Fréchet Differentiability of Lipschitz Functions and Porous Sets in Banach Spaces (AM-179)

Joram Lindenstrauss; David Preiss; Jaroslav Tišer

Joram Lindenstrauss, David Preiss and Jaroslav Tišer

Fréchet Differentiability of Lipschitz Functions and Porous Sets in Banach Spaces (AM-179)

In: Annals of Mathematics Studies, 179

Princeton University Press | 2012

https://doi.org/10.1515/9781400842698

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About this book

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.

Author information

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 Tier is associate professor of mathematics at Czech Technical University in Prague.

Reviews

[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

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

Frontmatter Free access PDF PDF	i
Contents Free access PDF PDF	vii
Chapter One: Introduction Access restricted Content is available PDF PDF	1
Chapter Two: Gâteaux differentiability of Lipschitz functions Access restricted Content is available PDF PDF	12
Chapter Three: Smoothness, convexity, porosity, and separable determination Access restricted Content is available PDF PDF	23
Chapter Four: ε-Fréchet differentiability Access restricted Content is available PDF PDF	46
Chapter Five: Γ-null and Γn-null sets Access restricted Content is available PDF PDF	72
Chapter Six: Férchet differentiability except for Γ-null sets Access restricted Content is available PDF PDF	96
Chapter Seven: Variational principles Access restricted Content is available PDF PDF	120
Chapter Eight: Smoothness and asymptotic smoothness Access restricted Content is available PDF PDF	133
Chapter Nine: Preliminaries to main results Access restricted Content is available PDF PDF	156
Chapter Ten: Porosity, Γn- and Γ-null sets Access restricted Content is available PDF PDF	169
Chapter Eleven: Porosity and ε-Fréchet differentiability Access restricted Content is available PDF PDF	202
Chapter Twelve: Fréchet differentiability of real-valued functions Access restricted Content is available PDF PDF	222
Chapter Thirteen: Fréchet differentiability of vector-valued functions Access restricted Content is available PDF PDF	262
Chapter Fourteen: Unavoidable porous sets and nondifferentiable maps Access restricted Content is available PDF PDF	319
Chapter Fifteen: Asymptotic Fréchet differentiability Access restricted Content is available PDF PDF	355
Chapter Sixteen: Differentiability of Lipschitz maps on Hilbert spaces Access restricted Content is available PDF PDF	392
Bibliography Access restricted Content is available PDF PDF	415
Index Access restricted Content is available PDF PDF	419
Index of Notation Access restricted Content is available PDF PDF	423

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Fréchet Differentiability of Lipschitz Functions and Porous Sets in Banach Spaces (AM-179)

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