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THESE
PRESENTEE A
L’UNIVERSITE BORDEAUX 1
ECOLE DOCTORALE DE MATHEMATIQUES ET INFORMATIQUE
ET
L’UNIVERSITE CHARLES DE PRAGUE
FACULTE DE MATHEMATIQUES ET PHYSIQUE
par Anton n Prochazka
DOCTEUR
SPECIALITE: Mathematiques Pures
ANALYSE DANS LES ESPACES DE BANACH
These dirigee par: Robert Deville et Petr H ajek
Soutenue le 24 juin 2009 a l’Universite Charles de Prague
Devant la commission d’examen formee de:
R. DEVILLE Professeur, Universite Bordeaux 1 Directeur
P. HAJEK Directeur de recherche,
Academy of Sciences of the Czech Republic Directeur
G. LANCIEN Professeur, Universite de Franche-Comte
E. MATHERON Professeur, Universite d’Artois Rapporteur
L. ZAJICEK Professeur, Universite Charles de Prague President
de jury
V. ZIZLER Directeur de recherche,
Academy of Sciences of the Czech Republic RapporteurResume
Cette these traite quatre sujets di erents de la theorie des espaces de Banach: Le premier
est une caracterisation de la propriete de Radon-Nikodym en utilisant la notion du jeu
des points et tranches: Le deuxieme est une evaluation de l’indice de dentabilite prefaible
des espaces C(K) ou K est un compact du hauteur denombrable: Le troisieme est un
renormage des espaces non separables qui est simultanement LUC, lisse et approximable
par des normes d’une lissite plus elevee. Le quatrieme est une approche par le theoreme de
Baire aux principes variationnels parametriques. La these commence par une introduction
qui examine le contexte de ces resultats.
Analysis in Banach spaces
Summary
The thesis deals with four topics in the theory of Banach spaces. The rst of them is a
characterization of the Radon-Nikodym property using the notion of point-slice games.
The second is a computation of the w* dentability index of the spaces C(K), where
K is a compact of countable height. The third is a renorming result in nonseparable
spaces, producing norms which are di erentiable, LUR and approximated by norms of
higher smoothness. The fourth topic is a Baire cathegory approach to parametric smooth
variational principles. The thesis features an introduction which surveys the background
of these results.
Mots-cles
espace de Banach, caracterisation de la propriete de Radon-Nikodym, caracterisation de
superre exivite, l’indice de Szlenk, l’indice de dentabilite, approximation des normes,
LUC, lissite, principle variationnel lisse, minimisation, dependence continue des min-
imiseurs
Keywords
Banach space, Radon-Nikodym property characterization, superre exivity characteriza-
tion, Szlenk index, dentability index, approximation of norms, LUR, smoothness, smooth
variational principle, perturbed minimization, continuous dependence of minimizers
Institut de Mathematiques de Bordeaux,
Universite Bordeaux 1
351, cours de la Liberation - F 33405 TALENCETo BlancaAcknowledgment
This thesis has been written within the \cotutelle" program under the supervision of Petr
H ajek on the Czech side and Robert Deville on the French side. I am sincerely and deeply
grateful to both my advisors for many things. They have introduced many beautiful areas
of modern analysis to me, they have shared their outstanding knowledge with me, guided
me through my research, always pointing me in the right direction. With both Robert
and Petr we have spent a lot of time discussing mathematics or just friendly talking.
They have been ready to meet me and answer my questions virtually any time. Thanks
are due to Petr for putting me in contact with Robert and thanks are due to Robert for
the warm welcome he gave me when I rst came to France. Also, I thank Robert for his
sel ess help with the translation of the introduction of this thesis to French.
I would like to thank Gilles Lancien for his kind invitation to Besan con to collaborate
on the topic which now forms Chapter 3 of this thesis. I have been honored by this
opportunity, I took great pleasure in working with Gilles, and I am grateful for the
inspiring discussions and the time he dedicated to me.
For friendly encouragement and excellent working conditions I thank the members of
the Department of Topology and Functional Analysis of the Academy of Sciences of the
Czech Republic, the Department of Mathematical Analysis of Charles University and the
Institut de Mathematiques de Bordeaux.
I would like to thank the French Government and the Fond Mobility of Charles Uni-
versity for the nancial support of my stay in France. The research presented in this
thesis was also supported by the grants: Institutional Research Plan AV0Z10190503,
A100190502, A100190801, GA CR 201/07/0394.
I thank my parents, sister and grandparents for always encouraging and supporting
me no matter how far from them I was.
The one who deserves my sincerest thanks is Blanca who believed in me and inspired
me during all my studies. For your love and patience, this work is dedicated to you.
Tony Proch azka
Bordeaux, April 2009
56Contents
1 Introduction 9
1.1 Introduction (English) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
1.1.1 The Radon-Nikodym property . . . . . . . . . . . . . . . . . . . . . 9
1.1.2 Quantitative aspects of the Radon-Nikodym property . . . . . . . . 11
1.1.3 The Radon-Nikodym property in dual spaces . . . . . . . . . . . . . 12
(! )11.1.4 Spaces C(K), K =; . . . . . . . . . . . . . . . . . . . . . . . . 14
1.1.5 Norms with good properties . . . . . . . . . . . . . . . . . . . . . . 15
1.1.6 A parametric variational principle . . . . . . . . . . . . . . . . . . . 18
1.2 General notions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
1.3 Uvod (cesky) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
1.3.1 Radonova-Nikodymo va vlastnost . . . . . . . . . . . . . . . . . . . 21
1.3.2 Kvantitativn rysy Radonovy-Nikodymo vy vlastnosti . . . . . . . . 23
1.3.3 Radonova-Nikodymo va vlastnost v du alech . . . . . . . . . . . . . . 24
(! )11.3.4 Prostory C(K), K =; . . . . . . . . . . . . . . . . . . . . . . . 26
1.3.5 Normy s dobrymi vlastnostmi . . . . . . . . . . . . . . . . . . . . . 26
1.3.6 Parametricky variacn princip . . . . . . . . . . . . . . . . . . . . . 30
1.4 Introduction (version fran caise) . . . . . . . . . . . . . . . . . . . . . . . . 32
1.4.1 La propriete de Radon-Nikodym . . . . . . . . . . . . . . . . . . . . 32
1.4.2 Aspects quantitatifs de la propriete de Radon-Nikodym . . . . . . . 34
1.4.3 La propriete de Radon-Nikodym dans les duaux . . . . . . . . . . . 35
(! )11.4.4 Les espaces C(K), K =; . . . . . . . . . . . . . . . . . . . . . . 37
1.4.5 Normes avec de bonnes proprietes . . . . . . . . . . . . . . . . . . . 37
1.4.6 Un principe variationnel parametrique . . . . . . . . . . . . . . . . 41
2 Games 45
2.1 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
2.1.1 Games, tactics, strategies . . . . . . . . . . . . . . . . . . . . . . . 45
2.1.2 Point-slice games . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46
2.1.3 (Small) slices inside slices . . . . . . . . . . . . . . . . . . . . . . . 47
2.2 No open winning tactics . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
2.3 Characterization of the RNP . . . . . . . . . . . . . . . . . . . . . . . . . . 49
2.3.1 "-slicings, "-tactics . . . . . . . . . . . . . . . . . . . . . . . . . . . 51
78 CONTENTS
2.3.2 Re ning "-slicings . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51
2.3.3 Stability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
2.3.4 Induction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54
2.4 Winning tactics and uniformly rotund norms . . . . . . . . . . . . . . . . . 56
2.5 Baire one functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58
(! )13 Weak dentability index of C(K), K =; 63
3.1 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63
3.2 Weak dentability index of C([0;]) . . . . . . . . . . . . . . . . . . . . . . 66
3.2.1 The upper estimate . . . . . . . . . . . . . . . . . . . . . . . . . . . 66
3.2.2 The lower . . . . . . . . . . . . . . . . . . . . . . . . . . . 68
3.3 The general case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71
4 Approximation of norms 73
4.1 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73
4.1.1 Functions that locally depend on nitely many coordinates . . . . . 73
4.1.2 Facts about convexity . . . . . . . . . . . . . . . . . . . . . . . . . 75
4.1.3 Projectional resolution of identity . . . . . . . . . . . . . . . . . . . 76
4.1.4 Approximation of norms . . . . . . . . . . . . . . . . . . . . . . . . 77
4.2 Main result . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80
4.3 About N . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82
4.4 About J . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84
4.4.1 A \nice" target space . . . . . . . . . . . . . . . . . . . . . . . . . . 85
4.4.2 Mapping X into the \nice" space . . . . . . . . . . . . . . . . . . . 88
4.4.3 The de nition of J . . . . . . . . . . . . . . . . . . . . . . . . . . . 92
k4.4.4 J is a limit of C -smooth norms . . . . . . . . . . . . . . . . . . . . 94
5 A parametric variational principle 97
5.1 Space of perturbations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98
5.2 The existence of an approximate minimum . . . . . . . . . . . . . . . . . . 101
5.2.1 Interesting facts about convex functions . . . . . . . . . . . . . . . 104
5.3 Parametric variational principle . . . . . . . . . . . . . . . . . . . . . . . . 106
5.4 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112
Index 119
List of Symbols 121
Bibliography 123Chapter 1
Introduction
1.1 Introduction (English)
We are going to deal here with four di erent topics in the theory of Banach spaces and it
is the aim of the introductory chapter to show which place in the theory they occupy.
Here and throughout, X is a real Banach space with a closed unit ball B and withX
the dual space X .
We start with a de nition of an elementary (yet fundamental) concept. Let A be a set
in a Banach space X. Let f2 X nf0g and a2R. If the set S =fx2A :f(x)>ag
is nonempty, it is called an open slice of A (or just a slice of A when no confusion may
arise). We denoteS (A) the set of all slices of A.o
With the notion of slice in hand we may de ne an important class of a Banach spaces
which we will meet constantly throughout this text. A Banach space X is said to have
the Radon-Nikodym property (RNP) if every bounded non-empty subset of X has slices
(nonempty by the de nition) of arbitrarily small diameter. More precisely, X has the
RNP if for every bounded non-empty subset A of X and every " > 0 there is a slice
S2S (A) such that diam(S)<".o
Because of the universal quanti er in the de nition, the RNP is naturally an isomor-
phic property. Its importance dwells in the fact that many familiar constructions on the
real line can be translated to the spaces with the RNP. An example of our claim is the
original, measure theoretic, de nition of the RNP which also explains the name of the
property: LetB be the Borel sets over [0; 1], be the Lebesgue measure on [0; 1]. A Banach
space X has the RNP if and only if every X-valued measure m on the probability space
([0; 1];B;) which is of nite total variation and absolutely continuous with respect to , is
R
1represented by a mappingf2L ([0; 1];X) by means of the equalitym(A) = f(x)d (x).
A
To see how various mathematicians contributed to this result we refer the reader to
the excellent monograph [Bou83]. The above theorem is a (rather heavy) means to see
910 CHAPTER 1. INTRODUCTION
nthat e.g. R enjoys the RNP.
Our rst result is a generalization of the elementary fact that bounded monotone
sequences of real numbers converge.
Theorem A. For a Banach space X it is equivalent:
(i) X has the RNP,
(ii) there exists a mapping F :X!X such that every bounded sequence (x )Xn
is convergent whenever it satis es
hF (x );xihF (x );x i for every n2N:n n n n+1
2Already in X = R , it is not obvious that (ii) holds, so this theorem is not really a
good tool to positively determine that some space has the RNP. On the other hand, it is
very useful as a su cient condition for the convergence of sequences in X with the RNP.
The implication (ii)) (i) is due to R. Deville and E. Matheron [DM07] who also proved
that ifX admits a uniformly rotund norm, then (ii) holds. The implication (i)) (ii) was
proved in full generality in [Pro09] and we will see the proof in Chapter 2.
LetS (X) be the set of all closed halfspaces of X, i.e. the sets of the formfx2 X :c
f(x)ag for somef2X nf0g anda2R. In fact, the above theorem is a reformulation
of a theorem operating with the notion of the point-closed halfspace game G(X;S (X))c
which we will now describe. There are two players { Player I and Player II. Player I starts
the game by choosing arbitrarily a point x 2X. Player II then plays a closed halfspace1
H containing the point x ; then Player I picks a point x 2H and Player’s II answer is1 1 2 1
a closed halfspace H which contains x (but not necessarily x ); then Player I chooses a2 2 1
point x in H (but not necessarily in H ); and so on. The above is called a run of the3 2 1
game G(X;S (X)). Player II wins a run if the resulting sequence (x ) is either Cauchyc n
or unbounded. A winning tactic for Player II is a mapping t : X !S (X) such thatc
it respects the rules of the game, i.e. x2 t(x), and such that Player II wins a run in
which he or she always chooses H := t(x ). It is easily seen (cf. Proposition 2.4) thatn n
(ii) in Theorem A is equivalent to saying that Player II has a winning tactic in the game
G(X;S (X)).c
In a more abstract setting, if K is any set in X andA is a collection of subsets of K
S
such thatK = A, we de ne the point-set game G(K;A) verbatim (see De nition 2.1).
The game design is due to J. Maly and M. Zeleny [MZ06] who also proved, for the
game G(B 2;flinesg), that Player II has a winning strategy { a decision rule represented
R
nby a sequence of mappings t : K ! A whose application, i.e. the choice H :=n n
t (x ;:::;x ), insures the victory of Player II. Note that any winning tactic of Player IIn 1 n
is automatically a winning strategy of Player II but the converse does not hold. Further,
it is proved in [DM07] that if X enjoys the RNP, then Player II has a winning strategy
in G(B ;S (B )). Surprisingly enough, we will show here the following:X o X