Fréchet Differentiability of Lipschitz Functions and Porous Sets in Banach Spaces 1st Edition by Joram Lindenstrauss, David Preiss, Jaroslav Tišer 0691153558 9780691153551

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ISBN 10: 0691153558

ISBN 13: 9780691153551

Author: Joram Lindenstrauss; David Preiss; Jaroslav Tišer

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.

Fréchet Differentiability of Lipschitz Functions and Porous Sets in Banach Spaces 1st Table of contents:

1 Introduction

1.1 Key notions and notation

2 Gâteaux differentiability of Lipschitz functions

2.1 Radon-Nikodým property

2.2 Haar and Aronszajn-Gauss null sets

2.3 Existence results for Gâteaux derivatives

2.4 Mean value estimates

3 Smoothness, convexity, porosity, and separable determination

3.1 A criterion of differentiability of convex functions

3.2 Fréchet smooth and nonsmooth renormings

3.3 Fréchet differentiability of convex functions

3.4 Porosity and nondifferentiability

3.5 Sets of Fréchet differentiability points

3.6 Separable determination

4 ε-Fréchet differentiability

4.1 ε-differentiability and uniform smoothness

4.2 Asymptotic uniform smoothness

4.3 ε-Fréchet differentiability of functions on asymptotically smooth spaces

5 Γ-null and Γn-null sets

5.1 Introduction

5.2 Γ-null sets and Gâteaux differentiability

5.3 Spaces of surfaces, and Γ- and Γn-null sets

5.4 Γ- and Γn-null sets of low Borel classes

5.5 Equivalent definitions of Γn-null sets

5.6 Separable determination

6 Fréchet differentiability except for Γ-null sets

6.1 Introduction

6.2 Regular points

6.3 A criterion of Fréchet differentiability

6.4 Fréchet differentiability except for Γ-null sets

7 Variational principles

7.1 Introduction

7.2 Variational principles via games

7.3 Bimetric variational principles

8 Smoothness and asymptotic smoothness

8.1 Modulus of smoothness

8.2 Smooth bumps with controlled modulus

9 Preliminaries to main results

9.1 Notation, linear operators, tensor products

9.2 Derivatives and regularity

9.3 Deformation of surfaces controlled by ωn

9.4 Divergence theorem

9.5 Some integral estimates

10 Porosity, Γn- and Γ-null sets

10.1 Porous and σ-porous sets

10.2 A criterion of Γn-nullness of porous sets

10.3 Directional porosity and Γn-nullness

10.4 σ-porosity and Γn-nullness

10.5 Γ1-nullness of porous sets and Asplundness

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

11 Porosity and ε-Fréchet differentiability

11.1 Introduction

11.2 Finite dimensional approximation

11.3 Slices and ε-differentiability

12 Fréchet differentiability of real-valued functions

12.1 Introduction and main results

12.2 An illustrative special case

12.3 A mean value estimate

12.4 Proof of Theorems 12.1.1 and 12.1.3

12.5 Generalizations and extensions

13 Fréchet differentiability of vector-valued functions

13.1 Main results

13.2 Regularity parameter

13.3 Reduction to a special case

13.4 Regular Fréchet differentiability

13.5 Fréchet differentiability

13.6 Simpler special cases

14 Unavoidable porous sets and nondifferentiable maps

14.1 Introduction and main results

14.2 An unavoidable porous set in ℓ1

14.3 Preliminaries to proofs of main results

14.4 The main construction, Part I

14.5 The main construction, Part II

14.6 Proof of Theorem 14.1.3

14.7 Proof of Theorem 14.1.1

15 Asymptotic Fréchet differentiability

15.1 Introduction

15.2 Auxiliary and finite dimensional lemmas

15.3 The algorithm

15.4 Regularity of f at x∞

15.5 Linear approximation of f at x∞

15.6 Proof of Theorem 15.1.3

16 Differentiability of Lipschitz maps on Hilbert spaces

16.1 Introduction

16.2 Preliminaries

16.3 The algorithm

16.4 Proof of Theorem 16.1.1

16.5 Proof of Lemma 16.2.1

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Tags: Joram Lindenstrauss, David Preiss, Jaroslav Tišer, Differentiability