On a characterization of spaces satisfying open mapping and equivalent theorems

TL;DR

This paper characterizes classes of topological vector spaces where open-mapping, continuous-inverse, and closed-graph theorems are equivalent, identifying the maximal classes with these properties.

Contribution

It identifies the largest classes of locally-convex topological vector spaces, namely barreled Ptak and infra-Ptak spaces, where key theorems are equivalent and preserved under common operations.

Findings

Barreled Ptak spaces form the largest class with these properties.

The class is closed under quotients, closed graphs, and continuous images.

Similar results hold for barreled infra-Ptak spaces.

Abstract

For classes of topological vector spaces, we analyze under which conditions open-mapping, continuous-inverse, and closed-graph properties are equivalent. Here, closure under quotients with closed subspaces and closure under closed graphs are sufficient. We show that the class of barreled Ptak spaces is exactly the largest class of locally-convex topological vector spaces, which contains all Banach spaces, is closed under quotients with closed subspaces, is closed under closed graphs, is closed under continuous images, and for which an open-mapping theorem, a continuous-inverse theorem, and a closed-graph theorem holds. An analogous, weaker result also holds for the strictly larger class of barreled infra-Ptak spaces.

Full text

We show that the class of barreled Pták spaces is exactly the largest class of locally-convex topological vector spaces, which contains all Banach spaces, is closed under quotients with closed subspaces, is closed under closed graphs, is closed under continuous images, and for which an open-mapping theorem, a continous-inverse theorem, and a closed-graph theorem holds.

An analogous, weaker result also holds for the strictly larger class of barreled infra-Pták spaces.

1. Introduction

This is a short paper in the field of topological vector spaces (t.v.s.), concerned with theorems on open mappings, continuous inverses, and closed graphs. These theorems have a long history, with many applications in different branches of functional analysis [Wer97, Mat98, Alt06, AV05]. Initially only formulated for Banach spaces, one line of research was to extend these theorems to very general classes of spaces [Ptá58, Ptá59, Ptá60, Ptá62, Ptá65, Ptá66, Ptá69, Ptá74, HM62, Hus62, Hus64a, Hus64b, Kri71, Sch71, Val78, Ada83, Ada86, Sim89, Rod91]. While this research states such theorems for linear mappings $u\colon E\to F$ with $E$ taken from one class $\mathcal{A}$ of t.v.s. and $F$ taken from a possibly different class $\mathcal{B}$, we approach the topic differently. We only allow $E$ and $F$ to come from the very same class of t.v.s. $\mathcal{C}$, and we ask, under which conditions on $\mathcal{C}$ the open-mapping theorem, the continuous-inverse theorem, and the closed-graph theorem are actually equivalent and hold. For the equivalence of these theorems for a class $\mathcal{C}$, the crucial insight is that $\mathcal{C}$ needs closure properties weaker than expected. Besides closure under quotients with closed subspaces, additionally, only closure under closed graphs is needed, not closure under closed finite products or closed subspaces. This insight leads to a characterization result, showing that the class of barreled Pták spaces is the natural habitat of these theorems, and that at least for locally-convex spaces, the barrier of being barreled and Pták cannot be overcome without losing important closure properties. As research in the 1960s considered Pták and barreled spaces already, this paper thus may explain, why research on these topics faded out in the 1970s.

2. Equivalences

In this work, we use notation, definitions, and results from the excellent textbook of Schaefer [Sch71]. Throughout, w.l.o.g. we assume that all topological vector spaces (t.v.s.) are $T_{0}$. Hence, they are fully regular. In particular, they are $T_{2}$. 111For the notions of $T_{0}$, $T_{2}$, and fully regular, and for the properties of t.v.s.being uniform and in case of $T_{0}$ being fully regular, we refer the reader to any topology textbook, e.g. [vQ01]. Recall that a map $u\colon E\to F$ is called closed, if the set $\mathrm{Graph}(u)=\{(e,u(e))\mid e\in E\}$ is a closed subset of $E\times F$. Map $u$ is called open, if for every open set $O\subseteq E$ the image $u(O)$ is open in $u(E)$.

We define three properties for a class $\mathcal{C}$ of t.v.s..

(O) Open-mapping property**: **

For every pair of t.v.s. $E$ and $F$ in $\mathcal{C}$ it holds that every surjective, linear, continuous map $u\colon E\to F$ is open.

(C) Continuous-inverse property**: **

For every pair of t.v.s. $E$ and $F$ in $\mathcal{C}$ it holds that every bijective, linear map $u\colon E\to F$ is continuous iff its inverse $u^{-1}$ is continuous.

(G) Closed-graph property**: **

For every pair of t.v.s. $E$ and $F$ in $\mathcal{C}$ it holds that every linear map $u\colon E\to F$ is closed iff it is continuous.

We say that a class $\mathcal{C}$ of t.v.s. is closed under closed graphs, if for every $E$ and $F$ in $\mathcal{C}$ and every linear, closed map $u\colon E\to F$ its graph $\mathrm{Graph}(u)$ is in $\mathcal{C}$. A class $\mathcal{C}$ of t.v.s. is closed under quotients with closed subspaces, if for every $E$ in $\mathcal{C}$ and $A$ a closed subspace of $E$, the quotient space $E/A$ is in $\mathcal{C}$. Furthermore, we say that a class $\mathcal{C}$ of t.v.s. has the OCG-equivalence property, if it is closed under quotients with closed subspaces, and if it is closed under closed graphs.

Theorem 1**.**

Let $\mathcal{C}$ be a class of ($T_{0}$) t.v.s. satisfying the OCG-equivalence property. Then properties (O), (C), and (G) are equivalent for $\mathcal{C}$.

The following arguments in the proof of the above theorem are well-known and thus not new. Presenting them needs justification. We give three reasons: (1) emphasis on where exactly the closure-properties of the class $\mathcal{C}$ are needed, (2) first-time crystal-clear presentation of these equivalences in this general setting, not found in textbooks in functional analysis, and (3) for the sake of completeness.

Proof.

(O) implies (C): Let $E$ and $F$ be t.v.s. in $\mathcal{C}$, and let $u\colon E\to F$ be bijective, linear, and continuous. By (O), $u$ is open. Hence, $u^{-1}$ is continuous. Analogously, argue for $u^{-1}$.

(C) implies (O): Let $E$ and $F$ be t.v.s. in $\mathcal{C}$, and let $u\colon E\to F$ be surjective, linear, and continuous. Subspace $N=u^{-1}(0)$ is closed by continuity of $u$. As $\mathcal{C}$ is closed by quotients with closed subspaces, $E/N$ is in $\mathcal{C}$. The induced map $u_{0}\colon E/N\to F$ is bijective and continuous. By (C), $u_{0}^{-1}$ is continuous. Hence, $u_{0}$ is open. Then finally, the map $u=p\circ u_{0}$ is open as composition of open maps, where $p\colon E\to E/N$ denotes the linear, continuous, and open projection.

(C) implies (G): Let $E$ and $F$ be t.v.s. in $\mathcal{C}$, and let $u\colon E\to F$ be linear. Define the bijective, linear map $v\colon E\to\mathrm{Graph}(u)$ by $v(e)=(e,u(e))$. Let $p_{E}$ and $p_{F}$ denote the linear, continuous projections from $E\times F$, respectively. If $u$ is continuous, then by Prop. 1 below, $\mathrm{Graph}(u)$ is closed. And if $\mathrm{Graph}(u)$ is closed, then it is in $\mathcal{C}$ by closure under closed graphs. As $v^{-1}=p_{E}\colon\mathrm{Graph}(u)\to E$ is bijective, linear, and continuous, the map $v$ is continuous by application of (C).

(G) implies (C): Let $E$ and $F$ be t.v.s. in $\mathcal{C}$. Define $s\colon E\times F\to F\times E$ by $s(x,y)=(y,x)$. Clearly, $s$ is a topological isomorphism. Let $u\colon E\to F$ be bijective and linear. By (G), the map $u$ is continuous iff $\mathrm{Graph}(u)$ is closed. This holds iff $\mathrm{Graph}(u^{-1})=s(\mathrm{Graph}(u))$ is closed. Again by (G), the former holds iff $u^{-1}$ is continuous. ∎

As we could only find proofs of the following proposition in the context of Banach spaces, we give a proof in full generality for the sake of completeness.

Proposition 1** (Folklore).**

If a map between topological $T_{2}$ spaces is continuous, then it is closed.

Proof.

Let $E$ and $F$ be topological $T_{2}$ spaces, and let $u\colon E\to F$ be a continuous map. Define map $v\colon E\times F\to E\times F$ by $v(e,g)=(e,u(e))$. Then $v$ is continuous and $v(E\times F)=\mathrm{Graph}(u)$. Consider an arbitrary point $(e,f)$ in the closure $\overline{\mathrm{Graph}(u)}$. Then there exists a filter $C$ containing $\mathrm{Graph}(u)$ and converging to $(e,f)$. By continuity of $v$, the image filter $v(C)$ converges to $v(e,f)=(e,u(e))$. As $E\times F$ is in $C$, we have $\mathrm{Graph}(u)$ in $v(C)$. The set of intersections of sets from $C$ and $v(C)$ (i.e., $C\cap v(C)=\{A\cap B\mid A\in C,B\in v(C)\}$) constitutes a filter base for a finer filter $D\supseteq C,v(C)$. Filter $D$ contains $\mathrm{Graph}(u)$ and converges both to $(e,f)$ and $(e,u(e))$, respectively. As $E\times F$ is $T_{2}$ as the product of two $T_{2}$ spaces, we have the uniqueness of the limit $(e,f)=(e,u(e))$. Hence, $(e,f)$ is in $\mathrm{Graph}(u)$, showing closedness of $\mathrm{Graph}(u)$. ∎

Note that a class $\mathcal{C}$ of t.v.s. is closed under closed graphs, if it is closed under finite products (i.e., with $E$ and $F$ in $\mathcal{C}$, we have $E\times F$ in $\mathcal{C}$) and closed under closed subspaces (i.e., with $E$ in $\mathcal{C}$, every closed subspace of $E$ is in $\mathcal{C}$). Main insight of above theorem is that the weaker property of closure under closed graphs suffices. Closure under finite products or closure under closed subspaces is not necessary.

It is well-known that the classes (all assumed $T_{0}$) of complete locally-convex spaces (l.c.s.), complete metrizable t.v.s. (Fréchet), Banach spaces, and nuclear spaces all satisfy the OCG-equivalence property.222For complete l.c.s.: finite products, [Sch71, II.5.2]; closed subspaces, [Sch71, I.2.1, II.6.1]; quotients under closed subspaces, [Sch71, I.2.3, II.6.1]. For complete metrizable spaces: finite products, [Sch71, I.2, Ex. 1(b)], [Bou95, Ch. II, §3.5, §3.9]; closed subspaces, [Sch71, I.2.1], [Bou95, Ch. II, §3.4, §3.9]; quotients under closed subspaces, [Sch71, I.2.3, I.6.3]. For Banach spaces: finite products, [Sch71, II.2.2], [Bou89, Ch. IX, §3.4]; closed subspaces, trivial; quotients under closed subspaces, [Sch71, I.2.3, II.2.3], [Bou89, Ch. IX, §3.4]. For nuclear spaces, see [Sch71, III.7.4]

In contrast, it is unclear if subclasses of barreled spaces, Pták spaces, or Baire spaces satisfy the property of OCG-equivalence, because in general, barreled spaces and Baire spaces are not closed under closed subspaces, and Pták spaces are not closed under finite products. At least, barreled spaces are closed under finite products and quotients with closed subspaces, see [Sch71, II.7.1 comment and Cor. 1], and Pták spaces are closed under closed subspaces and quotients with closed subspaces, respectively, see [Sch71, IV.8.2, IV.8.3 Cor. 3].

3. Characterization

For the notions of l.c.s., barreled space, Pták space, and infra-Pták space, we refer the reader to [Sch71, II.4, II.7, IV.8], respectively. For more information on barreledness and related properties, see also [Ada70, Val71a, Val71b, Val72a, Val72b, VD72, Val73, Val79, Val81, Sax74, Hol77, PC87].

Recall that a linear map $e\colon E\to F$ is called nearly-open, if for each [math]-neighborhood $U\subseteq E$, $u(U)$ is dense in some [math]-neighborhood in $u(E)$.

We say that a class $\mathcal{C}$ of t.v.s. is closed under continuous images, if for every $E$ in $\mathcal{C}$, every l.c.s. $F$, and every injective, linear, continuous, and nearly-open map $u\colon E\to F$, its image $u(E)$ is in $\mathcal{C}$.

Proposition 2**.**

The classes of Banach spaces, barreled Pták spaces, and barreled infra-Pták spaces are closed under continuous images.

Proof.

Let $F$ be an arbitrary l.c.s., and let $u\colon E\to F$ be an arbitrary injective, linear, continuous, and nearly-open map. Space $u(E)$ is l.c.s. as a subspace of $F$.

If $E$ is an (infra-)Pták space, then map $u$ is a topological homomorphism by [Sch71, IV.8.3, Thm.]. Hence, $u(E)$ is isomorphic to $E$ and thus an (infra-)Pták space.

If $E$ is a Banach space, then it is a Fréchet space, and thus a Pták space by the theorem of Krein-S̆mulian, see [Sch71, IV.6.4, Thm.]. By the above argument, $u(E)$ is isomorphic to $E$ and thus a Banach space.

We show that $u(E)$ is barreled, if $E$ is a barreled (infra)-Pták space. By [Sch71, IV.8.3, Thm.], map $u\colon E\to u(E)$ is an isomorphism. Let $B$ be an arbitrary Banach space, and let $v\colon u(E)\to B$ be an arbitrary linear and closed map. Then the composition map $v\circ u\colon E\to B$ is linear and closed, the latter because map $(u,\mathrm{id})\colon E\times B\to u(E)\times B$ is an isomorphism with $(u,\mathrm{id})\left(\mathrm{Graph}(v\circ u)\right)=\mathrm{Graph}(v)$. As $E$ is barreled, $B$ is $B_{r}$-complete, and $v\circ u$ is closed, map $v\circ u$ is continuous by the Thm. of Robertson-Robertson, [Sch71, IV.8.5, Thm.]. Hence, $v=(v\circ u)\circ u^{-1}$ is continuous. Finally, space $u(E)$ is barreled by the Thm. of Mahowald, [Sch71, IV.8.6]. ∎

Proposition 3**.**

The classes of Banach spaces, barreled Pták spaces, and barreled infra-Pták spaces are closed under closed graphs.

Proof.

The statement holds for Banach spaces, because Banach spaces are closed under finite products and closed subspaces.

Let $E$ and $F$ be arbitrary barreled (infra-)Pták spaces, and let $u\colon E\to F$ be an arbitrary linear and closed map. By the theorem of Robertson-Robertson, [Sch71, IV.8.5, Thm.], u is continuous. Note that the space $\mathrm{Graph}(u)$ is an l.c.s. as a closed subset of l.c.s. $E\times F$. Define the bijective and continuous map $v\colon E\to\mathrm{Graph}(u)$ by $v(e)=(e,u(e))$. The map $v$ is open and thus nearly-open, because its inverse $v^{-1}=p_{E}\colon\mathrm{Graph}(u)\to E$ is continuous. Now, $\mathrm{Graph}(u)$ is the continuous image of the barreled (infra-)Pták space $E$. The statement then follows from Prop. 2. ∎

Theorem 2** (Barreled Pták Characterization).**

The class of barreled Pták spaces is exactly the largest ($T_{0}$) class of l.c.s., which contains all Banach spaces, is closed under quotients with closed subspaces, is closed under closed graphs, is closed under continuous images, and for which an open-mapping theorem (O), a continuous-inverse theorem (C), or a closed-graph theorem (G) holds (and thus all of them).

Proof.

The classes of Banach spaces and of barreled Pták spaces both have the mentioned closure properties: they contain all Banach spaces, are closed under quotients with closed subspaces, are closed under closed graphs (Prop. 3), and are closed under continuous images (Prop. 2). It is well-known that property (O) holds for Banach spaces, and it also holds for barreled Pták spaces by [Sch71, IV.8.3, Cor. 1]. Consequently, for both of these classes, properties (O), (C) and (G) are equivalent (Thm. 1) and hold.

Let $\mathcal{C}$ be a maximal class of l.c.s. satisfying the assumed closure properties of the theorem. First of all, $\mathcal{C}$ satisfies all properties (O), (C), and (G), because it satisfies OCG-equivalence.

Let $E$ be an arbitrary l.c.s. in $\mathcal{C}$. We want to show that $E$ is barreled. Let $B$ be an arbitrary Banach space. We have $B$ in $\mathcal{C}$. Let $u\colon E\to B$ be an arbitrary linear, closed map. By (G), $u$ is continuous. Then by the theorem of Mahowald, [Sch71, IV.8.6], $E$ is barreled.

We want to show that $E$ is a Pták space. Let $F$ be an arbitrary l.c.s., and let $u\colon E\to F$ be an arbitrary linear, continuous, and nearly-open map. Subspace $N=u^{-1}(0)$ is closed, because $u$ is continuous. Hence, $E/N$ is in $\mathcal{C}$ by closure under quotients with closed subspaces. The map $u_{0}\colon E/N\to F$, associated with $u$, is injective, linear, continuous, and nearly-open. Thus, image $u(E)$ is in $\mathcal{C}$ by closure under continuous images. Applying (C) to bijective and continuous map $u_{0}\colon E/N\to u(E)$ yields that $u_{0}$ is open. Hence, $u_{0}$ is an isomorphism and thus $u$ a topological homomorphism by [Sch71, III, 1.2]. By [Sch71, IV.8.3, Thm.], $E$ is a Pták space.

Consequently, every space in $\mathcal{C}$ is a barreled Pták space. Finally, $\mathcal{C}$ must equal the class of barreled Pták spaces by maximality. ∎

We note the characterization of a barreled Pták space $E$ via its dual $E^{\prime}$: Here, every subspace $Q$ of $E^{\prime}$ is $\sigma(E^{\prime},E)$-closed, whenever $Q\cap A$ is $\sigma(E^{\prime},E)$-closed in $A$ for every equicontinuous subset $A$ of $E^{\prime}$, and every $\sigma(E^{\prime},E)$-bounded subset of $E^{\prime}$ is equicontinuous.

For a non-empty open subset $\Omega$ of $\mathbb{R}^{m}$, denote with $\mathcal{D}(\Omega)$ and $\mathcal{D}^{\prime}(\Omega)$ the spaces of test functions and distributions, respectively [Sch57, Sch58, Sch48]. Valdivia [Val77] showed that these spaces are not even infra-Pták. Hence, they fall out of the above framework. Maybe surprisingly, in sharp contrast, for the Schwartz space $\mathcal{S}$ of rapidly-decreasing and infinitely-differentiable functions, and the space of tempered distributions $\mathcal{S}^{\prime}$ the story is different. For a definition of these spaces, see e.g., [Sch71, III.8].

Proposition 4** (Maybe folklore).**

The Schwartz space $\mathcal{S}$ and the space of tempered distributions $\mathcal{S}^{\prime}$ are both barreled Pták spaces.

Proof.

As space $\mathcal{S}$ is a Montel space, [Sch71, IV.5.8], the strong dual $\left(\mathcal{S}^{\prime},\beta(\mathcal{S}^{\prime},\mathcal{S})\right)$ is a Montel space, [Sch71, IV.5.9]. As the strong topology $\beta(\mathcal{S}^{\prime},\mathcal{S})$ coincides with the topology of compact convergence $T_{c}$, $\mathcal{S}^{\prime}$ is a Montel space. Montel spaces are reflexive (by definition) and thus barreled, [Sch71, IV.5.6, Thm.]. Hence, $\mathcal{S}$ and $\mathcal{S}^{\prime}$ are barreled.

Space $\mathcal{S}$ is clearly a Frechét space, [Sch71, III.8]. Then by [Sch71, IV.8, Examples], both $\mathcal{S}$ and $\mathcal{S}^{\prime}$ are Pták spaces. ∎

In the same vein as above, we prove a characterization theorem for barreled infra-Pták spaces. For more information on infra-Pták spaces, see [Val75]. These spaces are more general then barreled Pták spaces. The missing closure under quotients with closed subspaces is exactly the differentiating property.

Theorem 3** (Barreled infra-Pták Characterization).**

The class of barreled infra-Pták spaces is exactly the largest ($T_{0}$) class of l.c.s., which contains all Banach spaces, is closed under closed graphs, is closed under continuous images, and for which an open-mapping theorem (O), a continuous-inverse theorem (C) or a closed-graph theorem (G) holds (and thus all of them).

Proof.

The classes of Banach spaces and of barreled infra-Pták spaces both have the mentioned closure properties: they contain all Banach spaces, are closed under closed graphs (Prop. 3), and are closed under continuous images (Prop. 2). It is well-known that properties (O), (C), and (G) hold for Banach spaces. Property (G) also holds for barreled infra-Pták spaces by [Sch71, IV.8.5, Thm.]. Property (G) implies (C) directly. We need to show (O). For this, let $u\colon E\to F$ be a surjective, linear, and continuous mapping between two barreled infra-Pták spaces $E$ and $F$. As $u$ is a surjective, linear map onto a barreled space, it is nearly open [Sch71, IV.8.2]. As $u$ is continuous and linear, its graph is closed. By Ptak’s general open mapping theorem [Sch71, IV.8.4], $u$ is open. Hence, (O) holds. Consequently, for both of these classes, all properties (O), (C), and (G) hold (and thus are equivalent).

Let $\mathcal{C}$ be a maximal class of l.c.s. satisfying the assumed closure properties of the theorem. First of all, $\mathcal{C}$ always satisfies property (C), because (O) and (G) imply (C) directly. As $\mathcal{C}$ is closed under closed graphs, (G) always holds for $\mathcal{C}$, too.

Let $E$ be an arbitrary l.c.s. in $\mathcal{C}$. We want to show that $E$ is barreled. Let $B$ be an arbitrary Banach space. We have $B$ in $\mathcal{C}$. Let $u\colon E\to B$ be an arbitrary linear, closed map. By (G), $u$ is continuous. Then by the theorem of Mahowald, [Sch71, IV.8.6], $E$ is barreled.

We want to show that $E$ is an infra-Pták space. Let $F$ be an arbitrary l.c.s., and let $u\colon E\to F$ be an arbitrary injective, linear, continuous, and nearly-open map. Then image $u(E)$ is in $\mathcal{C}$ by closure under continuous images. Applying (C) to bijective and continuous map $u\colon E\to u(E)$ yields that $u$ a topological homomorphism. By [Sch71, IV.8.3, Thm.], $E$ is an infra-Pták space.

Consequently, every space in $\mathcal{C}$ is a barreled infra-Pták space. Finally, $\mathcal{C}$ must equal the class of barreled infra-Pták spaces by maximality. ∎

Valdivia [Val84] was apparently the first, who gave an example of a space, which is infra-Pták but not Pták. Unfortunately, it is unclear, if this space is barreled or not. We give such an example below, showing that the above class of barreled infra-Pták spaces is strictly larger than the class of barreled Pták spaces. Surprisingly, for this we make use of considerations by Husain [Hus62], published twenty years earlier than Valdivia’s.

Proposition 5**.**

The dual space $\left(\mathbb{R}^{\mathbb{N}}\right)^{\prime}$ is barreled infra-Pták but not Pták.

Proof.

For Pták space $E=\mathbb{R}^{\mathbb{N}}$, its dual $(E^{\prime},t_{c})$ is reflexive and thus barreled. Here, strong topology $\beta$ and topology of uniform convergence on compact, convex sets $t_{c}$ coincide. It is not Pták [Hus62, Prop. 5]. As $E$ is a complete and metrizable l.c.s. (Fréchet), it is an S-space with CP property [Hus62, Remark after Thm. 1 and remark after Def. 2]. Hence, by [Hus62, Thm. 10] its dual $(E^{\prime},t_{c})$ is infra-Pták. ∎

Acknowledgements

We would like to thank Professor Olav Kristian Gunnarson Dovland (University of Agder, Norway) for his comments on a previous version of this paper. In addition, we would like to thank Professor Professor Mugnolo (FernUniversität Hagen, Germany) for his support. Finally, we would like to encourage the reader to give us feedback. Any help is appreciated very much!