Fréchet Differentiability of Lipschitz Functions and Porous Sets in Banach Spaces

by Joram Lindenstrauss, David Preiss, Jaroslav Tiser

Overview

This book makes a significant inroad into the unexpectedly difficult question of existence of Fréchet derivatives of Lipschitz maps of Banach spaces into higher dimensional spaces. Because the question turns out to be closely related to porous sets in Banach spaces, it provides a bridge between descriptive set theory and the classical topic of existence of derivatives of vector-valued Lipschitz functions. The topic is relevant to classical analysis and descriptive set theory on Banach spaces. The book opens several new research directions in this area of geometric nonlinear functional analysis.


The new methods developed here include a game approach to perturbational variational principles that is of independent interest. Detailed explanation of the underlying ideas and motivation behind the proofs of the new results on Fréchet differentiability of vector-valued functions should make these arguments accessible to a wider audience. The most important special case of the differentiability results, that Lipschitz mappings from a Hilbert space into the plane have points of Fréchet differentiability, is given its own chapter with a proof that is independent of much of the work done to prove more general results. The book raises several open questions concerning its two main topics.

Product Details

ISBN-13:	9780691153551
Publisher:	Princeton University Press
Publication date:	02/26/2012
Series:	Annals of Mathematics Studies , #179
Pages:	440
Product dimensions:	6.30(w) x 9.30(h) x 1.10(d)

About the Author

Joram Lindenstrauss is professor emeritus of mathematics at the Hebrew University of Jerusalem. David Preiss is professor of mathematics at the University of Warwick. Jaroslav Tišer is associate professor of mathematics at Czech Technical University in Prague.

Table of Contents

1 Introduction 1

1.1 Key notions and notation 9

2 Gateaux differentiability of Lipschitz functions 12

2.1 Radon-Nikodým property 12

2.2 Haar and Aronszajn-Gauss null sets 13

2.3 Existence results for Gateaux derivatives 15

2.4 Mean value estimates 16

3 Smoothness, convexity, porosity, and separable determination 23

3.1 A criterion of differentiability of convex functions 23

3.2 Fréchet smooth and nonsmooth renormings 24

3.3 Fréchet differentiability of convex functions 28

3.4 Porosity and nondifferentiability 31

3.5 Sets of Fréchet differentiability points 33

3.6 Separable determination 37

4 ε-Fréchet differentiability 46

4.1 ε-differentiability and uniform smoothness 46

4.2 Asymptotic uniform smoothness 51

4.3 ε-Fréchet differentiability of functions on asymptotically smooth spaces 59

5 Γ-null and Γ_n-null sets 72

5.1 Introduction 72

5.2 Γ-null sets and Gâteaux differentiability 74

5.3 Spaces of surfaces, and Γ - and Γ_n-null sets 76

5.4 Γ- and Γ_n-null sets of low Borel classes 81

5.5 Equivalent definitions of Γ_n-null sets 87

5.6 Separable determination 93

6 Fréchet differentiability except for Γ-null sets 96

6.1 Introduction 96

6.2 Regular points 97

6.3 A criterion of Fréchet differentiability 100

6.4 Fréchet differentiability except for Γ-null sets 114

7 Variational principles 120

7.1 Introduction 120

7.2 Variational principles via games 122

7.3 Bimetric variational principles 127

8 Smoothness and asymptotic smoothness 133

8.1 Modulus of smoothness 133

8.2 Smooth bumps with controlled modulus 141

9 Preliminaries to main results 156

9.1 Notation, linear operators, tensor products 156

9.2 Derivatives and regularity 157

9.3 Deformation of surfaces controlled by ω_n 161

9.4 Divergence theorem 164

9.5 Some integral estimates 165

10 Porosity, Γ_n- and Γ -null sets 169

10.1 Porous and σ-porous sets 169

10.2 A criterion of Γ_n-nullness of porous sets 173

10.3 Directional porosity and Γ_n-nullness 186

10.4 σ-porosity and Γ_n-nullness 189

10.5 Γ₁-nullness of porous sets and Asplundness 192

10.6 Spaces in which σ-porous sets are Γ-null 198

11 Porosity and ε-Fréchet differentiability 202

11.1 Introduction 202

11.2 Finite dimensional approximation 203

11.3 Slices and ε-differentiability 208

12 Fréchet differentiability of real-valued functions 222

12.1 Introduction and main results 222

12.2 An illustrative special case 225

12.3 A mean value estimate 230

12.4 Proof of Theorems 12.1.1 and 12.1.3 234

12.5 Generalizations and extensions 261

13 Fréchet differentiability of vector-valued functions 262

13.1 Main results 262

13.2 Regularity parameter 263

13.3 Reduction to a special case 269

13.4 Regular Fréchet differentiability 289

13.5 Fréchet differentiability 304

13.6 Simpler special cases 317

14 Unavoidable porous sets and nondifferentiable maps 319

14.1 Introduction and main results 319

14.2 An unavoidable porous set in l₁ 325

14.3 Preliminaries to proofs of main results 332

14.4 The main construction, Part I 339

14.5 The main construction, Part II 344

14.6 Proof of Theorem 14.1.3 347

14.7 Proof of Theorem 14.1.1 351

15 Asymptotic Fréchet differentiability 355

15.1 Introduction 355

15.2 Auxiliary and finite dimensional lemmas 359

15.3 The algorithm 363

15.4 Regularity of f at x_∞ 372

15.5 Linear approximation of / at x_∞ 380

15.6 Proof of Theorem 15.1.3 389

16 Differentiability of Lipschitz maps on Hilbert spaces 392

16.1 Introduction 392

16.2 Preliminaries 394

16.3 The algorithm 396

16.4 Proof of Theorem 16.1.1 403

16.5 Proof of Lemma 16.2.1 403

Bibliography 415

Index 419

Index of Notation 423

Editorial Reviews

"The book is well written—as one would expect from its distinguished authors, including the late Joram Lindestrauss (1936-2012). It contains many fascinating and profound results. It no doubt will become an important resource for anyone who is seriously interested in the differentiability of functions between Banach spaces."—-J. Borwein and Liangjin Yao, Mathematical Reviews Clippings

"[T]his is a very deep and complete study on the differentiability of Lipschitz mappings between Banach spaces, an unavoidable reference for anyone seriously interested in this topic."—-Daniel Azagra, European Mathematical Society

"We should be grateful to (the late) Joram Lindenstrauss, David Preiss, and Jaroslav Tiser for providing us with this splendid book which dives into the deepest fields of functional analysis, where the basic but still strange operation called differentiation is investigated. More than a century after Lebesgue, our understanding is not complete. But thanks to the contribution of these three authors, and thanks to this book, we know a fair share of beautiful theorems and challenging problems."—-Gilles Godefroy, Bulletin of the American Mathematical Society

From the Publisher