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

Joram Lindenstrauss, David Preiss and Jaroslav Tier

Published by Princeton University Press

Published in print February 2012 | ISBN: 9780691153551
Published online October 2017 | e-ISBN: 9781400842698
Fréchet Differentiability of Lipschitz Functions and Porous Sets in Banach Spaces (AM-179)

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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.

Keywords: descriptive set theory; Fréchet derivative; Lipschitz map; Banach space; higher dimensional space; Fréchet differentiability; Lipschitz function; nonlinear functional analysis

Book.  440 pages. 

Subjects: Mathematics

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Table of Contents

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

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

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Smoothness, Convexity, Porosity, and Separable Determination in Fréchet Differentiability of Lipschitz Functions and Porous Sets in Banach Spaces (AM-179)

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<i>Ε‎</i>-Fr ´Echet Differentiability in Fréchet Differentiability of Lipschitz Functions and Porous Sets in Banach Spaces (AM-179)

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Γ‎-Null and Γ‎<i>N</i>-Null Sets in Fréchet Differentiability of Lipschitz Functions and Porous Sets in Banach Spaces (AM-179)

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Fr ´Echet Differentiability Except For Γ‎-Null Sets in Fréchet Differentiability of Lipschitz Functions and Porous Sets in Banach Spaces (AM-179)

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

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

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

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Porosity, Γ‎<i>N</i>- and Γ‎-Null Sets in Fréchet Differentiability of Lipschitz Functions and Porous Sets in Banach Spaces (AM-179)

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