Extraction of critical points of smooth functions on Banach spaces
Miguel Garc\'ia-Bravo

TL;DR
This paper demonstrates how to approximate smooth functions on infinite-dimensional Banach spaces with functions that have no critical points, extending the results to spaces like $c_0$ and $l_p$, with applications in critical point theory.
Contribution
It introduces a method to approximate $C^1$ functions on Banach spaces by functions without critical points, generalizing to various infinite-dimensional spaces.
Findings
Existence of $C^1$ approximations with no critical points
Extension of results to $c_0$ and $l_p$ spaces
Approximation can be made arbitrarily close in derivative norm in $c_0$
Abstract
Let be an infinite-dimensional separable Hilbert space. We show that for every function , every open set with and every continuous function there exists a mapping such that for every , outside and has no critical points (). This result can be generalized to the case where or , . In the case it is also possible to get that for every .
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Extraction of critical points of smooth functions on Banach spaces
Miguel GarcΓa-Bravo
ICMAT (CSIC-UAM-UC3-UCM), Calle NicolΓ‘s Cabrera 13-15. 28049 Madrid SPAIN
(Date: September 2019)
Abstract.
Let be an infinite-dimensional separable Hilbert space. We show that for every function , every open set with and every continuous function there exists a mapping such that for every , outside and has no critical points (). This result can be generalized to the case where or , . In the case it is also possible to get that for every .
Key words and phrases:
Banach space, Morse-Sard theorem, approximation, critical point, diffeomorphic extraction
The author was supported by Programa Internacional de Doctorado de la FundaciΓ³n La Caixa-Severo Ochoa 2016 and partially suported by grant MTM2015-65825-P. .
1. Introduction and main results
Our goal in this paper is to prove the following result:
Theorem 1.1**.**
Let be one of the classical infinite-dimensional Banach spaces or with . Let be a function and a continuous function. Take any open set containing the critical set of points of , that is . Then there exists a function such that,
- (1)
* for all ;* 2. (2)
* for all ;* 3. (3)
* is surjective for all , i.e. has no critical points; and* 4. (4)
in the case that we also have that for all .
We can make be of class inside the open set , where denotes the order of smoothness of the space , or . A brief explanation of this fact can be found in Remark 4.2.
This theorem is a particular case of the following two more technical results.
Theorem 1.2**.**
Let be an infinite-dimensional Banach space with an unconditional basis and with a equivalent norm that locally depends on finitely many coordinates. Let be a function and a continuous function. Take any open set such that . Then there exists a function such that,
- (1)
* for all ;* 2. (2)
* for all ;* 3. (3)
* for all ; and* 4. (4)
* is surjective for all .*
Theorem 1.3**.**
Let be an infinite-dimensional Banach space with a strictly convex equivalent norm and with a -suppression unconditional basis , that is a Schauder basis such that for every and every we have that
[TABLE]
Let be a function and a continuous function. Then for every open set such that there exists a function such that,
- (1)
* for every .* 2. (2)
* for all .* 3. (3)
* is surjective for all .*
The case and , in Theorem 1.1 follow from Theorem 1.2 and Theorem 1.3 respectively. The reader can find the details of why this is so in Remark 4.3.
Note that the approximating function that we build does not have any critical point, hence it is an open mapping.
The classical Morse-Sard theorem [18, 22] states that for a given function , if then its set of critical values is of Lebesgue measure zero in . This set is defined to be the image of the set of critical points, which in turn is defined as .
In general if and are Banach spaces, for a differentiable mapping , stands for the set of points at which the differential is not surjective, and is thus the set of critical values of . In the case that is of infinite dimension a natural question appears: is it possible to know that is small in any sense by just assuming enough regularity conditions on ? Unfortunately the answer is no because there exist smooth functions so that their set of critical values contain intervals (see Kupkaβs counterexample [16]).
A weaker question that we can ask ourselves is if at least any continuous mapping can be uniformly approximated by another one with small critical set of values. Let us mention that for many applications of the Morse-Sard theorem this is sufficient.
The first result of that type was in the case of a continuous function , where is a separable Hilbert space. Eells and McAlpin proved in [11] that in such case can be uniformly approximated by a smooth function whose set of critical values is of measure zero. This was so called an approximate Morse-Sard result. However in [4], a much stronger result was obtained by Azagra and Cepedello-Boiso: every continuous mapping from , a separable Hilbert space, into can be uniformly approximated by smooth mappings with no critical points. HΓ‘jek and Johanis [14] established a similar result for in the case that is a separable Banach space which contains and admits a smooth bump function. Also in the case , Azagra and JimΓ©nez-Sevilla [7] were able to characterize the class of separable Banach spaces such that any continuous can be uniformly approximated by another one of class without any critical point, as those Banach spaces with separable dual.
Let us comment finally about the very recent paper [3]. In this work, due to Azagra, Dobrowolski and the author, many of the previous results are generalized. It is proved that for the case of , , and a quotient of , any continuous function can be uniformly approximated by a smooth one with no critical points, where is denoting the order of smoothness of the space (see [3, Theorems 1.6, 1.7] for more details).
In the present paper we consider a different approach to this problem. Suppose that our given continuous function is already of class and we know that its set of critical points is included in some open set . The question is, are we able not only to uniformly approximate by another function without critical points but also to make be equal to outside ?
The key will be to use a -fine approximating result for the function , and this is provided by the results of [19, 5]. This corresponds to Section 3 of the paper.
As a matter of fact, in [19], Moulis was already able to relate -fine approximations with approximate Morse-Sard type results. She proved that for every function , where is an infinite-dimensional separable Hilbert space and is a separable Hilbert space, and every continuous function there exists a function such that , for every and such that has empty interior in . Obviously we strengthen this conclusion by being able to get and considering other Banach spaces, not necessarily Hilbertian. On the other hand for the Hilbert case we cannot write as the target space an infinite-dimensional Banach space as Moulis does and also we do not get the approximation in the derivatives.
The proof of both Theorems 1.2 and 1.3 will follow these two steps:
- β’
Step 1: Firstly we construct a function such that and and such that either is the empty set for the case of Theorem 1.2, or is locally contained in a finite union of complemented subspaces of infinite codimension in for the case of Theorem 1.3.
- β’
Step 2: We extend the function to the whole space by letting it be equal to outside . Because of the -fine approximation of Step 1 this extension is still of class on . For the case of Theorem 1.2 we are done. For the case of Theorem 1.3 we must find a diffeomorphism which will be the identity outside and such that refines (in other words, is limited by ), where is an open cover of by open balls chosen in such a way that if then
[TABLE]
The existence of such a diffeomorphism follows by a result of Section 2, which is a consequence of some results on extractibility theory from the paper [3, Section 2]. Then, the mapping has no critical point, is equal to outside and satisfies for all .
Let us fix now some notations and definitions.
We call the unconditional basis of and the associated biorthogonal functionals. Let also be the natural projections defined as and let be the unconditional constant for the basis. This constant is defined to be the least number such that for every and every ,
[TABLE]
Note that for every . We shall not confuse with the suppression unconditional constant , defined as the least number such that for all (equivalent finite) set , , where represents the projection . We have the relation . Observe also that in the statement of Theorem 1.3 it is required that .
We say that the norm locally depends on finitely many coordinates if for every there exists a natural number , an open neighbourhood of , some functionals and a function such that
[TABLE]
for every . In particular we will make use of the fact that if the norm is of class and we take , then
[TABLE]
for every .
A function is said to be limited by an open cover provided that the set refines ; that is, for every , we may find a such that both and are in .
When we say that a closed set is locally contained in a finite union of complemented subspaces of infinite codimension we mean that for every there exists an open neighbourhood of and some closed subspaces complemented in and of infinite codimension such that
[TABLE]
Finally for a function , where we write its FrΓ©chet derivative at a point by , where each is a continuous linear functional on . If is -valued we sometimes simply write for its derivative.
We will also use indistinctly the symbol to denote the norm in , and the euclidean norm in .
2. A comment about the strong extraction property
In the proof of Theorem 1.3 we will need the following.
Proposition 2.1**.**
Let be a Banach space with a smooth norm. Take an open cover of an open set and a closed set that is locally contained in a finite union of complemented subspaces of infinite codimension in . Then there exists a diffeomorphism which is the identity outside and is limited by .
To achieve this we will use some recent results on diffeomorphic extraction of closed sets that appear in [3, Section 2]. In that paper the next definitions are introduced.
Definition 2.2**.**
A subset of Banach space has the strong extraction property with respect to an open set if , is relatively closed in , and for every open set , every subset relatively closed in there exists a diffeomorphism from onto which is the identity on . If in addition for any we can ask the diffeomorphism not to move points more than (that is, for all ) we will say that has the -strong extraction property with respect to .
We will also say that such a closed set has locally the strong (or -strong) extraction property if for every point there exists an open neighbourhood of such that has the strong (-strong respectively) extraction property with respect to every open set for which is a relatively closed subset of .**
We have the following properties.
Lemma 2.3**.**
Let us suppose that have the -strong extraction property with respect to an open set of . Then
- (1)
For every set , relatively closed in , has the -strong extraction property with respect to ; 2. (2)
For every open subset , has the -strong extraction property with respect to . 3. (3)
* has the -strong extraction property with respect to .*
Proof.
- (1)
This follows directly from the definition. 2. (2)
See [3, Lemma 2.22 (2)]. 3. (3)
Take relatively closed in and an open set . We want to find a diffeomorphism from onto which is the identity on and does not move points more than .
Define the sets and , which are relatively closed in and satisfy . In particular by (1) they have the -strong extraction property with respect to .
- (a)
There exists a diffeomorphism which is the identity on and does not move points more than . 2. (b)
For the open set , using (2) we know that has the -strong extraction property with respect to . Hence there exists a diffeomorphism , which is the identity on and does not move points more than .
Observe that
[TABLE]
Hence we can define a diffeomorphism
[TABLE]
which is the identity on and does not move points more than .
β
For this kind of sets the following abstract extractibility result holds.
Theorem 2.4**.**
[3, Theorem 2.24]** Let be a Banach space and be a closed subset of which has locally the -strong extraction property. Let be an open subset of and be an open cover of . Then there exists a diffeomorphism from onto which is the identity on and is limited by .
Proof of Proposition 2.1.
For every there exists an open neighbourhood of and some closed subspaces complemented in and of infinite codimension such that
[TABLE]
If admits an equivalent smooth norm it is known (see for instance [3, Theorem 1.4]) that given a complemented subspace of infinite codimension and the open set , then has the -strong extraction property with respect to any open set for which is a relatively closed subset of . Therefore thanks to Lemma 2.3 the set has the -strong extraction property with respect to any open set for which is relatively closed on .
Now, using Lemma 2.3 , the set has the -strong strong extraction property with respect to any open set for which is relatively closed on . And this means that has locally the -strong strong extraction property. To conclude the proof apply Theorem 2.4, noting that we have and hence .
β
For more information about diffeomorphic extraction of closed sets in Banach spaces see for instance [8, 24, 20, 21, 10, 1, 2, 3].
3. -fine approximation controlling the set of critical points
Let us proceed with Step 1 of the scheme of the proof of the main Theorems 1.2 and 1.3, described in the introduction. We intend to prove the following two theorems.
Theorem 3.1**.**
Let be an infinite-dimensional Banach space with an unconditional basis and with a equivalent norm that locally depends on finitely many coordinates. Let be an open subset of , a function and a continuous function. Then there exists a function such that
- (1)
* for every .* 2. (2)
* for every .* 3. (3)
, i.e. has no critical points.
Theorem 3.2**.**
Let be an infinite-dimensional Banach space with a strictly convex equivalent norm and with a -suppression unconditional basis (in particular -unconditional with ). Let be an open subset of , a function and a continuous function. Then there exists a function such that:
- (1)
* for every .* 2. (2)
* for every .* 3. (3)
* is locally contained in a finite union of complemented subspaces of infinite codimension in .*
The proofs of these results appear in Subsections 3.1 and 3.2 respectively, following the ideas of the papers [19, 5].
However, we must previously introduce an important result that is an easier and slightly different version of [5, Lemma 5]. The proof will mainly be the same but here we want also to study the structure of the critical set of points of the approximating function and we do not care if the approximating function has more regularity than the initial function. If the given function is , it is enough for the approximating function to be as well.
For the readers convenience we present a self-contained proof, even though the arguments are the same as in [19, 5].
Lemma 3.3**.**
Let and be a Banach spaces. Suppose that is infinite-dimensional and has a -unconditional basis and a equivalent norm. Take an open set of . For every open ball with , and for every function and numbers with , there exists a function such that for , we have
- (1)
. 2. (2)
. 3. (3)
For every there exists and a neighbourhood of such that
[TABLE]
for every and , where are functions.
Proof.
Choose . Let be a smooth function such that if , if and .
For every we define the functions and ,
[TABLE]
where . We denote by the zero operator.
Fact 3.4**.**
The mapping is well-defined, smooth on , and has the following properties:
- (1)
* for all ;* 2. (2)
* for all ;* 3. (3)
.
Proof.
For any , because and the are uniformly bounded, there exists a neighbourhood of and an such that for all and , and so . Thus is a well-defined smooth map. We next compute and estimate its derivative.
We have that
[TABLE]
If and
[TABLE]
where are functions, defined by .
Looking at the expression of we compute its derivative for every ,
[TABLE]
Observe that we have proved of Lemma 3.3.
Now since , and the derivative of the norm always has norm one, for all and all we get that
[TABLE]
For a fixed , define to be the smallest integer with . Then for any , and , and so, for every ,
[TABLE]
proving .
We next estimate .
[TABLE]
which proves . Lastly, property is immediate from and the choice of . β
Going back to the proof of Lemma 3.3 define
[TABLE]
which is a function. Firstly we have that for every ,
[TABLE]
using the Lipschitzness of in .
Secondly for every ,
[TABLE]
The proof of the Lemma is now complete.
β
3.1. Proof of Theorem 3.1
Proof of Theorem 3.1.
Using the openness of , the continuity of and , the separability of and the assumption that the norm locally depends on finitely many coordinates, we find a covering
[TABLE]
of such that
- (i)
with for every . 2. (ii)
for every . 3. (iii)
for every . 4. (iv)
For every there exist a number , some linear functionals , and a function such that
[TABLE]
for every .
Now for every choose functions with bounded derivative so that for and for . We precisely take where is and and . It must be noted here that despite the fact that the norm is not differentiable at the origin, the functions are for every because in a neighbourhood of they are constantly one.
We introduce the following constants,
[TABLE]
Next define for every ,
[TABLE]
One can easily check that we have the following properties:
- β’
For every there exists such that and hence for every and .
- β’
for every .
- β’
for every and .
In particular is a partition of unity which is subordinate to .
For every we apply the previous Lemma 3.3 for each ball , the function and the constants and for and respectively. Note that we can apply the Lemma 3.3 because
[TABLE]
The resulting functions from the proof of the lemma will be called . In particular we have
[TABLE]
and
[TABLE]
for every .
Let us define finally
[TABLE]
where is a continuous linear surjective operator which we next construct. Define inductively such that for each , is a non-null element of satisfying that
[TABLE]
(note that it is the span, not the closed span); which can never fill the whole space because Banach spaces of infinite dimension can not have a countable Hamel basis. We also impose that their norms are small enough, more precisely,
[TABLE]
An important property that derives from this definition of is that the set is linearly independent and hence is a surjective linear operator. We also have that
[TABLE]
Using the expression (3.3) let us check that properties and of the statement of the main theorem are satisfied for this choice of .
Firstly if , then and
[TABLE]
Therefore for every ,
[TABLE]
We have proved .
In order to show and , let us analyze what the derivative of looks like, and inspect its critical set.
Claim 3.5**.**
For every there exist and a neighbourhood of such that:
- (i)
For every ,
[TABLE]
[TABLE] 2. (ii)
For every and , has the form
[TABLE]
Proof.
Recall that for every there is such that for every and every . So expression (3.3) becomes
[TABLE]
for all . Computing the derivative we get
[TABLE]
for every .
For every , by of Lemma 3.3, we can find a neighbourhood of and a number such that such that for every ,
[TABLE]
Define then β
Using equation (3.6) of Claim 3.5, we can write
[TABLE]
for every . Let us try to estimate all these quantities. Applying inequality (3.1) and the bound of given by (3.4) we get
[TABLE]
for every . On the other hand by our choice of the partition of unity, and using (3.2) and again (3.4) we have that for every ,
[TABLE]
We also know that the norm of is bounded by as was indicated when stating the properties of the partition of unity. This fact together with these previous computations allow us to conclude that
[TABLE]
for every . We have then proved of Theorem 3.1.
Let us focus now on studying the critical set of points of .
Use Claim 3.5 to choose a vector for which there exist numbers and a neighbourhood such that and of the claim hold. Define also
[TABLE]
Take and . Our goal is to find a vector such that . Once we prove this we will get of Theorem 3.1.
With fixed, looking at the formula (3.7) of Claim 3.5, we are interested in the expression of the bounded linear operators for . Let be the least number such that , that is (observe that necessarily ), then we write equation (3.6) as
[TABLE]
We want to find a vector for which
[TABLE]
Let us pay attention to the vectors and , for and . For simplicity let us rename these vectors as . Each of these elements , , belongs to some ball (for each we associate a unique , not necessarily equal to ). So by using property (iv) from the beginning of the proof there exists a finite number of continuous linear functionals and a function such that
[TABLE]
We intend to take a vector , so that for every .
For every , let us introduce the finite set of functionals
[TABLE]
By the definition of we have that , which is equivalent to saying that Therefore there exists an element such that and for every .
For every , take and define , so we have
[TABLE]
Moreover, for every , for every , and for every . Furthermore, writing in coordinates, we have that .
Recall that , so
[TABLE]
where are functions. Hence with our choice of we have for every .
On the other hand, looking at formula (3.7) of Claim 3.5, we also get for every .
Finally we also have for every , because for every .
Putting all these facts together, we have proved that and consequently the critical set of points of is empty.
β
3.2. Proof of Theorem 3.2
The essence of the proof will be close to the one of the previous subsection. However there are some important changes. Here we do not rely on a norm that locally depends on finitely many coordinates, but on the property of the basis of being -suppression unconditional, which will provide us with the necessary tools to approximate the function and its derivative by another function with a small critical set of points.
Proof of Theorem 3.2.
has a separable dual, so it does not contain copies of and since it has an unconditional basis, by [17, Theorem 1.c.9] we know that the basis is also shrinking, that is, .
Using the openness of , the continuity of and , and the facts that and , we find a covering
[TABLE]
of and continuous linear functionals for every such that:
- (i)
with for every . 2. (ii)
for all . 3. (iii)
for every . 4. (iv)
. 5. (v)
For every ,
[TABLE]
for some , , where is an increasing sequence of natural numbers. Note that we allow some or to be null.
At this point we proceed exactly as in the previous subsection, defining the partition of unity subordinate to , and also the constants and . We also apply Lemma 3.3, exactly in the same way as before, but now to the function and the constants and for and respectively, obtaining .
We define finally
[TABLE]
where is a continuous linear surjective operator that will be defined in the following paragraph.
Choose a family of pairwise disjoint subsets of natural numbers such that each has infinite elements and, if we denote , then is infinite. Write also as a pairwise disjoint union of sets, each of them having again infinite elements. For every and we choose satisfying that
[TABLE]
Define and also assume with no loss of generality that
[TABLE]
Following the computation made for proving Theorem 3.1 in the previous subsection, we can check that for every ,
[TABLE]
which proves .
To analyze the derivative of and its set of critical points in order to show and we also have at our disposal the following.
Claim 3.6**.**
For every there exist and a neighbourhood of such that:
- (i)
For every ,
[TABLE]
[TABLE] 2. (ii)
For every and , has the form
[TABLE]
Proof.
Follow the proof of Claim 3.5. β
Using equation (3.10) of Claim 3.6, a straightforward calculation as in the previous subsection gives
[TABLE]
for every . We have thus proved of Theorem 3.2.
It remains to study the critical set of .
Take a vector . By Claim 3.6 there exist numbers and a neighbourhood such that (i) and (ii) of the claim hold. Define also
[TABLE]
Let us divide the set in another disjoint infinite family of subsets , each of them having infinite elements. Consider also the set
[TABLE]
and define .
In order to establish of Theorem 3.2 our goal is to show that if
[TABLE]
and then there exists a vector such that . Indeed for every we would have found a neighbourhood such that
[TABLE]
Fix and look at the formula of given by property (i) of Claim 3.6. We are interested in the expression of the continuous linear operators for . Let be the least number such that , that is (observe that necessarily ), then we may write equation (3.10) as
[TABLE]
We need to find a vector for which
[TABLE]
By definition of there exist such that and . Furthermore the vectors have their -coordinates non-null because we had . This implies that the -coordinates of all the vectors in the set (see expression (3.12)) are non-null.
We will need the following:
Fact 3.7**.**
For every and every we have that
[TABLE]
Proof.
This is a consequence of the facts that the norm is strictly convex and the basis is -suppression unconditional. For details see for example [3, Fact 4.5]. β
Consequently we can assure that
[TABLE]
for every . For every , let us define . Since , we can write
[TABLE]
so
[TABLE]
On the other hand for every . In particular we can find an element
[TABLE]
Otherwise we would have which implies that for every , a contradiction with the definition of .
Let us now mix all these previous ingredients together. The vector we are looking for is
[TABLE]
We obviously have , so it remains to check that , that for every and that for every .
For the , recall that . So we have that
[TABLE]
where are functions. The elements belong to the set so it is clear that for every .
For the , using (3.11) and the facts that the elements belong to the set and that the coordinates , we conclude that for every .
The fact that is clear since
[TABLE]
for every and .
Finally we also have for every , because and for every .
We have proved that and consequently the critical set of points of is locally contained in a finite union of complemented subspaces of infinite codimension in . β
4. Main result
Theorems 3.1 and 3.2 above give us an approximation of a function and of its derivative by another function which has a nice critical set of points . In the case of Theorem 3.1 the term nice means we are in the best situation where . And in the case of Theorem 3.2 the term nice will mean for us that the closed set has the -strong extraction property with respect to , that is, there exists a diffeomorphism such that is the identity outside and refines a given open cover of . With these functions at our disposal, and with the help of Proposition 2.1 we can prove our main Theorems 1.2 and 1.3.
Proofs of Theorems 1.2 and 1.3.
Firstly we choose another function such that and for every . This is doable because in every separable Banach space with a equivalent norm, every closed set is the zero set of a function 111Wells proved in his thesis [23] that if a separable Banach space admits a smooth Lipschitz bump function, that is a non-null function with bounded derivative and bounded support, then every closed set of is the zero set of some function. Since a Banach space admitting an equivalent norm has a smooth Lipschitz bump function our statement is correct..
By Theorems 3.1 or 3.2 there exists a function such that
- (1)
for every ; 2. (2)
for every ; 3. (3)
in the case of Theorem 1.2, or is locally contained in subspaces of infinite codimension in in the case of Theorem 1.3.
Let us extend now this function to the whole space by letting it be equal to outside . We keep calling this extension by and it is important to note that this function is still of class . The only points where this fact could not be clear are those from the boundary of . However the FrΓ©chet derivative of at those points exists and is because
[TABLE]
Here we are using the facts that is FrΓ©chet differentiable in and that and for every .
We have just shown that is FrΓ©chet differentiable on , but it remains to show that it is . Straightforwardly for every ,
[TABLE]
and
[TABLE]
by the continuity of , property of Theorems 3.1 and 3.2 and because .
- (1)
Case of Theorem 1.2: Define and we obtain that
[TABLE]
for all and for every . Besides, it is clear that does not have any critical point. 2. (2)
Case of Theorem 1.3: We will extract the critical set in the following way. Observe that is a closed set included in (note that because is surjective for every ), and by of Theorem 3.2 is locally contained in a finite union of complemented subspaces of infinite codimension. Using Proposition 2.1, there exists a diffeomorphism which is the identity outside and is limited by the open cover that we next define. Recall that we have
[TABLE]
for all . Since and are continuous, for every there exists so that if then . We set .
Finally, let us define
[TABLE]
Since is limited by we have that, for any given , there exists such that , and therefore that is, we have that
[TABLE]
We obtain that
[TABLE]
for all . Furthermore is the identity outside so for every . Besides, it is clear that does not have any critical point: since , we have that the linear map is surjective for every , and is a linear isomorphism, so is surjective for every .
β
The following corollary should be compared with [6, Theorem 1.1], [4, Theorem 1.5] or [7, Corollary 8]. These results are related with the failure of Rolleβs theorem in infinite-dimensional Banach spaces.
Corollary 4.1**.**
Let be a Banach space satisfying the conditions of Theorem 1.2 (in particular ). Then for every open set there exists a bump function whose support is the closure of and does not have any critical point in .
Remark 4.2**.**
We could have gotten that the approximating function is of class (where is the order of smoothness of the space ) inside the open set . To achieve this one should get a version of Lemma 3.3 exactly as in [5, Lemma 5]. Doing this we would get from that lemma that the functions are of class . Hence the approximating function from Theorems 3.1 and 3.2,
[TABLE]
is a function of class on .
Moreover, we can find an extracting diffeomorphism of class by Proposition 2.1, hence will be a mapping on .**
Remark 4.3**.**
- (1)
The space satisfies the conditions of Theorem 1.2. The supremum norm in locally depends on finitely many coordinates, so applying [13, Theorem 1] one gets the existence of an equivalent smooth norm on that locally depends on finitely many coordinates. The space , with a metrizable countable compactum, also satisfies the conditions of Theorem 1.2. 2. (2)
The space satisfies the conditions of Theorem 1.3. For every the canonical norm of is
[TABLE]
With this expression it is easy to check that the basis is in fact -suppression unconditional with unconditional constant . It is also a norm of class , where is defined as follows: if , ; if , , and is equal to the integer part of if .
5. Acknowledgement
The author wants to thank professor Daniel Azagra for his help and many suggestions in writing this paper.
The reference list from the paper itself. Each links out to its DOI / PubMed record.
- 1[1] D. Azagra, Diffeomorphisms between spheres and hyperplanes in infinite-dimensional Banach spaces , Studia Math. 125 (1997) no. 2, 179β186.
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