Fréchet differentiability of Lipschitz functions and porous sets in Banach spaces / Joram Lindenstrauss, David Preiss, Jaroslav Tiser.

Format:

Book

Author/Creator:

Lindenstrauss, Joram, 1936-

Contributor:

Preiss, David.

Tišer, Jaroslav, 1957-

Series:

Annals of mathematics studies ; no. 179.

Annals of mathematics studies ; no. 179

Language:

English

Subjects (All):

Banach spaces.

Calculus of variations.

Functional analysis.

Physical Description:

1 online resource (436 p.)

Edition:

Course Book

Place of Publication:

Princeton : Princeton University Press, 2012.

Language Note:

English

Summary:

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.

Contents:

Frontmatter

Contents

Chapter One: Introduction

Chapter Two: Gâteaux differentiability of Lipschitz functions

Chapter Three: Smoothness, convexity, porosity, and separable determination

Chapter Four: ε-Fréchet differentiability

Chapter Five: Γ-null and Γn-null sets

Chapter Six: Férchet differentiability except for Γ-null sets

Chapter Seven: Variational principles

Chapter Eight: Smoothness and asymptotic smoothness

Chapter Nine: Preliminaries to main results

Chapter Ten: Porosity, Γn- and Γ-null sets

Chapter Eleven: Porosity and ε-Fréchet differentiability

Chapter Twelve: Fréchet differentiability of real-valued functions

Chapter Thirteen: Fréchet differentiability of vector-valued functions

Chapter Fourteen: Unavoidable porous sets and nondifferentiable maps

Chapter Fifteen: Asymptotic Fréchet differentiability

Chapter Sixteen: Differentiability of Lipschitz maps on Hilbert spaces

Bibliography

Index

Index of Notation

Notes:

Description based upon print version of record.

Includes bibliographical references and indexes.

ISBN:

9786613379955

9781283379953

1283379953

9781400842698

1400842697

OCLC:

769343169