness. Show that

a) A linear map $T\colon X\to Y$ is continuous iff it is bounded iff it maps the open unit ball about [math] into one of finite radius $r$, in symbols $TB_{1}\subset B_{r}$.

b) Now consider the vector space ${\mathcal{L}}(X,Y)$ that consists of all bounded linear operators $T\colon X\to Y$ with addition $T+S\colon x\mapsto Tx+Sx$ and scalar multiplication $\alpha T\colon x\mapsto\alpha Tx$ for $\alpha\in{\mathbb{R}}$. Taking the infimum of all radii $r>0$ of balls $B_{r}\supset TB_{1}$ still containing the image under $T$ of the unit ball defines a norm

[TABLE]

called the operator norm. Alternatively, it is given by

[TABLE]

[Hints: a) Cf. Rudin (1991, 1.29).]

By Exercise A.1.57 the normed vector space $\left({\mathcal{L}}(X,Y),\mathopen{\|}\cdot\mathclose{\|}\right)$ carries a natural metric $d_{\|}$ and is a locally convex TVS. The metric topology ${\mathcal{T}}_{\|}={\mathcal{T}}_{\|}(X,Y)$ is called the operator norm topology or theoperator norm!topology uniform topology, also indicated by ${\mathcal{L}}_{\|}(X,Y)$.

Convention: Whenever we speak of ${\mathcal{L}}(X,Y)$ as a normed vector space it is automatically endowed with the operator norm topology.

Proposition A.1.59** (Operator norm topology is bounded-open topology).**

For normed vector spaces $X$ and $Y$ the bounded-open and the operator norm topology on ${\mathcal{L}}(X,Y)$ coincide, in symbols $\mathcal{T}_{\mathrm{b}}={\mathcal{T}}_{\|}$.

Proof.

Let $\mathopen{\|}\cdot\mathclose{\|}$ be the operator norm on ${\mathcal{L}}(X,Y)$. Balls are centered at [math].

${\mathcal{T}}_{\|}\subset\mathcal{T}_{\mathrm{b}}\colon$ It suffices to show that norm open balls $B_{r}:=\{\mathopen{\|}\cdot\mathclose{\|}<r\}$ are open with respect to $\mathcal{T}_{\mathrm{b}}$. This means that $B_{r}$ must contain together with any element $S$ a whole $\mathcal{T}_{\mathrm{b}}$-open neighborhood $S+{\mathcal{B}}_{A,U}$ where ${\mathcal{B}}_{A,U}=\{T\in{\mathcal{L}}(X,Y)\mid T(A)\subset U\}$ with $A\subset X$ bounded and $0\ni U\subset Y$ open; cf. (A.1.3).

To see this abbreviate $s:=\mathopen{\|}S\mathclose{\|}\in[0,r)$ and let $A$ be the closed unit ball in $X$ and $U$ the open ball in $Y$ of radius $\frac{r-s}{2}$. For $T\in{\mathcal{B}}_{A,U}$ we get

[TABLE]

Hence the $\mathcal{T}_{\mathrm{b}}$-open neighborhood $S+{\mathcal{B}}_{A,U}$ of $T$ is contained in $B_{r}$.

$\mathcal{T}_{\mathrm{b}}\subset{\mathcal{T}}_{\|}\colon$ By translation invariance of both topologies it suffices to show121212 We could have localized to [math] already in order to prove ${\mathcal{T}}_{\|}\subset\mathcal{T}_{\mathrm{b}}$ above.

that each element ${\mathcal{B}}_{A,U}\in{\mathcal{B}}(0)\subset{\mathcal{L}}(X,Y)$ of the local basis of $\mathcal{T}_{\mathrm{b}}$ at [math] contains an open ball $B_{r}\in{\mathcal{T}}_{\|}$ about [math]. Given $A\subset X$ bounded and $0\in U\subset Y$ open, pick open balls $A\subset B_{r}\subset X$ and $B_{\varepsilon}\subset U\subset Y$. Then $\mathopen{\|}T\mathclose{\|}_{{\mathcal{L}}(X,Y)}<r={\varepsilon}/R$ implies that $T\in{\mathcal{B}}_{A,U}$. Indeed $TA\subset TB_{R}=RTB_{1}\subset RB_{{\varepsilon}/R}=B_{\varepsilon}\subset U$. ∎

Sequential convergence properties

Lemma A.1.60** **(Convergence in compact-open topology means

convergence of the orbit through each point).

Let $X$ and $Y$ be normed vector spaces. Consider operators $(T_{\nu})_{\nu\in{\mathbb{N}}}\subset{\mathcal{L}}(X,Y)\ni T$. Then $T_{\nu}\to T$ in $\mathcal{L}_{\mathrm{c}}(X,Y)$ iff for each domain element $\xi$ the image sequence $T_{\nu}\xi$ converges to $T\xi$ in $Y$.

A.2 Analysis on Banach spaces

All vector spaces are over the real numbers. Throughout any linear structure is with respect to the real numbers and, as a rule of thumb, by $X$ and $Y$ we denote normed linear spaces and by $E$ and $F$ Banach spaces. In the context of linear spaces subspace means linear subspace.

A.2.1 Banach spaces

Definition A.2.1**.**

A Cauchy sequence is a sequence $x_{\nu}$ in a normed linear space $X$ such that $\mathopen{\|}x_{n}-x_{m}\mathclose{\|}\to 0$ whenever $n,m\to\infty$. The norm is called complete if every Cauchy sequence converges (admits a limit). A linear space $E$ endowed with a complete norm131313 also called a complete normed linear space

is called a Banach space. Any closed linear subspace $F\subset E$ endowed with the norm of $E$ is a Banach space itself, called a Banach subspace.

Relevant examples of Banach spaces are enlisted in Theorem A.3.1.

Direct sum and topological complements

Definition A.2.2** (Direct sum).**

The direct sum of Banach spaces $X\oplus Y$ is the set of pairs $\{(x,y)\mid x\in X,\,y\in Y\}$ which is equipped with and complete under the norm $\mathopen{\|}(x,y)\mathclose{\|}:=\mathopen{\|}x\mathclose{\|}+\mathopen{\|}y\mathclose{\|}$.141414 Alternatively, use any of the equivalent norms $\mathopen{\|}(x,y)\mathclose{\|}_{p}^{p}:=\mathopen{\|}x\mathclose{\|}^{p}+\mathopen{\|}y\mathclose{\|}^{p}$ for $1\leq p<\infty$ or $\mathopen{\|}(x,y)\mathclose{\|}_{\infty}:=\max\{\mathopen{\|}x\mathclose{\|},\mathopen{\|}y\mathclose{\|}\}$.

Definition A.2.3** (Banach space complement).**

A closed subspace $X$ of a Banach space $Z$ is said to be complemented if there is a closed subspace $Y$ of $Z$ such that $X\cap Y=\{0\}$ and $X+Y=Z$. In this case we write $X\oplus Y=Z$ and call $Y$ a Banach space complement or a topological complement of $X$, one also says that the Banach space $X$ splits.

Example A.2.4** (Not every closed subspace is complemented).**

Consider the Banach space \ell^{\infty}:=\{\text{x\colon{\mathbb{N}}\to{\mathbb{R}}$,$\nu\mapsto x_{\nu}, bounded}\} of bounded real sequences equipped with the sup norm. The subspace $c_{0}$ of sequences that converge to zero is closed, but does not admit a topological!complement: There is no closed subspace $d$ such that $c_{0}\oplus d=\ell^{\infty}$; see Whitley (1966).

Quotient spaces

Definition A.2.5** (Quotient space).**

Suppose $X$ is a normed linear space and $A\subset X$ is a closed linear subspace. The quotient space of $X$ by $A$ is the set of cosets151515 equivalently, the set of equivalence classes $\{[x]\mid x\in X\}$ where $x\sim y$ if $x-y\in A$

denoted and defined by

[TABLE]

The function $X/A\to[0,\infty)$ given by the distance of any point representing the coset $x+A$ to the closed subspace $A$, namely

[TABLE]

is called the quotient norm. Often we use the shorter notation $\left\|x+A\right\|$.

Exercise A.2.6**.**

a) Check that the operations $\alpha(x+A):=\alpha x+A$ for $\alpha\in{\mathbb{R}}$ and $(a+A)+(b+A):=(a+b)+A$ are well defined on $X/A$ and endow the set of cosets with the structure of a linear space. Here closedness of $A$ is actually not needed. b) Check that $\mathopen{\|}x+A\mathclose{\|}_{X/A}=\mathopen{\|}y+A\mathclose{\|}_{X/A}$ whenever $x+A=y+A$ or, equivalently, whenever $x-y\in A$. c) Show that the function $x+A\mapsto\left\|x+A\right\|$ is a norm on the linear space $X/A$.

[Hint: c) Non-degeneracy ($\left\|x+A\right\|=0$ $\Rightarrow$ $x\in A$) relies on closedness of $A$.]

Proposition A.2.7** (Quotient Banach spaces).**

Suppose $E$ is a Banach space and $A$ is a closed subspace. Then the following is true.

• (i)

The quotient norm on $E/A$ is complete.

• (ii)

The map between Banach spaces defined by

[TABLE]

is linear,* surjective, continuous, and of norm $\mathopen{\|}\pi\mathclose{\|}\leq 1$ at most one. It is called the** projection onto the quotient space $E/A$.*

• (iii)

Suppose, in addition, that $E$ is reflexive, then $E/A$ is reflexive.

Proof.

(i) Given a Cauchy sequence $x_{\nu}+A$ in the coset space $E/A$, by the Cauchy property it suffices to extract a subsequence that converges to a limit element $e+A$ in $E/A$. Forgetting sequence members, if necessary, there is a subsequence, still denoted by $x_{\nu}+A$, that satisfies

[TABLE]

Thus there is a sequence of points $a_{\nu}\in A$ whose distance to $x_{\nu+1}-x_{\nu}$ satisfies $\mathopen{\|}x_{\nu+1}-x_{\nu}-a_{\nu}\mathclose{\|}<1/2^{\nu}$. Consider the partial sum sequence $z_{\nu+1}:=a_{\nu}+\dots+a_{1}\in A$. As the sequence $x_{\nu}-z_{\nu}$ is Cauchy in $E$, indeed

[TABLE]

it admits a limit $e$ in the Banach space $E$. It follows that the sequence $x_{\nu}+A$ converges to $e+A$ in $E/A$ and we are done. Indeed

[TABLE]

(ii) The map $\pi$ is linear by definition of addition in the coset space $E/A$. Surjectivity is obvious. To see continuity and $\mathopen{\|}\pi\mathclose{\|}\leq 1$, given $x\in E$, pick $a=0\in A$ to get that $\mathopen{\|}\pi(x)\mathclose{\|}:=\inf_{a\in A}\mathopen{\|}x-a\mathclose{\|}\leq\mathopen{\|}x\mathclose{\|}$. (iii) Brezis (2011, Prop. 11.11). ∎

For more details about quotients see e.g. Brezis (2011, §11.2) or Rudin (1987, §18.14).

A.2.2 Linear operators

Given normed linear spaces $X$ and $Y$, recall that a linear map $T\colon X\to Y$ is continuous iff it is continuous at one point iff it is bounded; see e.g. Reed and Simon (1980, Thm. I.6). To be bounded means that the operator norm of $T$, defined by

[TABLE]

is finite. By ${\mathcal{L}}(X,Y)$ we denote the linear space of continuous linear operators $T\colon X\to Y$. Juxtaposition $ST\colon X\to Y\to Z$ denotes composition. The invertible elements $T$ of ${\mathcal{L}}(X,Y)$, that is $TS={{\mathchoice{\rm 1\mskip-4.0mul}{\rm 1\mskip-4.0mul}{\rm 1\mskip-4.5mul}{\rm 1\mskip-5.0mul}}}$ and $ST={{\mathchoice{\rm 1\mskip-4.0mul}{\rm 1\mskip-4.0mul}{\rm 1\mskip-4.5mul}{\rm 1\mskip-5.0mul}}}$ for some (unique) $S\in{\mathcal{L}}(Y,X)$, are called isomorphisms or **toplinear isomorphisms161616 A toplinear isomorphism is a continuous linear bijection whose inverse is continuous, too. The notion makes sense in the general context of topological vector spaces. ** to emphasize context. In case of Banach spaces $E$ and $F$ invertible elements of ${\mathcal{L}}(E,F)$, aka toplinear isomorphisms, aka isomorphisms, are precisely the continuous linear bijections; cf. e.g. Lang (2001, I §2) and Lang (1993, IV §1). (The inverse is continuous by the closed graph theorem.)

Abbreviate ${\mathcal{L}}(X):={\mathcal{L}}(X,X)$. By ${\mathcal{L}}^{k}(X,Y)$ we denote the linear space of $k$-fold multilinear maps $T\colon X\oplus\dots\oplus X\to Y$. If the norm of $Y$ is complete, then the operator norm is complete, so ${\mathcal{L}}(X,Y)$ is a Banach space. Thus the dual space $X^{*}={\mathcal{L}}(X,{\mathbb{R}})$ of a normed linear space is a Banach space.

Unique extension

Theorem A.2.8** (B.L.T. theorem).**

Suppose $T$ is a bounded linear map from a normed linear space $X$ to a complete normed linear space $F$. Then $T$ extends uniquely to a bounded linear map $\tilde{T}$ from the completion of $X$ to $F$.

Proof.

See e.g. Reed and Simon (1980, Thm. I.7). ∎

Compact operators and projections

Definition A.2.9** (Compact operator).**

A linear operator $S\colon X\to Y$ between normed linear spaces is called compact if for every bounded sequence in the domain, the image sequence has a convergent subsequence or, equivalently, if it maps bounded sets to pre-compact sets (sets whose closure is compact). Compact linear operators are automatically continuous.

Definition A.2.10** (Projection).**

A continuous linear operator $P\colon X\to X$ is called a projection if it is idempotent, in symbols $P\circ P=P$.

Exercise A.2.11** (Continuous projections split).**

Let $E$ be a Banach space and $P\in{\mathcal{L}}(E)$ a projection. Then the image $F:={\rm im\,}P$ is closed and complemented by the closed image $G:={\rm im\,}Q$ of the continuous projection $Q:={{\mathchoice{\rm 1\mskip-4.0mul}{\rm 1\mskip-4.0mul}{\rm 1\mskip-4.5mul}{\rm 1\mskip-5.0mul}}}-P$, that is

[TABLE]

[Hint: Kernels of continuous maps are closed and ${\rm im\,}P=\ker Q$ and vice versa.]

Principle of uniform boundedness

The Hahn–Banach theorem and the Banach–Steinhaus theorem are two pillars of functional analysis. The latter is also known as the principle of uniform boundedness. Its proof is based on the Baire category theorem which requires a non-empty complete metric space, for instance a Banach space $E$.

Theorem A.2.12** (Banach–Steinhaus).**

Suppose $E$ is a Banach space. Let ${\mathcal{F}}$ be a family of bounded linear operators $T\colon E\to Y$ to some normed linear space. Suppose that the ${\mathcal{F}}$-orbit through each point $x\in E$, namely each set

[TABLE]

is a bounded subset of $Y$. Then the operator norm is uniformly bounded along the family ${\mathcal{F}}$: There is a constant $c_{\mathcal{F}}\geq 0$ such that

[TABLE]

Proof.

See e.g. Reed and Simon (1980, Thm. III.9). ∎

Recall that ${\mathcal{L}}(E,F)$ carries the operator norm topology. How to utilize the principle of uniform boundedness is illustrated in the proof of

Proposition A.2.13**.**

Suppose $E_{1},E_{0},F_{0}$ are Banach spaces and $U_{1}\subset E_{1}$ is an open subset. Then the following is true.

• a)

Let the map $\Phi\colon U_{1}\oplus E_{0}\to F_{0}$, $(x,\eta)\mapsto\Phi(x,\xi)=:\Phi(x)\eta$, be continuous and, moreover, linear in the second variable. Then the induced map

[TABLE]

*is continuous. (The target carries the compact-open topology.)*171717 The target carrying the compact-open topology means that a sequence $T_{\nu}\in{\mathcal{L}}(E_{0},F_{0})$ converges to an element $T\in{\mathcal{L}}(E_{0},F_{0})$ iff for each domain element $\xi$ the sequence $T_{\nu}\xi$ converges to $T\xi$ in $F_{0}$; see Lemma A.1.60. **

• b)

If $S\colon E_{1}\to E_{0}$ is a compact linear operator, then the induced map

[TABLE]

is continuous. (The target carries the norm topology.)

• c)

For $\Phi\colon U_{1}\oplus E_{0}\to F_{0}$ and $S\colon E_{1}\to E_{0}$ as in a) and b) the induced map

[TABLE]

is continuous.

Proof.

a) Proposition A.1.39 a).

b) Given $T\in{\mathcal{L}}(E_{0},F_{0})$ and a sequence $T_{\nu}\in{\mathcal{L}}(E_{0},F_{0})$ with $T_{\nu}\zeta\to T\zeta$ in $F_{0}$ for each $\zeta\in E_{0}$, assume by contradiction that there is a constant ${\varepsilon}>0$ and a sequence in $E_{1}$ of bounded norm, say $\mathopen{\|}\xi_{\nu}\mathclose{\|}_{E_{1}}=1$, such that $\mathopen{\|}T_{\nu}S\xi_{\nu}-TS\xi_{\nu}\mathclose{\|}_{F_{0}}\geq{\varepsilon}$. Because the linear operator $S\colon E_{1}\to E_{0}$ is compact, there is $\eta\in E_{0}$ and subsequences, still denoted by $T_{\nu}$ and $\xi_{\nu}$, such that $S\xi_{\nu}\to\eta$ in $E_{0}$. Hence

[TABLE]

Contradiction. Here the two inequalities are obtained by first adding twice zero and applying the triangle inequality, then using the definition of the operator norm. It remains to understand the vanishing of the three limits. For limit three this is obvious and limit two vanishes by hypothesis. Concerning limit one consider the family ${\mathcal{F}}:=\{T_{\nu}\}_{\nu\in{\mathbb{N}}}\mathop{\cup}\{T\}\subset{\mathcal{L}}(E_{0},F_{0})$. Each ${\mathcal{F}}$ orbit

[TABLE]

is bounded in $F_{0}$, even compact, as $T_{\nu}\zeta\to T\zeta$ by hypothesis. By the Banach–Steinhaus Theorem A.2.12 the family ${\mathcal{F}}$ is bounded in the operator norm.

c) The composition of continuous maps is continuous. But composing the continuous maps (A.2.9) and (A.2.10) is the map (A.2.11). ∎

Dual spaces and Reflexivity

Definition A.2.14** (Dual space).**

Given a normed linear space $X$, its dual space is the Banach space $X^{*}:={\mathcal{L}}(X,{\mathbb{R}})$ of continuous linear functionals $\lambda\colon X\to{\mathbb{R}}$.

Theorem A.2.15** (Hahn–Banach).**

Suppose $V$ is a linear subspace, closed or not, of a Banach space $X$ and $\lambda\in V^{*}$ is a continuous linear functional on $V$. Then there is a linear functional $\Lambda\in X^{*}$ that extends $\lambda$ and such that

[TABLE]

Proof.

See e.g. Brezis (2011, Cor. 1.2). ∎

Definition A.2.16** (Reflexive).**

A normed linear space $X$ is called reflexive if the canonical isometric linear map $J\colon X\to(X^{*})^{*}$ given by evaluation

[TABLE]

is surjective; see e.g. Bühler and Salamon (2018, §2.4). (Note that any linear isometry is injective.)

Remark A.2.17**.**

We highly recommend Brezis (2011, §3.5).

• a)

Kakutani’s Theorem: Reflexivity of a Banach space $E$ is equivalent to compactness of the closed unit ball of $E$ in the weak topology.

• b)

Closed linear subspaces of reflexive Banach spaces are reflexive.

• c)

A uniformly convex Banach space, so any Hilbert space, is reflexive.

Example A.2.18** (Non-reflexive Banach spaces).**

• (i)

The closed linear subspace $c_{0}$ of the Banach space $\ell^{\infty}$ in Example A.2.4 is not reflexive; see e.g. Salamon (2016, Exc. 4.37). Hence $\ell^{\infty}$ is not reflexive either by Remark A.2.17 b).

• (I)

More generally, let $C^{0}_{\rm bd}(X)$ be the space of bounded continuous functions on a locally compact topological space $X$ endowed with the sup norm (e.g. $\ell^{\infty}$). Then the Banach space $C^{0}_{\rm bd}(X)$ is reflexive iff $X$ is a finite set. See e.g. Conway (1985) (III $\S$11 Exc. 2 and V $\S$4 Exc. 3).

• (ii)

Consequently $C^{0}(Q)$ is not reflexive for compact manifolds $Q$ of $\dim Q\geq 1$. Neither is $C^{k}(Q)$ for $k\in{\mathbb{N}}$; this follows by reduction to the case $k=0$ using the graphs maps of differentials, see e.g. Weber (2017a, App. A).

The following theorem can be viewed as a substitute in the Banach space universe of the orthogonal projections available in the Hilbert space world.

Theorem A.2.19** (Projection theorem for reflexive Banach spaces).**

Let $E$ be a reflexive Banach space and $C\subset E$ a closed convex subset. For every $x\in E$ there is an element $y\in C$ which minimizes the distance to $x$, that is

[TABLE]

Proof.

The proof uses Kakutani’s theorem, see e.g. Brezis (2011, Cor. 3.23). ∎

Arzelà–Ascoli – convergent subsequences

Theorem A.2.20** (Arzelà–Ascoli Theorem).**

*Suppose $(Q,d)$ is a compact metric space and $C(Q)$ is the Banach space of continuous functions on $Q$ equipped with the sup norm. Then the following is true. A subset ${\mathcal{F}}$ of $C(Q)$ is pre-compact if and only if the family ${\mathcal{F}}$ is equicontinuous 181818 $\forall{\varepsilon}>0$ $\exists\delta>0$ such that $\mathopen{|}f(x)-f(y)\mathclose{|}<{\varepsilon}$ whenever $d(x,y)<\delta$ and $f\in{\mathcal{F}}$.

and pointwise bounded 191919 $\sup_{f\in{\mathcal{F}}}\mathopen{|}f(x)\mathclose{|}<\infty$ for every $x\in Q$. .*

For a proof see e.g. Rudin (1991, Thm. A.5) or Salamon (2017, App. C). By Theorem A.2.20 this generalizes to maps taking values in a metric space.

A.2.3 Calculus

An efficient presentation of the Fréchet derivative in Banach spaces $E,F$ is given in §1.1 of Ambrosetti and Prodi (1993) where §2.2 deals with the implicit function theorem (IFT). We follow Lang (1993, PART FOUR).

Fréchet or total derivative $df(x)$

Consider Banach spaces $E$ and $F$ and let $U$ be open in $E$. One says that a map $f\colon U\to F$ is differentiable at a point $x$ of $U$ if there is a continuous linear map $D\colon E\to F$ and a map $\psi$ defined for all sufficiently small elements $h$ in $E$ and with values in $F$ such that

[TABLE]

and such that $f$ near $x$ is given by the sum

[TABLE]

Set $h=0$ to see that it makes sense to set $\psi(0):=0$. Equivalently, denoting $o(h):=\left\|h\right\|\psi(h)$ the condition becomes

[TABLE]

Exercise A.2.21**.**

a) Differentiability at $x$ implies continuity at $x$. b) If $D\in{\mathcal{L}}(E,F)$ satisfies (A.2.12), then it is uniquely determined by $f$ and $x$.

Definition A.2.22** (Derivative and differential).**

Let $f\colon E\supset U\to F$ be differentiable at a point $x\in U$. Then the unique continuous linear operator $D$ satisfying (A.2.12) is called the (Fréchet) derivative of $f$ at $x$ and denoted by $df(x):=D\in{\mathcal{L}}(E,F)$. If $f$ is differentiable at every point of $U$ one says that $f$ is differentiable on $U$. In this case the map

[TABLE]

into the Banach space of continuous linear maps ${\mathcal{L}}(E,F)$ endowed with the operator norm is called the (Fréchet) differential of $f$. If $df$ is continuous one says that $f$** is of class $C^{1}$**, in symbols $f\in C^{1}(U,F)$. Higher derivatives

[TABLE]

are defined iteratively. If they exist and are continuous for $\ell=0,\dots k$, one says that $f$ is of class $C^{k}$. Here ${\mathcal{L}}^{\ell}(E,F)$ denotes the Banach space of $k$-fold multilinear maps $E\oplus\dots\oplus E\to F$. One says that $f$ is a smooth map, or of class $C^{\infty}$, if $f$ is of class $C^{k}$ for every $k\in{\mathbb{N}}_{0}$.

Gâteaux or all-directional derivative ${\partial}f(x)$

A map $f\colon E\subset U\to F$ between Banach spaces with $U$ open is said Gâteaux differentiable at $x\in U$ if for each $\xi\in E$ the directional derivative

[TABLE]

exists and defines a continuous linear map ${\partial}f(x)\colon E\to F$, $\xi\mapsto{\partial}_{\xi}f(x)$.

Exercise A.2.23**.**

Show that a) (Fréchet) differentiable implies Gâteaux differentiable, but b) not vice versa.

[Hint: b) Define $f\colon{\mathbb{R}}^{2}\to{\mathbb{R}}$ by $f(0,0):=0$ and by $f(u,v):=u^{4}v/(u^{6}+v^{3})$ off the origin. Show that ${\partial}_{\zeta}f(0,0)=0$ for every $\zeta\in{\mathbb{R}}^{2}$. So each directional derivative not only exists, but also the map $\zeta\mapsto{\partial}_{\zeta}f(0,0)$ is linear. Is $f$ continuous at $(0,0)$?]

A.2.4 Banach manifolds

Roughly speaking, a Banach manifold is a topological space (Hausdorff and paracompact) which is locally modeled on some Banach space such that all transition maps between the local models are differentiable. Differentiability of maps between Banach manifolds is defined in terms of differentiability of the corresponding maps between the local model Banach spaces. We recommend the book by Lang (2001) concerning differential geometry on Banach manifolds.

Suppose $X$ is a topological space and $k\in{\mathbb{N}}_{0}$ or $k=\infty$. A Banach chart $(V,\phi,E)$ for $X$ consists of a Banach space $E$ and a homeomorphism

[TABLE]

between open subsets. Two charts are called $C^{k}$** compatible** if the transition map

[TABLE]

is a $C^{k}$** diffeomorphism** (an invertible $C^{k}$ map with $C^{k}$ inverse). A $C^{k}$** Banach atlas for $X$** is a collection ${\mathcal{A}}$ of pairwise $C^{k}$ compatible Banach charts for $X$ such that the chart domains form a cover $\{V_{i}\}_{i}$ of $X$. Two atlases are called equivalent if their union forms an atlas.

Given such pair $(X,{\mathcal{A}})$, then $X$ is connected iff it is path connected. Furthermore, for $k\geq 1$ connectedness of $X$ implies that all model Banach spaces $E_{i}$ in the charts of ${\mathcal{A}}$ are isomorphic to one and the same Banach space $E$. In this case we say that $(X,{\mathcal{A}})$ is modeled on $E$.

Remark A.2.24** (Starting from just a set $X$).**

Alternatively starting with just a set $X$ one can construct a $C^{k}$ Banach atlas as follows. Choose a collection of bijections (the future coordinate charts)

[TABLE]

from a subset $V$ of $X$ onto an open subset $U$ of a Banach space $E$. There are two requirements: Firstly, the sets $V$ of all the charts must cover $X$ and, secondly, for each pair of charts the set $\phi(V\mathop{\cap}\widetilde{V})$ must be open in $E$. The notions $C^{k}$ compatibility and $C^{k}$ Banach atlas are unchanged. Given a $C^{k}$ Banach atlas ${\mathcal{A}}$, consider the collection ${\mathcal{B}}\subset 2^{X}$ of all subsets $\phi^{-1}(U^{\prime})$ of $X$ where $(\phi,V)$ runs through all charts of ${\mathcal{A}}$ and $U^{\prime}\subset E$ runs through all open subsets of $\phi(V)$. One checks that ${\mathcal{B}}$ is a basis of a topology and endows $X$ with that topology. Then ${\mathcal{A}}$ is an atlas on the topological space $X$ in the earlier sense. For an application see Exercise 2.8.6.

Definition A.2.25**.**

A $C^{k}$** Banach manifold** is a paracompact Hausdorff space $X$ endowed with an equivalence class of $C^{k}$ Banach atlases. If $k=0$ one speaks of a topological and if $k=\infty$ of a smooth Banach manifold. We often abbreviate smooth Banach manifold by Banach manifold. In case all model spaces are Hilbert spaces one speaks of a Hilbert manifold.

Definition A.2.26** (Maps between Banach manifolds).**

A continuous map

[TABLE]

between Banach manifolds is said to be of class $C^{k}$ if for all charts $\phi\colon X\supset V\to E$ and $\psi\colon Y\supset W\to F$ the chart representative $\psi\circ f\circ\phi^{-1}$ is of class $C^{k}$ as a map between open subsets of the Banach spaces $E$ and $F$.

A.3 Function spaces

Theorem A.3.1** (Properties of $L^{p}$ and Sobolev spaces).**

---

{labeling}*(separable)**

Fischer–Riesz Theorem: The spaces $L^{p}({\mathbb{R}},{\mathbb{R}}^{n})$ with norm $\mathopen{\|}\cdot\mathclose{\|}_{p}:=\mathopen{\|}\cdot\mathclose{\|}_{L^{p}}$ are Banach spaces whenever $1\leq p\leq\infty$.

*The spaces $L^{p}({\mathbb{R}},{\mathbb{R}}^{n})$ are separable202020 A topological space is called separable if it admits a dense sequence.

for $1\leq p<\infty$, but not separable for $p=\infty$.*

The spaces $L^{p}({\mathbb{R}},{\mathbb{R}}^{n})$ are reflexive for $1<p<\infty$, but not reflexive for $p=1,\infty$.

The Sobolev spaces $W^{k,p}({\mathbb{R}},{\mathbb{R}}^{n})$ have analogous properties in $p$.

For proofs of the three properties of $L^{p}$ see e.g. Theorems 4.8, 4.13, and 4.10, respectively, in Brezis (2011), for $W^{k,p}$ see Brezis (2011, Prop. 8.1). Concerning Sobolev spaces see also Adams and Fournier (2003).

Index

• (extension) Lemma 2.4.14, Definition 2.4.6

• (level operators) Lemma 2.4.14

• (ptw diff’able) Lemma 2.4.14

• (ptw diff) Definition 2.4.6

• (${\mathrm{sc}}$-isomorphism) Definition 2.3.24

• (${\mathrm{sc}}$-splittings) Definition 2.3.24

• ($Tf$ is ${{\mathrm{sc}}}^{0}$) Definition 2.4.6

• $(U,C,E)$ ${\mathrm{sc}}$-triple Definition 2.4.1

• $(U\triangleright F)_{m,k}$ non-symmetric product Definition 3.4.3

• $\mathopen{|}\cdot\mathclose{|}_{m}$ norm in Banach scale level $E_{m}$ Definition 2.1.7

• $\mathopen{|}S\mathclose{|}$ number of elements of finite set $S$ §A.1

• $2^{S}$ collection of all subsets of set $S$ §A.1

• $\bigcup\emptyset=\emptyset_{S}$, $\bigcap\emptyset=S$, for $\emptyset\subset 2^{S}$ Exercise A.1.2

• ${\partial}_{\xi}f$ directional derivative §A.2.3

• $\emptyset_{S}$ empty set in ambient universe $S$ §A.1

• $\mathopen{\|}T\mathclose{\|}$ operator norm Exercise A.1.58

• $\pi\colon Y\to X$ sc-bundle Definition 3.4.1

• $\subset$, $\subsetneq$ §A.1

• $C_{\mathrm{c}}(M,N)$ is $C(M,N)$ endowed with compact-open topology §A.1.1

• $\mathop{\mathrm{index}}T:=\dim\ker T-\dim{\rm coker\,}T$ Fredholm index §2.3.2

• ${\mathrm{sc}}^{0}$ scale continuous Definition 2.4.2

• ${\mathrm{sc}}^{1}(U,V)$ Remark 2.4.7

• ${\mathrm{sc}}^{k}$ scale differentiability Definition 2.4.15

• ${\mathrm{ssc}}^{k}$-manifold §2.8

• $A^{\cap E}$ Banach scale generated $A$ Definition 2.1.14

• $A^{k}$ shifted scale with levels $(A^{k})_{m}:=A_{k+m}$ Definition 2.1.6

• associated

• base retract Definition 3.4.7

• base retraction Definition 3.4.7

• atlas

• strong bundle – Definition 3.4.13

• ${\mathcal{A}}^{Y^{[0]}}_{X}$ induced sc-bundle atlas Exercise 3.4.16

• ${\mathcal{A}}^{Y}_{X}$ strong bundle atlas Definition 3.4.13

• b-topology item -

• ${\mathcal{B}}(0)$ family of all basic collections ${\mathcal{B}}_{A,V}$ §A.1.2

• $B^{\cap A}$ subscale of $A$ generated by $B\subset A$ Definition 2.1.4

• $B^{A}={\rm Map}(A,B)$ set of all maps $f\colon A\to B$ §A.1

• Baire’s Theorem Theorem A.1.54

• Banach

• manifold Definition A.2.25

• space Definition A.2.1

• splits Definition A.2.3

• subscale Definition 2.1.13

• subspace Definition A.2.1

• Banach scale Definition 2.1.7

• completion – Example 2.2.7

• reflexive – Definition 2.1.11

• separable – Definition 2.1.11

• Banachable space footnote 13

• base retract

• associated – Definition 3.4.7

• base retraction

• associated – Definition 3.4.7

• basic

• collection Definition A.1.33

• open sets §A.1.1

• basis of the topology Definition A.1.5

• boundary point §2.8

• bounded

• linear map between TVS’s Definition A.1.29

• linear operator §A.2.2

• subset of TVS Definition A.1.29

• bounded-open topology item -

• bundle chart

• strong – Definition 3.4.12

• $B_{x}({\varepsilon})$ open ball of radius ${\varepsilon}$ about $x$ Definition A.1.42

• c-topology item -

• is the compact-open topology §A.1.2

• $C(M,N)$ continuous maps from $M$ to $N$ Definition A.1.19

• $C^{1}(U,F)$ Definition A.2.22

• Cauchy sequence Definition A.1.50, Definition A.2.1

• chart

• Banach – §A.2.4

• $C^{k}(\SS^{1})$ is a Banach scale Exercise 2.2.2

• $C^{k}_{\rm bd}({\mathbb{R}})$ is not a Banach scale Example 2.2.1

• closed set Definition A.1.3

• codimension §2.3

• codomain §A.1

• cokernel §2.3.2

• collection §A.1

• compact

• linear operator Definition A.1.29, Definition A.2.9

• pre- – §A.1.1

• set §A.1.1

• compact-open topology footnote 17, item -, §A.1.1, item (c)

• compatible topology Definition A.1.26

• complement

• of Banach subspace Definition A.2.3

• topological – Example A.2.4

• complete

• metric space Definition A.1.52

• norm Definition A.2.1

• completion scale Example 2.2.7

• constant scale Definition 2.1.3

• continuity

• diagonal – in norm Remark 2.4.8

• horizontal – in compact-open topology Remark 2.4.8

• w.r.t. compact-open topology item (c)

• continuous

• at [math] Exercise A.1.32

• at a point Definition A.1.19

• continuously scale differentiable Definition 2.4.6

• convergence in topological space Definition A.1.18

• convex set footnote 11

• corner point of complexity $k\geq 2$ §2.8

• counter-examples:

• bump running to infinity Example 2.2.1

• not a Banach scale Example 2.2.1, Example 2.2.7

• not complemented Example A.2.4

• sub-basis only Remark A.1.25

• cover §A.1.1

• sets §A.1.1

• $d_{\infty}$ supremum metric Proposition A.1.45

• decompress Chapter 3

• decompressing domain Definition 3.2.3

• decompression of $f$ Chapter 3

• degeneracy index §2.7

• stratification §3.3.2

• dense Definition A.1.9

• dense subset Definition A.1.9

• derivative Definition A.1.40

• diagonal – Definition 2.4.6

• directional – §A.2.3

• Gâteaux – §A.2.3

• of $f$ at $x$ Definition A.2.22

• on Banach space Definition A.2.22

• on TVS Definition A.1.40

• $Df(x)$ ${\mathrm{sc}}$-derivative Definition 2.4.6

• $df(x)\in{\mathcal{L}}(E,F)$ derivative of $f$ at $x$ Definition A.2.22

• diagonal

• continuity in norm Remark 2.4.8

• derivative Definition 2.4.6

• differential item (iii)

• map Definition 2.4.4

• diffeomorphism §A.2.4

• differentiable

• at a point §A.2.3

• map between TVS Definition A.1.40

• differential

• diagonal – item (iii)

• of $f$ Definition A.1.40, Definition A.2.22

• direct sum

• of Banach spaces Definition A.2.2

• directional derivative §A.2.3

• distance function §A.1.3

• $D_{\ell}f(x):=Df(x)\mid_{E_{\ell}}\colon E_{\ell}\to F_{\ell}$ item (b)

• domain §A.1

• double

• filtration Definition 3.4.3

• scale Definition 3.4.3

• dual space Definition A.2.14

• $d_{X}$ degeneracy index on M-polyfold $X$ Definition 3.3.14

• empty set §A.1

• equicontinuous Definition A.1.48, Theorem A.2.20

• exponential

• law §A.1

• map §A.1

• $f^{-1}V$ pre-image §A.1.1

• $f\colon A\to B$ map §A.1

• $f\colon O\to O^{\prime}$ ${\mathrm{sc}}$-smooth retract map Definition 3.2.3

• $f\colon U\to V$ is of class ${\mathrm{sc}}^{k}$ Convention 2.4.3

• ${\mathcal{F}}(q)$ orbit through $q$ Theorem A.1.55

• $F\oplus G$ direct sum Banach scale Exercise 2.1.8

• family §A.1

• equicontinuous Theorem A.2.20

• equicontinuous – Definition A.1.48

• of all basic collections §A.1.2

• pointwise bounded Theorem A.2.20

• filter base footnote 9

• finite set §A.1

• Floer homology Chapter 1

• Fréchet derivative

• on Banach space Definition A.2.22

• on TVS Definition A.1.40

• Fredholm

• index §2.3.2, Definition 2.3.24

• operator §2.3.2

• freedom of speech Figure 3.3

• function §A.1

• Gâteaux

• derivative §A.1.2

• differentiable §A.1.2

• at $x\in U$ §A.2.3

• graph of a map §A.1

• growth function

• of Floer PDE Example 2.2.10

• Hausdorff (or $T_{2}$) Definition A.1.21

• topology Definition A.1.21

• hereditary properties §A.1.1

• Hessian

• scale – Exercise 2.4.17

• Hilbert manifold Definition A.2.25

• horizontal continuity in compact-open topology Remark 2.4.8

• implicit function theorem (IFT) §A.2.3

• index

• degeneracy – §2.7

• indices

• lifting – Lemma 2.5.5

• induced

• map Definition 2.4.4

• ${\mathrm{sc}}$-map of height $\ell$ Definition 2.4.4

• scale Definition 2.1.4

• topology §A.1.1

• interior point §2.8

• isomorphism

• of trivial-strong-bundles Definition 3.4.5

• toplinear §A.2.2

• iterated

• tangent bundle §2.4

• tangent map §2.4

• kernel Banach scale §2.3.2

• ${\mathcal{L}}(E,F)$ bounded linear operators with operator norm topology §A.2.2

• ${\mathcal{L}}(X,Y$ cont.lin.ops. §A.1.2

• $\mathcal{L}_{\mathrm{b}}(X,Y)$ bounded-open topology item -

• $\mathcal{L}_{\mathrm{c}}(E,F)$ compact-open topology Convention 2.5.1

• $\mathcal{L}_{\mathrm{c}}(X,Y)$ compact-open topology item -

• Lemma

• Slice – §A.1.2

• level Definition 2.1.7

• of a scale Definition 2.1.1

• operator Chapter 2, item (ii), Lemma 2.4.14

• of ${\mathrm{sc}}$-derivative $Df(x)$ item (b)

• preserving item (i), item (i)

• regularity §2.3.2, Proposition 2.3.25

• levels of sc-manifolds §2.8

• lifting indices Lemma 2.5.5

• linear operator

• bounded Exercise A.1.58

• ${\mathcal{L}}^{k}(X,Y)$ $k$-fold multilinear maps §A.2.2

• local basis of the topology at $x$ Definition A.1.11

• local generator

• of sub-M-polyfold Definition 3.3.12

• locally

• convex TVS §A.1.2

• finite §A.1.1

• loop space Example 2.8.4

• $\mathcal{L}_{\mathrm{p}}(X,Y)$ point-open topology item -

• $\mathcal{L}_{\mathfrak{S}}(X,Y)$ Theorem A.1.35

• $\mathcal{L}_{\mathrm{sc}}(E,F)$ ${\mathrm{sc}}$-operators Definition 2.3.3

• $\mathcal{L}_{\mathrm{sc}}^{+}$ ${\mathrm{sc}}^{+}$-operators Definition 2.3.11

• M-polyfold Definition 3.3.2

• atlas Definition 3.3.1

• atlases

• equivalent Definition 3.3.1

• charts Definition 3.3.1

• diffeomorphism Definition 3.3.3

• levels Definition 3.3.5

• sub –

• local generator Definition 3.3.12

• sub- – Definition 3.3.12

• tame – Definition 3.3.17

• M-polyfold charts

• compatible – Definition 3.3.1

• transition map Definition 3.3.1

• M-polyfold map

• “freedom of speech” Figure 3.3

• sc-smooth – Definition 3.3.3

• manifolds

• methods to define – §3.3

• map

• between sets §A.1

• continuous – Definition A.1.19

• metric Definition A.1.41

• space Definition A.1.41

• complete – Definition A.1.52

• topology Definition A.1.42

• moduli spaces Chapter 1

• neighborhood §A.1.1

• symmetric – item (iii)

• Nemitski operator §A.1.2

• non-symmetric product Definition 3.4.3

• norm Definition A.1.56

• operator – Exercise A.1.58, §A.2.2

• semi- – Definition A.1.56

• normed vector space Definition A.1.56

• $O$ ${\mathrm{sc}}$-smooth retract Definition 3.2.1

• open

• cover §A.1.1

• sets Definition A.1.3

• open problem Exercise 3.3.4

• operator

• compact linear – Definition 2.1.7

• sc-linear item (i)

• operator norm Exercise A.1.58, §A.2.2

• topology §A.1.4

• orbit

• family ${\mathcal{F}}$- – Theorem A.1.55

• p-topology item -

• is the point-open topology §A.1.2

• $p\colon K\to O$ trivial-strong-bundle retract Definition 3.4.9

• $p\colon TO\to O^{1}$ tangent bundle of ${\mathrm{sc}}$-retract Chapter 3

• $P\colon Y\to X$ strong bundle Definition 3.4.14

• paracompact Definition A.1.21

• parametrized

• solutions Chapter 1

• part of $B$ in level $A_{m}$ Definition 2.1.4

• partial quadrant Definition 2.4.1

• point-open topology item -

• points §A.1

• of regularity $m$ Definition 2.1.1

• pointwise bounded Theorem A.2.20

• polyfolds Chapter 1

• pre-basis on a set Definition A.1.16

• pre-compact §A.1.1

• set Definition A.1.29

• sets Definition A.2.9

• pre-compact set Definition A.1.29

• pre-image §A.1.1

• principal part of section Definition 3.4.11

• projection Definition A.2.10, Definition 2.3.6, §3.1

• onto quotient space item (ii)

• theorem for Banach spaces Theorem A.2.19

• quotient

• norm Definition A.2.5, §2.3

• space Definition A.2.5

• projection onto – item (ii)

• refinement §A.1.1

• reflexive Definition A.2.16

• Banach scale Definition 2.1.11

• regularity Lemma 2.4.12

• level – §2.3.2, Proposition 2.3.25

• regularizing §2.3.2, Proposition 2.3.25

• retract §3.1

• retract map

• decompressing a – Definition 3.2.3

• sc-smooth – Chapter 3

• strong – Definition 3.4.10

• retraction §3.1

• revolution

• the – happens here Chapter 2

• ${\mathfrak{S}}$-family §A.1.2

• ${\mathfrak{S}}$-topology on ${\mathcal{L}}(X,Y)$ Theorem A.1.35

• sc abbreviates

• scale Chapter 2

• scale continuous Chapter 2

• sc-Banach space Definition 2.1.7

• tangent bundle of – §2.4

• sc-bundle Definition 3.4.1

• atlases

• induced – Exercise 3.4.16

• sc-bundles over M-polyfolds §3.4

• sc-charts §2.8

• sc-compatible – §2.8

• sc-compact operator Definition 2.3.10

• sc-complement Definition 2.3.14, Definition 2.3.14

• sc-continuous map Definition 2.4.2

• sc-derivative

• $Df(x)$ Definition 2.4.6

• of $f$ at $x$ Definition 2.4.6

• sc-diffeomorphism Theorem 2.7.2, §2.8

• between sc-manifolds Definition 2.8.3

• sc-differentiable

• strongly – §2.8

• sc-direct sum Exercise 2.1.8

• sc-Fredholm operator Definition 2.3.24, Definition 2.3.31

• sc-Hessian item (b)

• sc-isomorphism Definition 2.3.8

• sc-manifold Definition 2.8.2

• Hilbert – Definition 2.8.2

• levels of – §2.8

• shifted – §2.8

• tangent bundle of – Exercise 2.8.6

• tangent vector to – §2.8

• sc-operator Definition 2.3.3

• sc-projection Definition 2.3.6

• induced sc-splitting Exercise 2.3.16, Lemma 2.3.7

• sc-retract Definition 3.2.1

• tame – Definition 3.3.17

• tangent bundle of – Chapter 3

• sc-section

• of strong bundle Definition 3.4.17

• of trivial-strong-bundle Definition 3.4.11

• principal part of – Definition 3.4.11

• sc-smooth Definition 2.4.15

• map Definition 2.8.3

• retract

• tangent bundle of – Lemma 3.2.8

• retract map Chapter 3, Definition 3.2.3

• retraction Definition 3.2.1

• tame – Definition 3.3.15

• splicing §3.2.1

• sc-splitting Definition 2.3.14

• sc-structure Definition 2.1.7

• sc-subspace Definition 2.1.14

• sc-triple Definition 2.4.1

• tangent – Lemma 3.2.5

• sc+-operator Definition 2.3.11

• sc+-section

• of strong bundle Definition 3.4.17

• of trivial-strong-bundle Definition 3.4.11

• scale Definition 2.1.1

• Banach space Definition 2.1.7

• Banach sub– Definition 2.1.13

• bounded Definition 2.3.3

• continuous Definition 2.3.3, Definition 2.4.2

• double – Definition 3.4.3

• Hessian Exercise 2.4.17

• linear operator item (i)

• structure Definition 2.1.1

• on a Banach space Definition 2.1.7

• scale calculus

• history Chapter 1

• motivated by shift map Chapter 1

• second countable Definition A.1.8

• section

• of strong bundle Definition 3.4.17

• of trivial-strong-bundle Definition 3.4.11

• principal part of – Definition 3.4.11

• semi-norm Definition A.1.56

• separable footnote 20, Definition A.1.9

• Banach scale Definition 2.1.11

• separating points Definition A.1.21

• set

• finite – §A.1

• shift map Chapter 1

• shifted scale Definition 2.1.6

• Slice Lemma §A.1.2

• smooth

• map Definition A.2.22

• points Definition 2.1.1

• Sobolev spaces

• Hilbert space valued – Definition 2.2.9

• solutions

• parametrized Chapter 1

• unparametrized Chapter 1

• splicing

• core §3.2.1

• sc-smooth – §3.2.1

• splitting

• of Banach space Definition A.2.3

• $\SS^{1}$ unit circle §2.2

• ssc-manifold §2.8

• stratification

• degeneracy index – §3.3.2

• strong bundle Definition 3.4.14

• atlas Definition 3.4.13

• chart Definition 3.4.12

• compatible – Definition 3.4.13

• sc-section of – Definition 3.4.17

• ${\mathrm{sc}}^{+}$-section Definition 3.4.17

• section of – Definition 3.4.17

• trivial Definition 3.4.4

• total space §3.4.1

• trivial– Definition 3.4.9

• strong retract map Definition 3.4.10

• strongly scale differentiable §2.8

• sub-basis of the topology Definition A.1.7

• sub-M-polyfold Chapter 3, Definition 3.3.12

• sublevels Definition 2.1.1

• subscale Definition 2.1.2

• Banach – Definition 2.1.13

• generated by $B$ Definition 2.1.4

• subset

• bounded – Definition A.1.29

• topology §A.1.1

• subspace §A.1.1, §A.2

• superlevels Definition 2.1.1

• supremum metric Proposition A.1.45

• symmetric

• neighborhood item (iii)

• $T(U,C,E):=(TU,TC,TE)$ Definition 2.4.5

• ${\mathcal{T}}^{\cap A}:={\mathcal{T}}\mathop{\cap}A$ subset topology §A.1.1

• $T_{2}$ – Hausdorff topological space Definition A.1.21

• $T_{4}$ – normal topological space Definition A.1.21

• $T{\mathcal{A}}$ M-polyfold atlas for $TX$ §3.3

• tame

• M-polyfold Definition 3.3.17

• retract Definition 3.3.17

• retraction Definition 3.3.15

• tangent bundle §2.4

• iterated – §2.4

• of sc-manifold Exercise 2.8.6

• of ${\mathrm{sc}}$-smooth retract Lemma 3.2.8

• of ${\mathrm{sc}}$-triple Definition 2.4.5

• tangent map Definition 2.4.6

• iterated – §2.4

• of M-polyfold map Definition 3.3.10

• tangent vector to sc-manifold §2.8

• target §A.1

• ${\mathcal{T}}_{d}$ metric topology Definition A.1.42

• $Tf\colon TO\to TO^{\prime}$ Definition 3.2.11

• $Tf\colon TU\to TV$ tangent map Definition 2.4.6

• theorem

• projection – for Banach spaces Theorem A.2.19

• Theorem of

• Arzelà–Ascoli Theorem A.1.55, Theorem A.2.20

• Baire Theorem A.1.54

• Banach–Steinhaus Theorem A.2.12

• Fischer-Riesz Theorem A.3.1

• Kakutani item a)

• time shift Chapter 1

• $T_{m}\colon E_{m}\to F_{m}$ level operator item (ii)

• $TO\to O^{1}$ tangent bundle of ${\mathrm{sc}}$-smooth retract $O$ Lemma 3.2.8

• toplinear isomorphism §A.2.2

• topological

• complement Example A.2.4

• space Definition A.1.3

• Hausdorff (or $T_{2}$) Definition A.1.21

• normal (or $T_{4}$) Definition A.1.21

• paracompact Definition A.1.21

• subspace of – §A.1.1

• vector space Definition A.1.26

• topology Definition A.1.3

• basis for some – Theorem A.1.14

• basis of the – Definition A.1.5

• bounded-open – item -

• c- – §A.1.2

• compact-open – footnote 17, item -, item (c)

• compatible with vector space operations Definition A.1.26

• discrete – Remark A.1.25

• indiscrete – Exercise A.1.15

• induced – §A.1.1

• local basis of – Definition A.1.11

• on $TX$ §3.3

• p- – §A.1.2

• point-open – item -

• pre-basis for a – Definition A.1.16

• ${\mathfrak{S}}$- – Theorem A.1.35

• sub-basis of the – Definition A.1.7

• subset – §A.1.1

• trivial – Exercise A.1.15

• transition map §A.2.4

• vector bundle – §3.4

• trivial strong bundles Definition 3.4.4

• trivial-strong-bundle

• isomorphism Definition 3.4.5

• map Definition 3.4.5

• retract Definition 3.4.9, Definition 3.4.9

• retraction Definition 3.4.7

• sc-section of – Definition 3.4.11

• ${\mathrm{sc}}^{+}$-section Definition 3.4.11

• section of – Definition 3.4.11

• total space §3.4.1

• TVS

• bounded linear map between – Definition A.1.29

• locally convex – §A.1.2

• topological vector space Definition A.1.26

• $T_{x}X$ M-polyfold tangent space §3.3

• $U^{1}\oplus E^{0}$ footnote 13

• $U^{\rm C}:=X\setminus U$ complement Definition A.1.3

• uniform

• convergence §A.1.3

• on compact subsets Exercise A.1.47

• topology §A.1.4

• unparametrized

• solutions Chapter 1

• $U_{x}\subset X$ means $x\in U\subset X$ §A.1.1

• vector space

• normed – Definition A.1.56

• $W^{k,p}(\SS^{1})$ is a Banach scale Example 2.2.3

• $W^{k,p}_{\delta_{k}}({\mathbb{R}})$ is a Banach scale Exercise 2.2.4

• $X^{*}:={\mathcal{L}}(X,{\mathbb{R}})$ dual space Definition A.2.14

• $X^{k}$ shifted sc-manifold structure §2.8

• $X_{m}$ level $m$ of sc-manifold §2.8

• $x_{n}\to z$ in topological space Definition A.1.18