Relatively Open Partitionings in Functional Analysis and in Dierential Inclusions

Transcription

1 Relatively Open Partitionings in Functional Analysis and in Dierential Inclusions DSci Dissertation - Project Nadezhda Ribarska

2 2 The dissertation consists of two chapters. The rst of them is devoted to the investigation of the concepts of fragmentability and countable cover by sets of small local diameter which proved to be useful in the theory of weak Asplund spaces and renorming theory as part of the Geometry of Banach Spaces. The second one deals with existence theory of local solutions of dierential inclusions with possibly non-convex right-hand side. It is striking that similar set structures (namely, relatively open partitionings) play signicant role in both topics, thus unifying this work. The dissertation is based on the following papers: 1. N.K.Ribarska, A note on fragmentability of some topological spaces, C.R.Acad. Bulgare Sci., 43(1990), N.K.Ribarska, The dual of a G ateaux smooth Banach space is weak star fragmentable, Proceedings of the AMS, 114, no.4 (1992), Ribarska N., Three space property for σ-fragmentability, Mathematika, 45(1998), N.K.Ribarska, On the property countable cover by sets of small local diameter, Studia Mathematica 140, 2 (2000), N.K.Ribarska, V.D.Babev, A stability property for locally uniformly rotund renorming, J.Math.Anal.Appl. 350(2009), M.I.Krastanov, N.K.Ribarska, Ts.Y.Tsachev, Dierential Inclusions with non-convex right-hand side, SIAM J. on Optimization 18(2007), no 3, Bogdan Georgiev, Nadezhda Ribarska, On a Sweeping Process with the Cone of Limiting Normals, C.R.Acad. Bulgare Sci, 66 (2013), no 5, Bogdan Georgiev, Nadezhda Ribarska, On Sweeping Process with the Cone of Limiting Normals, Set-Valued and Variational Analysis, 21 (2013), no 4, M.I.Krastanov, N.K.Ribarska, Viability and an Olech type result, Serdica Math. J., 39 (2013), no 3-4, Mira Bivas, Nadezhda Ribarska, Projection process with denable right-hand side, submitted. as well as on an unpublished (but cited) manuscript.

3 Contents 1 Fragmentability and countable cover Introduction Fragmentability of some topological spaces When σ-fragmentability implies fragmentability? Three space property Weak star fragmentability of a dual ball β-dierentiability and β-smallness A stability property for σ-fragmentability Countable cover by sets of small local diameter Gruenhage spaces and descriptive spaces A stability property for SLD A stability property for LUR renorming Dierential inclusions Preliminaries and technical lemmata Preliminaries in non-smooth analysis Some technical lemmata Colliding on a set and ε-solutions Colliding on a set and some examples The basic assumption A theorem on existence of ε-solutions Sweeping process History and statement of the problem O-minimal structures A positive result and a Lemma An existence result in the classical sweeping process Projection process Example Existence result

4 4 CONTENTS Application Viability - locally nite case Invariant ε-approximations and some properties An existence theorem Viability - general case Partial invariant ε-approximations Extended Euler curves and an existence theorem An Olech type result

5 Chapter 1 Fragmentability and countable cover 1.1 Introduction This chapter is devoted to a part of geometry of Banach spaces which lies on the edge between analysis, general topology and optimization. Some of the most important and celebrated results in Banach space theory establish deep relations between the topological and the linear structure of the space. When a Banach space E is given, the topological spaces (E, w) and (E, w ) are of special interest (as well as the space endowed the pointwise topology when E is C(X) for some compact X). It turns out that functional analysis may give birth to notions and constructions which could interest general topologists as well. One of the centers of this dissertation is the idea of "fragmentability". The following denition rst appeared in the paper of J.E.Jayne and C.A.Rogers [51] in 1985: Denition Let X be a topological space and ρ be a metric dened on it. Let ε be a positive real. We say that ρ ε-fragments X if each nonempty subset Y of X has a nonempty relatively open subset of ρ-diameter less than ε. X is said to be fragmented by the metric ρ, if ρ ε-fragments it for every ε > 0. Let us point out that fragmentability (or some related notions) appeared naturally in dierent situations before One of the most famous examples in this direction is the geometrical characterization of the Radon- Nikod ym property: 5

6 6 CHAPTER 1. FRAGMENTABILITY AND COUNTABLE COVER Theorem For a Banach space (X, ) the following assertions are equivalent: (i) X is dentable, that is for every nonempty bounded subset M of X and for every ε > 0 there are x X and a R such that the slice M {x > a} is nonempty and has diameter less than ε; (ii) X has the Radon-Nikod ym property, that is for every absolutely continuous vector measure of bounded variation dened on the Lebesgue measurable subsets of [0, 1) and taking values in X there exists a measurable function g : [0, 1) X such that for every x X the composition x g is Lebesgue integrable and x, τ(e) = for every Lebesgue measurable E [0, 1). E x, g(t) dλ(t) The close connection between the Radon-Nikod ym property and the differentiability of convex functions is one of the classical (and striking) results in the area. Let us recall some denitions: Denition The Banach space E is called Asplund (weak Asplund) if every convex function dened on an open convex subset of E is Fr echet (G ateaux) dierentiable at each point of a dense G δ subset of its domain. The reader is referred to the book of Phelps [67] for more information about Asplund spaces. The following result belongs to Namioka, Phelps and Stegall: Theorem The Banach space E is Asplund if and only if its dual has w -RNP (equivalently, the dual norm fragments the bounded subsets of E ). While Asplund spaces are extensively studied and have several beautiful and useful characterizations, weak Asplund spaces, though important for optimization as well, have none. The investigation of weak Asplund spaces is one of the main motivations for this work. Questions related to Radon-Nikod ym property are just a part of the topics where fragmentability appears. Jayne and Rogers introduced the notion while studying the existence of "nice" selectors for upper semicontinuous compact-valued maps (see [51] and [43]):

7 1.1. INTRODUCTION 7 Theorem Let F : X Y be an upper semicontinuous map with nonempty compact values from the metric space X into the fragmented Hausdor topological space Y. Then there exists a selector f for F which is σ- discrete and Borel class 1 with respect to the topology in Y, and the set of points of discontinuity of f is a set rst category in X. The conclusion of Theorem does not depend on a xed metric which fragments Y but only on the existence of such a metric. The same is true in some of the theorems proved by Christensen and Kenderov in [23] as far as single-valuedness only as conclusion is concerned. Denition The topological space X is said to be fragmentable if there exists a metric on it which fragments X. It is straightforward to extend the reasoning of Christensen and Kenderov and to obtain that every minimal usco correspondence from a Baire space to a fragmentable space is single-valued on a residual set. Thus, using the same result, one gets that any Banach space E whose dual, endowed with the weak star topology, is fragmentable, is weak Asplund. In [72] and [75] the known sucient conditions for weak Asplundness of a Banach space have been proved to actually yield the fragmentability of (E, w ). Also, it has been proved in [72] that if X is a fragmentable Hausdor compact space then (C(X), w ) is fragmentable as well (and therefore C(X) is weak Asplund). In [53] Kalenda found the rst example of a compact nonfragmentable space which belongs to the Stegall's class. Then Kenderov, Moors and Scier in [57] established an example of a weak Asplund space whose dual (with the weak star topology) is not fragmentable. Still, the class of Banach spaces with fragmentable dual is a very large, well behaving (because of the nice stability properties of the class of fragmentable spaces) and widely used subclass of the class of weak Asplund spaces. In [72] an internal characterization of fragmentable spaces has been found. It involves partitions of the space of particular kind. These set structures are the common point between the rst and the second chapter of this dissertation: Denition A well ordered family U = {U ξ : 0 ξ ξ 0 } of subsets of the topological space X is said to be a relatively open partitioning of X, if (i) U 0 = ; ) (ii) U ξ is contained in X \( η<ξ U η and is relatively open in it for every ξ, 0 < ξ < ξ 0 ; (iii) X = ξ<ξ 0 U ξ.

8 8 CHAPTER 1. FRAGMENTABILITY AND COUNTABLE COVER The reader is referred to [72] for some of the properties of the relatively open partitionings. Denition A family U of subsets of the topological space X is said to be a σ-relatively open partitioning of X, if U = n=1 U n, where U n, n = 1, 2,... are relatively open partitionings of X. U is said to separate the points of X, if whenever x and y are two dierent elements of X there exists n such that x and y belong to dierent elements of the partitioning U n. In this case we say that X admits a separating σ-relatively open partitioning. The following internal characterization of fragmentable spaces can be found in [72]: Denition The topological space X admits a separating σ-relatively open partitioning if, and only if, there exists a metric which fragments X. The second section in this chapter is devoted to further investigation of some properties of fragmentable and σ-fragmentable (see Denition below) spaces. In the third section the related property "countable cover by sets of small local diameter (see Denition 1.3.2) is studied. Closely related to it is the result presented in the forth section, but it is moved in a separate section because of its technical complexity. As a conclusion to this introduction I would like to point out the motivation for the research presented here. For the rst chapter the aim was "nd a reasonable characterization of weak Asplund spaces". For the second chapter the question is "nd reasonable necessary and sucient conditions a dierential inclusion with upper semicontinuous possibly non-convex righthand side to have local solution". In both cases only some partial results have been obtained. 1.2 Fragmentability of some topological spaces The notion of σ-fragmentability has been introduces by Jayne, Namioka and Rogers in [51]. Denition Let X be a topological space and let ρ be a metric on it. We say that ρ σ-fragments X if for every positive real ε there exist countably many subsets X i ε of X such that X = i=1 X i ε and ρ ε-fragments X i ε for every positive integer i.

9 1.2. FRAGMENTABILITY OF SOME TOPOLOGICAL SPACES When σ-fragmentability implies fragmentability? The rst theorem in this subsection has been published in [74] and soon afterwards I noticed that the same construction works in a slightly dierent situation (see Proposition in this subsection), but the result is not published. Compare the assumptions of Theorem and Proposition to the assumptions (a) and (b) of Proposition These investigations already have their development in the work of some young Spanish mathematicians. Theorem Let (X, τ) be a topological space and let ρ be a metric which σ-fragments it. If ρ is lower semi-continuous, then (X, τ) is a fragmentable space. Proof. We will construct a σ-relatively open partitioning U = n=1 i=1 U ni of X. Let us x the positive integer n. Then for ε = 1/n > 0 the space X can be represented as i=1 X i ε so that ρ ε-fragments each X i ε. Now we will dene a relatively open partitioning U ni = {U ni ξ : 0 ξ < ξ ni 0 } in such a way that each element U of U ni either has ρ-diameter not greater than ε or U does not intersect Xε. i We put U0 ni =. If we have constructed Uβ ni for all β < α, where α is a xed ordinal number, we consider the sets D = X \ U ni and A = Xε i D. If D is empty, we put ξ ni 0 = α and nish the process. If D \ Ā, we put U ni α = D \ Ā. It is relatively open in D, nonempty and does not intersect Xε. i If D = Ā then A and there exists an open set U such that U A and ρ-diam(u A) ε because ρ ε-fragments Xε. i Now the lower semi-continuity of the metric ρ gives ρ-diamu A ε. As U is open, we have Ā U A U D. We put U ni α = Ā U in this case. It is a nonempty relatively open subset of D and its ρ-diameter is not greater than ε. Thus the construction of U ni is nished. We will prove now that U separates the points of X. Let x and y be two dierent elements of X. If n is a positive integer such that ρ(x, y) > ε = 1/n, there exists i N with x Xε. i Then U ni separates x and y. Indeed, if we assume the contrary, there exists an ordinal number α < ξ0 ni such that Uα ni contains both x and y. As x Xε, i we have Uα ni Xε i and so ρ-diam ( Uα ni ) ε. This is a contradiction to ρ(x, y) > ε. β<α β

10 10 CHAPTER 1. FRAGMENTABILITY AND COUNTABLE COVER Therefore X admits a separating σ-relatively open partitioning U and thus it is a fragmentable space. Proposition Let (X, τ) be a Hausdor topological space and let ρ be a metric which σ-fragments it. If τ ρ (the topology, generated by ρ) is stronger than the original topology τ, then (X, τ) is a fragmentable space. Proof. Let the σ-relatively open partitioning U be constructed as above. We have to prove that it separates the points of X under the new assumptions. Indeed, let x and y be two dierent elements of X. Since X is Hausdor, there exists an open V with x V V y. Now τ ρ τ implies that there exists a positive integer n such that {z X : ρ(x, z) < 1/n} V. Let ε = 1/n and let i N be such that x Xε. i Then, if Uα ni of U ni containing x, we have that is the element x U ni α {z X : ρ(x, z) < 1/n} V y. Therefore U ni α separates x and y and the proof is complete Three space property The aim of this subsection is to prove a stability result about σ-fragmentability and countable cover of sets of small local diameter (see Denition 1.3.2), namely the so called three space property. A partial result in this direction is obtained in [56] using a game characterization of σ-fragmentable spaces. The proof presented here follows the scheme of the proof of the decomposition method in [61]. This result establishes (in particular) the three space property for the notion of countable cover by sets that are union of slices of diameter less than ε for every ε > 0 (called sjnr, see Denition 1.4.8), which is stronger than countable cover of sets of small local diameter (in that the relatively open sets in the denition should be slices). In [61] Molto, Orihuela and Troyanski proved that a Banach space has the above property if, and only if, it admits an equivalent locally uniformly rotund norm. The scheme works easily for countable cover..., but the case of σ- fragmentability requires additional ideas. The other ingredient of the proof in this case is constructing relatively open partitionings. Theorem Let E be a Banach space, H be its closed subspace and F = E/H. Let F and H be σ-fragmentable (i.e., the spaces, equipped with their

11 1.2. FRAGMENTABILITY OF SOME TOPOLOGICAL SPACES 11 weak topologies, are σ-fragmented by their norm). Then E is σ-fragmentable also. Proof. We will denote by T : E F the canonical projection. The Bartle-Graves theorem (cf. for example [27]) implies the existence of a continuous map B : F E such that T (Bω) = ω for every ω F. Let us x a positive ε > 0. We will seek for a countable cover of E such that any nonempty set A, which is contained in one member of the cover, has a nonempty relatively weak open subset of diameter not greater than 2ε. The norm-to-norm continuity of B yields that for every positive integer j and every element ω F there exists η j (ω) > 0 such that Bω B ω < 1/j whenever ω F and ω ω < η j (ω). Let us denote by F (j) k the set {ω F : η j (ω) > 1/k} for arbitrary j, k N. We have F = k=1 F (j) k for every xed positive integer j. On the other hand, if we x k N, the σ-fragmentability of F gives F = m=1 F m (1/k) so that for every m N and every nonempty S F m (1/k) there exists a weak open in F set U with U S and diam(u S) < 1/k. Now we have ( F = F m (1/k) F (j) ) k. k=1 m=1 Let S F m (1/k) F (j) k be nonempty. Then S F m (1/k) implies the existence of a weak open set U F, U S, diam(u S) < 1/k. Let ω and ω be arbitrary elements of U S. Then ω ω < 1/k < η j (ω) because ω S F (j) k and hence Bω B ω < 1/j. If (m, k) N 2 is denoted by a countable index l and the set F m (1/k) F (j) k is denoted by F j l, this means that for every positive integer j we have a countable cover F = l=1 F j l of F so that for every l N and for every nonempty subset S of F j l there exists a weak open subset U of F with U S and diam B(U S) 1/j. The σ-fragmentability of the other Banach space H implies the existence of a countable cover H = n=1 H n with the property that for every positive integer n and for every nonempty subset S of H n there exists a weak (in H) open set V with V S and diam(v S) < ε. Let us x n N and construct a relatively open partitioning {Hn ξ : 0 ξ < ξ n } of (H n, w) in the following way. We put Hn 0 =. The whole set H n has a nonempty relatively weak open subset of diameter less than ε. Let it have the appearance V H n where k V = {y H : h s (y) > α s }, s=1

12 12 CHAPTER 1. FRAGMENTABILITY AND COUNTABLE COVER α s reals and h s continuous linear functionals on H of norm one, s = 1, 2,..., k. It is obvious by the Hahn-Banach theorem that we can consider instead of h s their continuations to the whole E with the same norm, so without loss of generality h s E, h s = 1, s = 1, 2,..., k. Let y n1 V H n be arbitrary. Then is positive and hence the set α n1 = min {h s (y n1 ) α s : s = 1, 2,..., k} Ṽ = k {y H : h s (y) > α s + 1 } 2 αn1 s=1 is still weak open in H and contains y n1, thus Ṽ H n ; Ṽ V implies diam(ṽ H n) diam(v H n ) < ε. Let us write Hn 1 = Ṽ H n and let us assign to it the positive integer j(h 1 n) = min { i N : 1 i αn1 2 and ε-diam(v H n) 1 i Suppose we have constructed Hn η for every ordinal 1 η < ξ. Then we consider the remainder Rn ξ = H n \ η<ξ Hn η. }. If R ξ n = we stop the process and put ξ n = ξ. If not, R ξ n is a nonempty subset of H n and so there exists a weak open set V nξ with V nξ R ξ n and diam(v nξ R ξ n) < ε. Let V nξ have the form V nξ = k nξ s=1 { } y H : h nξ s (y) > αs nξ where αs nξ are reals and h nξ s are continuous linear functionals of norm one on E, s = 1, 2,..., k nξ. Let us x an arbitrary y nξ V nξ Rn. ξ Then { } α nξ = min h nξ s (y nξ ) αs nξ : s = 1, 2,..., k nξ is positive. We consider the weak open set Ṽ nξ = k nξ s=1 { y H : h nξ s (y) > α nξ s αnξ }.

13 1.2. FRAGMENTABILITY OF SOME TOPOLOGICAL SPACES 13 Now we put Hn ξ to be the set Ṽ nξ Rn. ξ It is nonempty because it contains y nξ and its diameter is less than ε because it is contained in V nξ Rn. ξ We assign to this set the positive integer { j(hn) ξ 1 = min i N : i αnξ 2 and ε-diam(v nξ Rn) ξ 1 }. i Thus we nished the construction of the relatively open partitioning {Hn ξ : 1 ξ < ξ n } of H n assigning to each of its members a positive integer j(hn). ξ It is clear that without loss of generality we could assume that H n are disjoint (i.e. H n1 H n2 = if n 1 n 2 ). Therefore for our xed ε > 0 we have a partitioning {Hn ξ : n N, 1 ξ < ξ n } of H. As every point x H belongs to exactly one member of the above partitioning, let say Hn ξx x, we ( correctly dene j(x) = j Hn ξx x ). Then we put Hn j = {x H n : j(x) = j}. So we have a countable partitioning {Hn} j n,j=1 of H, H = n,j=1 Hn, j which we will use to dene our countable cover of E in the following way: { } E j nl = x E : x B(T x) Hn j and T x F j. Indeed, for every x E we have x B(T x) H and so there exist positive integers n, j with x B(T x) Hn. j But for this j we have F = l=1 F j l, hence there exists a positive integer l with T x F j l, thus x Ej nl. In this way we had divided E into countably many parts E j nl and E = E j nl. n=1 j=1 l=1 We will prove that each nonempty subset of E j nl has a nonempty relatively open subset of diameter not greater than 2ε. In the rest of the proof we will x the positive integers n, j, l and a nonempty subset A of E j nl. Let us consider the nonempty set T (A) F. The inclusion A E j nl implies the inclusion T (A) F j l and hence there exists a weak open in F set U with U T (A) and diam[b(u T (A))] 1/j. We put C = A {z E : T z U}. The set {z E : T z U} is weak open in E, because for every continuous functional h in F and real number α we have {z E : h(t z) > α} = {z E : (T h)(z) > α} l

14 14 CHAPTER 1. FRAGMENTABILITY AND COUNTABLE COVER and T h E is continuous functional as well. Now U T (A) yields C and, of course, C A E j nl. For the sake of shortening the notation, let Ψ(x) = x B(T x) for every x E. We have now that Ψ(C) is nonempty and is contained in Hn. j As Hn j H n = ξ<ξ n Hn, ξ we can dene ξ C { } = min ξ [1, ξ n ) : Hn ξ Ψ(C) Then Ψ(C) is contained in the remainder n = H n \ R ξ C H η n η<ξ C and its intersection with H ξ C n (as Ψ(C) Hn, j i.e., j(x) = j for every x Ψ(C)). appearance H ξ C n = Ṽ Rξ C n Ṽ = s=1 is nonempty. The latter yields j(h ξ C n where k {y H : h s (y) > α s + 1 } 2 α,. Let H ξ C n ) = j have the h s = 1 and α s reals, s = 1, 2,..., k. In this notation by the construction of the partitioning the positive real number α should be such that 1/j α/2 and, if V is the set k s=1 {y H : h s (y) > α s }, then the norm diameter of V R ξ C n is less than ε and 1/j ε diam in C with Ψ(x) Ṽ (indeed, Ψ(C) Hξ C n ( V R ξ C n ). Let us x a point x = Ψ(C) Ṽ and such a x exists). Now we are ready to dene the set W = {z E : T z U} ( k s=1 { z E : h s (z) > α s α + h s(b(t x))} ). It is clear that W is a weak open subset of the Banach space E. Moreover, A W = C ( k s=1 { z E : h s (z) > α s α + h s(b(t x))} ). The fact that the xed point x belongs to C, together with (by Ψ(x) Ṽ ) h s (x) = h s (x B(T x)) + h s (B(T x)) = = h s (Ψ(x)) + h s (B(T x)) > α s α + h s(b(t x)),

15 1.2. FRAGMENTABILITY OF SOME TOPOLOGICAL SPACES 15 gives that x is in A W, hence the latter is nonempty. It remains to estimate the diameter of the nonempty relatively weak open set A W. We had already xed a point x in it and let y be another one, arbitrarily picked from A W. We are interested in estimating x y. It is natural to use the representations x = (x B(T x)) +B(T x) = Ψ(x) +B(T x) and y = Ψ(y) + B(T y), so x y Ψ(x) Ψ(y) + B(T x) B(T y). It is easy to estimate the second term, because x, y A W C gives T x, T y U T (A) and diam(b(u T (A))) 1/j yields On the other hand y W yields B(T x) B(T y) 1/j. h s (Ψ(y)) = h s (y) h s (B(T y)) > α s α + h s(b(t x)) h s (B(T y)) = But the norm of h s is one and so Therefore = α s α + h s(b(t x) B(T y)). h s (B(T x) B(T y)) B(T x) B(T y) 1/j. h s (Ψ(y)) > α s α 1 j α s by the choice of j = j(h ξ C n ). Thus we have obtained h s (Ψ(y)) > α s for every s = 1, 2,..., k, hence Ψ(y) V. The inclusions Ψ(y) Ψ(A W ) Ψ(C) R ξ C n and the above statement imply Ψ(y) V R ξ C n. Similarly Ψ(x) Ṽ V, Ψ(x) Ψ(C) R ξ C n give Ψ(x) V R ξ C n and hence ( ) Ψ(x) Ψ(y) diam V R ξ C n. Therefore ( ) x y Ψ(x) Ψ(y) + B(T x) B(T y) diam V R ξ C n + 1/j = ( ( )) = ε + (1/j) ε diam V R ξ C n ε by the choice of j. The point y was arbitrary in A W, so from the above inequality we have diam(a W ) 2ε which completes the proof.

16 16 CHAPTER 1. FRAGMENTABILITY AND COUNTABLE COVER Remark The proof of the theorem does not give any information whether σ-fragmentability by some kind of topologically representable sets (cf. [48]) is a three space property. Corollary Let E be a Banach space, H be its closed subspace and F = E/H. Let H and F with their weak topologies have countable covers of small local norm diameter. Then so does E. To prove the corollary it is sucient to check that the countably many subsets E j nl of E, built in the proof of Theorem 1.2.4, work and it is straightforward. As a matter of fact, in this situation constructing of the relatively open partitionings {Hn ξ : 0 ξ < ξ n }, n = 1, 2,... can be trivially avoided thus simplifying the proof. The last remark together with the norm-to-norm continuity of the Bartle-Graves mapping shows that, in spite of the case of σ-fragmentability, having countable cover of sets of small local diameter that are norm closed (or dierences of norm closed sets) is a three space property Weak star fragmentability of a dual ball This subsection is devoted to the strengthening of two famous sucient conditions a Banach space to be weak Asplund. The rst one is a theorem of Asplund published in 1968 (see [2]): Theorem Let E be a Banach space. If the dual space E admits a dual strictly convex norm then E is weak Asplund. Since, in this case, the corresponding norm in E is G ateaux dierentiable (o zero), it was natural to ask if the assertion of the above theorem of Asplund is valid for every Banach space E with a norm which is G ateaux dierentiable at each nonzero point (i.e., a G ateaux smooth norm). This long standing question was answered positively by Preiss, Phelps and Namioka in [68]: Theorem Let E be a Banach space. smooth then E is weak Asplund. If the norm of E is G ateaux The connection between the notions fragmentability and weak Asplund is revealed by the following statement, which was proved in [23]: Theorem Let E be a Banach space. If E with its weak-star topology is fragmentable then E is weak Asplund.

17 1.2. FRAGMENTABILITY OF SOME TOPOLOGICAL SPACES 17 It was not clear, however, whether this result implies the above mentioned theorems of Asplund and Preiss-Phelps-Namioka, respectively. Theorem below shows that this is the case for the theorem of Asplund. Combining it with the techniques developed in [68] it is shown in Theorem that the Preiss-Phelps-Namioka theorem can also be considered as a corollary of Theorem (In fact, Theorem in this subsection is a further step after Theorems 1.6 and 2.2 of [68] and its proof uses the same construction.) Our approach needs no external notions like games and mappings. It provides additional information about the intrinsic structure of (E, w ). Theorem Let E be a strictly convex Banach space and τ be a Hausdor locally convex topology on it. If the strictly convex norm is lower semicontinuous with respect to τ, then (E, τ) is a fragmentable space. Proof. Let be the strictly convex norm on E, which has τ-closed balls. We will denote by B the closed -ball centered at 0. If n is a positive integer, we can construct a relatively open partitioning V n = {V n s : 0 s < n 2 + 2} of E setting V0 n =, V 1 n = E \ (nb), V n 2 s < n and Vn n 2 +1 ( s = n s 2 n ) ( B \ n s 1 n ) B for = (1/n)B. Now we will "fragment" every V n s to convex parts, i.e. we will construct a relatively open partitioning U ns with convex elements of every Vs n in V n using the following lemma: Lemma Let (in terms of the theorem) C and D be two τ-closed convex subsets of E with C D. Then there exists a relatively open partitioning of (D \ C, τ) consisting of convex sets. Proof of the Lemma. We will construct the relatively open partitioning U ( = {U ξ : 0 ξ < ξ 0 } inductively so that at each step both U ξ and ) D \ ζ ξ U ζ are convex sets. We put U 0 = (of course D is convex). If ( ) we have constructed U ζ for every ζ < ξ, we consider A = D \ ζ<ξ U ζ. It is convex because of the induction hypothesis if ξ is a successor ordinal and as an intersection of convex sets if ξ is a limit ordinal. If A \ C is empty, we nish the process. If not, we can strictly separate a point x A \ C from the τ-closed convex set C by a τ-closed hyperplane. We put U ξ to be the intersection of A with the τ-open halfspace, which contains x. Then it

18 18 CHAPTER 1. FRAGMENTABILITY AND COUNTABLE COVER is a ( convex nonempty τ-relatively open subset of A \ C. Moreover, the set ) D \ ζ ξ U ζ is convex, too. We denote by U n a renement of V n consisting of convex sets (it may be obtained by alphabetically ordering the elements of U ns, s [0, n 2 + 2); see Proposition 1.5 in [72]). Then U = n=1 U n separates the points of E. Indeed, let us assume the contrary, i.e. there exist two dierent elements x and y of E which can not be separated by U. This means that x and y belong to one and the same element of U n for every n. The convexity of these sets yields that the whole segment [x, y] is contained in the same element and therefore no two points from [x, y] can be separated by U. But every two points with dierent norms can be separated by V n for some positive integer n and, moreover, by its renement U n. Hence, all points in the segment [x, y] have equal norms or [x, y] lies on the surface of a -ball. This contradicts the strict convexity of. It is natural for the topology τ (appearing in the assumptions of Theorem ) to be the weak topology of a Banach space or the weak star topology of a dual Banach space or the pointwise topology of a space of continuous functions. Combining Theorem and Theorem we immediately get another proof of the theorem of Asplund The slight improvement in the corollary below comes from the good stability properties of the class of all fragmentable spaces (see [72], Proposition 2.8). Corollary Let E be a Banach space and E admit an equivalent dual strictly convex norm. Then every closed linear subspace F of E is weak Asplund. Now we turn to strengthening the Preiss-Phelps-Namioka theorem: Theorem Let E be a Banach space and be a G ateaux smooth norm on it. Then the topological space (E, w ) is fragmentable. Proof. Step 1. Let us denote by B the dual unit ball. As E = m=1 mb, it suces to prove that (B, w ) is a fragmentable space. Step 2. Basic step of the construction. Let U be a subset of B, p be an equivalent norm on E, and ε, β be two positive numbers. Then we construct a relatively open partitioning

19 1.2. FRAGMENTABILITY OF SOME TOPOLOGICAL SPACES 19 V = {V ξ : 0 ξ < ξ 0 } of (U, w ) and associate to every element V of V a nonnegative real number s V, an element e V of E with p(e V ) = 1, and a new equivalent norm p V on E in the following way. We put V 0 =, s V0 = 0, and e V0, p V0 be arbitrary. If we have constructed ( V η for every ordinal number η < ξ, we consider ) the set R ξ = U \ η<ξ V η. If R ξ =, we put ξ 0 = ξ and nish the process. If not, we put s Vξ = sup{p (x ) : x R ξ } where p is the norm conjugate to p. If s Vξ = 0, it means that R ξ = {0}. Then we put V ξ = {0}, ξ 0 = ξ + 1, and e Vξ, p Vξ be arbitrary and nish again. If s Vξ > 0, there exists an element e Vξ of E with p(e Vξ ) = 1 such that the set { } V ξ = x R ξ : x, e Vξ > (1 ε)s Vξ is nonempty. Moreover, V ξ is w -relatively open in R ξ. We put { } x q Vξ (x) = inf λevξ : λ R foe every x E and p 2 V ξ = p 2 + β.q Vξ. Thus the relatively open partitioning V is constructed. Step 3. Construction of the σ-relatively open partitioning U = n=1 U n of (B, w ). Following the construction in [68] we x two sequences of positive numbers 1/2 > ε 1 > ε 2 >... and β 1 > β 2 >... such that ε k 0, βk 2 < 3, and (ε k ) 1/2 /β k <. Applying the basic step with U := B, p :=, ε := ε 1, and β := β 1, we obtain a relatively open partitioning U 1 = {Uξ 1 : 0 ξ < ξ 1 } of (B, w ), and for every element U U 1 we get s U 0, e U E with e U = 1 and an equivalent norm p U on E. If we have constructed the relatively open partitionings U 1, U 2,..., U n, we consider a xed element Uξ n of U n = {Uξ n : 0 ξ < ξ n }. Then we apply the basic step to Uξ n with p := p U n ξ, ε := ε n+1, and β := β n+1. So we get a partitioning Uξ n = {U ξη n : 0 η < ηn ξ } of U ξ n to every element of which a nonnegative real, an equivalent norm, and a p U n ξ -norm-one vector are associated. Now U n+1 = {Uξ n : 0 ξ < ξ n } is a relatively open partitioning of B if its elements Uξη n, 0 ξ < ξ n, 0 η < ηξ n, are ordered lexicographically (see [72], Proposition 1.5). Obviously U n+1 is a renement of U n.

20 20 CHAPTER 1. FRAGMENTABILITY AND COUNTABLE COVER Step 4. Proof that U is separating. Let us assume the contrary, i.e. there exist two dierent points x 1 and x 2 in B such that they belong to one and the same element V n of U n for every positive integer n. Let us denote by s n, p n and e n respectively the nonnegative real number, the equivalent norm and the unit vector associated to V n by the construction. Our assumption implies that neither V n is empty nor V n = {0} and hence s n > 0 for every n N. The construction gives us that the following properties hold: (i) B = V 0 V 1 V 2... n=1 V n {x 1, x 2 }; (ii) V n {x V n 1 : x, e n > (1 ε n )s n } for every n 1; (iii) s n = sup { p n 1 (x ) : x V n } > 0; (iv) p n 1 (e n ) = 1 for every n 1; (v) p 0 =, p 2 n = p 2 n 1 + β n.q 2 n, where q n (x) = inf { x λe n : λ R} for every x E. Further the proof is contained in [68] (the last half of the proof of Theorem 1.6) with the same notation except for the indexation of the smooth norms which is moved back by one. Remark Theorem 2.2 of [68] states that the dual (E, w ) of a smooth space E belongs to the Stegall's class S, i.e. every minimal usco correspondence from a Baire space into it is single-valued on a residual subset of its domain. (See Denition for minimal usco in the next subsection. For additional information about S see for example [80], [81], [54], [19].) Lemma 5.10 from [65] (or Proposition 2.5 from [72]) shows that fragmentable spaces are in S and so Theorem immediately yields Theorem 2.2 of [68] β-dierentiability and β-smallness The results in this subsection generalize the theorems from Ÿ3 of [68]. We include them here because the view to fragmentability through maps is important to us.

21 1.2. FRAGMENTABILITY OF SOME TOPOLOGICAL SPACES 21 Denition Let X and Y be Hausdor topological spaces and F : X Y be a multivalued map. F is said to be upper semicontinuous at x 0 X if for every open set V Y that contains F (x 0 ) there exists a neighborhood U of x 0 such that F (x) V for every x U. We say that F is upper semicontinuous if it is upper semicontinuous at every point x X. If, in addition, F (x) is nonempty and compact for every x in X, the correspondence F is called usco. The graphs of all usco correspondences from X into Y can be ordered by inclusion. Minimal elements of this relation are graphs of usco correspondences from X into Y, which will be called minimal usco. For minimal usco mappings the following property holds: for every open subset U of X and every open subset V of Y such that F (U) V there exists a nonempty open subset W of U with F (W ) V (see [23]). Denition Let β be a family of nonempty bounded subsets S of the Banach space E satisfying: (a) λs β whenever λ R and S β; (b) the union of nitely many members of β is in β; (c) the union of the members of β is all of E. We say that a continuous convex function f on an open convex subset D of E is β-dierentiable at x D if for all S β the limit of the dierence quotient lim t 0 (f(x + ty) f(x))/t exists uniformly for y in S. Natural choices for β are all nite sets (G ateaux dierentiability), all weakly compact sets (Hadamard dierentiability), or all bounded sets (Fr echet dierentiability). If the norm is β-dierentiable at every x 0, it is said to be β-smooth. We say that U E is a neighborhood of x E if there exists S β with x + S 0 U, where S 0 = {z E : z, z 1 for all z S}. It is easy to check that thus we have introduced a linear topology on E. We will call it β as well. This topology is closely related to β-dierentiability by the following fact: The subdierential mapping f of a continuous convex function f : D R (D is an open subset of E) is single-valued and β-upper semicontinuous at the point x 0 D if and only if f is β-dierentiable at x 0 (see [68], Proposition 3.1). We introduce the following notion:

22 22 CHAPTER 1. FRAGMENTABILITY AND COUNTABLE COVER Denition Let X be a topological space and U = n=1 U n be a σ- relatively open partitioning of X. Let τ be a second topology on X. We say that U is τ-small if for every x X and for every neighborhood W of x there exist a positive integer n and an element U of U n with x U W. Remark Let ρ be a metric on the topological space X and τ ρ be the topology generated by ρ. If ρ fragments X then there exists a separating τ ρ -small σ-relatively open partitioning of X. On the other hand, if X admits a separating τ ρ -small σ-relatively open partitioning, then ρ fragments all compact subsets of X. The following proposition justies introducing the notion of τ-smallness. It is a generalization of Proposition 2.5 in [72]. Proposition Let (X, τ 1 ) be a topological space, τ 2 be a second topology on X and suppose X admits a separating τ 2 -small τ 1 -σ-relatively open partitioning. Then every minimal usco map ϕ : B (X, τ 1 ) from a Baire space B into (X, τ 1 ) is single-valued and τ 2 -upper semicontinuous at every point of a residual subset of B. Proof. Let U = n=1 U n, U n = {Uξ n : 0 ξ < ξ n } be a separating τ 2 -small τ 1 -σ-relatively open partitioning of X. We put B n = {b B : V neighborhood of b U U n, ϕ(v ) U} for every n N. Obviously the B n are open subsets of B. We will prove that they are dense. Indeed, let V be an arbitrary nonempty open subset of B and n be a positive integer. We consider the ordinal number { } ˆξ = min ξ : Uξ n ϕ(v ). ( ) The minimality of ϕ and ϕ(v ) η<ˆξ+1 U η n, η<ˆξ+1 U η n open yield the existence of a nonempty open subset W of V with ϕ(w ) η<ˆξ+1 U η n. Hence ϕ(w ) U ṋ ξ because of the choice of ˆξ and so W B n, i.e., B n V. Now we will prove that ϕ is single-valued and τ 2 -continuous at every point b in the residual subset G = n=1 B n of B. As ϕ(b) is contained in one element of U n for every n, no two points from ϕ(b) can be separated by U. Therefore ϕ(b) is a singleton (U is separating). Let U be a τ 2 -open neighborhood of ϕ(b). Then there exist n and α n < ξ n with ϕ(b) U n α n U because U is τ 2 -small. But b G B n, and hence there exists a neighborhood V of b with ϕ(v ) U n α n. So ϕ(v ) U and ϕ is τ 2 -upper semicontinuous at b.

23 1.2. FRAGMENTABILITY OF SOME TOPOLOGICAL SPACES 23 Corollary Let E be a Banach space and β be a family of its subsets as described in Denition If (E, w ) admits a separating β-small σ-relatively open partitioning, then every continuous convex function on an open subset of E is β-dierentiable on a residual subset of its domain. One can prove this using Proposition 2.5, Lemma 2.5 of [23] or Lemma 3.5 of [68]. Theorem Suppose that the norm in E is β-smooth. Then (E, w ) admits a separating σ-relatively open partitioning. Because of Proposition , this theorem is a generalization of Theorem 3.4 in [68] but its proof is the same. One should skip the beginning of the proof in [68] and use the subsequent argument to show that the σ- relatively open partitioning constructed in the proof of Theorem is actually β-small. Combining the results from this subsection one can obtain once more the following Corollary Let E be a Banach space whose norm is Fr echet smooth. Then the dual norm fragments the bounded subsets of (E, w ) A stability property for σ-fragmentability This subsection is devoted to Theorem Let X and Y be two compact spaces. Suppose that C(X) and C(Y ) are σ-fragmentable in the respective pointwise topology by the uniform norm. Then the same holds true for the space of the continuous functions C(X Y ) on their cartesian product. To prove the above stated result we will need also a game characterization of σ-fragmentability due to Kenderov and Moors. Kenderov and Moors introduced in [55] and studied in [56] the following topological game. Two players α and β alternatively select subsets of the topological space T. The player β starts the game by selecting an arbitrary non-empty subset B 1 of T. Then α chooses some nonempty subset A 1 of B 1 which is relatively open in B 1. In general, if the selection A n of the player α is already specied, the player β makes the next move by selecting an arbitrary nonempty set contained in A n. In return, α selects a nonempty relatively open subset A n+1 of B n+1. Continuing this alternative selection of sets in T, a sequence B 1 A 1 B 2... B n A n...

24 24 CHAPTER 1. FRAGMENTABILITY AND COUNTABLE COVER is generated which we call a play and denote by p = (B 1, A 1, B 2,..., B n, A n,...). A nite stage of the game, when α is to move, will be called a partial play and denoted by p n = (B 1, A 1, B 2,..., A n 1, B n ). A mapping ω which assigns to each partial play p n, n 1 some nonempty relatively open subset of the last element of p n, is called a strategy for the player α. Given the strategy ω, we call a play p = (B 1, A 1, B 2,..., B n, A n,...) an ω-play if A n = ω(p n ) for every n 1. So far we just dened the rules of the game, i.e. we specied the conditions a sequence (nite sequence) of sets in T should satisfy to be considered as a play (a partial play) of the game. Now we want to state who is the winner of a given play p of the game. The player α is said to be the winner i p contains no more than one point. We need only to say that a strategy ω for the player α is called winning, if every ω-play is won by α and we are ready to state the following theorem, proved in [55]: Theorem A topological space T is fragmentable if and only if the player α has a winning strategy in the above described game on T. Having in mind this theorem, we will refer to the above described game as the fragmentability game on T. Kenderov and Moors in [56] proved that after a slight modication in the winning rule the fragmentability game can be used to characterize σ- fragmentability as well. More precisely, they proved Theorem Let T be a topological space and ρ be a metric on it which is lower semicontinuous (with respect to the topology of T ). Then the following are equivalent: (a) T is σ-fmgmented by ρ; (b) there exists a metric on T which fragments it and generates a topology stronger than the topology generated by ρ; (c) there exists a strategy ω for the player α in the fragmentability game on T such that for every ω-play p either p =, or lim (ρ-diam (p n)) = 0. n

25 1.2. FRAGMENTABILITY OF SOME TOPOLOGICAL SPACES 25 We will apply this theorem to the space (C(K), p) of all continuous functions on some compact space endowed with its pointwise topology and the metric ρ will be the norm metric. It worths to point out that we are not using the full power of the results proved in [56]. There is a further equivalence (in this particular case for T and ρ) to the condition that there exists a metric on (C(K), p) which fragments it and generates a topology stronger than the pointwise one. In what follows we will use a little free the word strategy, meaning some rule for α how to behave in some situations, or using a mathematical language, our partial strategies will be mappings dened not necessarily on the whole set of partial plays of the game, but on some proper subset of it. All of the subsequent lemmata provide such partial rules, which apply from some xed partial play on. In all cases at every move some condition is checked. If the condition becomes true, we switch on another partial strategy. If the condition is never satised, the intersection of the respective play becomes empty. If not explicitly stated something other, from now on the words the fragmentability game will mean the fragmentability game on (C(X Y ), p). Lemma Let ε be a xed positive number. Let us be given also a partial play p 0 of the fragmentability game which ends with the nonempty set B 0, corresponding to a β-move. Then there exists a strategy ω of the player α such that either for each ω-play extending p 0 its intersection is empty, or following ω after nitely many steps we arrive to a nonempty set B, for which there exist nitely many points y 0, y 1,..., y n in Y such that for every function f in B and for every y Y there exists j {0, 1,..., n} with whenever x X, or in other words, f(x, y) f(x, y j ) ε f(, y) f(, y j ) C(X) ε. Proof of the Lemma. We will dene the partial strategy ω rst choosing an arbitrary point y 0 in Y and checking if f(, y) f(, y 0 ) C(X) ε whenever f is in B 0 and y is in Y. If this is true, we dene ω(p 0 ) = B 0 and we are done. If no, there exist a function g B 0 and points y 1 Y, x 1,0 X such that g(x 1,0, y 0 ) g(x 1,0, y 1 ) > ε.

26 26 CHAPTER 1. FRAGMENTABILITY AND COUNTABLE COVER Then we put ω(p 0 ) = {f B 0 : f(x 1,0, y 0 ) f(x 1,0, y 1 ) > ε}. Note that ω(p 0 ) is a nonempty and relatively pointwise open subset of B 0. Let us denote the partial play (p 0, ω(p 0 ), B 1 ) by p 1, where B 1 is any response of the player β. Thus we nished our rst step. Let p k be a partial play and its last element be B k. Let so far we have picked elements y 0, y 1,..., y k of Y and elements x i,j of X, i > j, i, j {0, 1,..., k} in such a way that f(x i,j, y i ) f(x i,j, y j ) > ε whenever f is in B i. Now we have two possibilities: either for every y Y and for every function f B k there exists j {0, 1,..., k} with f(, y) f(, y j ) C(X) ε, or not. If yes, we are done, the nonempty set B in the formulation of the lemma being B k and n being k. If no, there exist a function g B k and points y k+1 in Y, x k+1,j in X, j {0, 1,..., k} such that g(x k+1,j, y k+1 ) g(x k+1,j, y j ) > ε holds true. Then the player α plays the set ω(p k ) = {f B k : f(x k+1,j, y k+1 ) f(x k+1,j, y j ) > ε}. j k This set is a nonempty pointwise relatively open subset of the previous move, so we are keeping the rules of the game. We denote the so obtained continuation of the partial play p k by p k+1 = (p k, ω(p k ), B k+1 ) where B k+1 is the last move of β. The so constructed partial strategy ω satises the requirements of the lemma. Indeed, we stop at a nite stage only after checking the respective conclusion. Let us have played innitely many moves according to ω and the correspondent ω-play be p with partial plays p k, k 0. We must show that the intersection of the sets forming p is empty. Let us assume the contrary, i.e. there exists a function f C(X Y ) belonging to p. Then the argument of Ahmed Bouziad in [10] proving his Fact 1 provides a contradiction. The next lemma is the same as the previous one, stated for the compact space X instead of Y.

27 1.2. FRAGMENTABILITY OF SOME TOPOLOGICAL SPACES 27 Lemma Let ε be a xed positive number. Let us be given also a partial play p 0 of the fragmentability game which ends with the nonempty subset B 0 of the space C(X Y ). Then there exists a strategy ω of the player α such that either for each ω-play extending p 0 the intersection is empty, or following ω after nitely many steps we arrive to a nonempty set B, for which there exist nitely many points x 0, x 1,..., x m in X such that for every function f in B and for every x X there exists i {0, 1,..., m} with f(x, ) f(x i, ) C(Y ) ε. In the next lemma we make use of the assumption of σ-fragmentability of C(Y ). Lemma Let ε be a xed positive number and x X. Let us be given also a partial play p 0 of the fragmentability game which ends with the nonempty B C(X Y ), corresponding to a β-move. Then there exists a strategy ω of the player α such that either for each ω-play extending p 0 the intersection is empty, or following ω after nitely many steps we arrive to a nonempty set A such that the set {f( x, ) C(Y ) : f A} has norm diameter in C(Y ) less than or equal to ε. Proof of the lemma. Let π(a) = {f( x, ) C(Y ) : f A} for each subset A of the space C(X Y ). Then π(b) is a nonempty subset of C(Y ) and so it can be considered as the rst move of the player β in the fragmentability game on C(Y ). Let the partial play consisting of this single move be p 0. Now we know that there exists a strategy ω for α in this game which satises the condition (c) of Theorem with ρ being the norm metric on C(Y ). We will "lift" it to the whole space in order to construct the required strategy in the game on C(X Y ). Namely, we put ω(p 0 ) = {f B : f( x, ) ω( p 0 )}. It is straightforward to check that this is a correct α-move. Let B 1 be the response of the player β. Thus p 1 = (p 0, ω(p 0 ), B 1 ) is a partial play in the fragmentability game on C(X Y ) and p 1 = ( p 0, ω( p 0 ), π(b 1 )) is a partial play in the fragmentability game on C(Y ). We proceed in this way constructing simultaneously two sequences {p k } k 0, { p k } k 0 of partial plays on C(X Y ) and C(Y ) respectively so that (i) ω(p k ) = {f B k : f( x, ) ω( p k )}; (ii) p k+1 = (p k, ω(p k ), B k+1 );

28 28 CHAPTER 1. FRAGMENTABILITY AND COUNTABLE COVER (iii) p k+1 = ( p k, ω( p k ), π(b k+1 )). Note that in (i) we have a correct denition of ω(p k ) and in (iii) the response of the player β in the game on C(Y ) is dened via the response of β in the game on C(X Y ). Both denitions are compatible with the rules of the game. We will prove that the so dened strategy ω has the required properties. Let p be an ω-play, continuation of p 0, with partial plays {p k } k 0. Let p be the corresponding ω-play with partial plays { p k } k 0 so that the properties (i), (ii), (iii) hold true. Since ω satises (c) of Theorem , either p is empty, or the diameters of the sets forming p go to zero. If the rst option holds, it is impossible p, because π B k π (B k ) = p. k 0 k 0 If the second option holds true, there exists a positive integer n such that C(Y ) -diam( ω( p n )) is less than ε. Then C(Y ) -diam {f( x, ) C(Y ) : f ω(p n )} ε. So this strategy either leads to an empty set, or after nitely many moves leads to a set, whose trace on the line { x} Y has C(Y ) -diameter not greater than ε. Again we state a symmetric lemma now utilizing the σ-fragmentability of C(X). Lemma Let ε be a xed positive number and ỹ Y. Let us be given also a partial play p 0 of the fragmentability game which ends with the nonempty B C(X Y ), corresponding to a β-move. Then there exists a strategy ω of the player α such that either for each ω-play extending p 0 the intersection is empty, or following ω after nitely many steps we arrive to a nonempty set A such that the set {f(, ỹ) C(X) : f A} has norm diameter in C(X) less than or equal to ε. Lemma Let ε be a xed positive number and p 0 be a partial play with last element B 0. Then there exists a strategy of the player α which either leads to an empty set, or after nitely many steps provides a nonempty set B whose norm diameter in C(X Y ) is not greater than 7ε. Proof of the lemma. The construction of the partial strategy required consists in subsequent applying the partial strategies constructed