Fréchet Differentiability of Lipschitz Functions and Porous Sets in Banach Spaces

Joram Lindenstrauss, David Preiss, Jaroslav Tišer

• Publisher: Princeton University Press
• Year: 2012
• Print ISBN: 9780691153568, 9780691153551
• eBook ISBN: 9781400842698
• Topic: Mathematics
• Subtopic: Applied Mathematics
• Index: Mathematics

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.

Table of contents

• Cover
• Title
• Copyright
• Dedication
• Contents
• 1 Introduction
• 2 Gâteaux differentiability of Lipschitz functions
• 3 Smoothness, convexity, porosity, and separable determination
• 4 ε-Fréchet differentiability
• 5 Γ-null and Γn-null sets
• 6 Fréchet differentiability except for Γ-null sets
• 7 Variational principles
• 8 Smoothness and asymptotic smoothness
• 9 Preliminaries to main results
• 10 Porosity, Γn- and Γ-null sets
• 11 Porosity and ε-Fréchet differentiability
• 12 Fréchet differentiability of real-valued functions
• 13 Fréchet differentiability of vector-valued functions
• 14 Unavoidable porous sets and nondifferentiable maps
• 15 Asymptotic Fréchet differentiability
• 16 Differentiability of Lipschitz maps on Hilbert spaces
• Bibliography
• Index
• Index of Notation