Fréchet Differentiability of Lipschitz Functions and Porous Sets in Banach Spaces (AM-179) (1)
Fréchet Differentiability of Lipschitz Functions and Porous Sets in Banach Spaces (AM-179) (1)
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Abstract
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.
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Front Matter
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One
Introduction
Joram Lindenstrauss and others
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Two
Gâteaux differentiability of Lipschitz functions
Joram Lindenstrauss and others
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Three
Smoothness, convexity, porosity, and separable determination
Joram Lindenstrauss and others
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Four
ε-Fréchet differentiability
Joram Lindenstrauss and others
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Five
Γ-null and Γn-null sets
Joram Lindenstrauss and others
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Six
Fréchet differentiability except for Γ-null sets
Joram Lindenstrauss and others
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Seven
Variational principles
Joram Lindenstrauss and others
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Eight
Smoothness and asymptotic smoothness
Joram Lindenstrauss and others
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Nine
Preliminaries to main results
Joram Lindenstrauss and others
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Ten
Porosity, Γn- and Γ-null sets
Joram Lindenstrauss and others
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Eleven
Porosity and ε-Fréchet differentiability
Joram Lindenstrauss and others
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Twelve
Fréchet differentiability of real-valued functions
Joram Lindenstrauss and others
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Thirteen
Fréchet differentiability of vector-valued functions
Joram Lindenstrauss and others
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Fourteen
Unavoidable porous sets and nondifferentiable maps
Joram Lindenstrauss and others
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Fifteen
Asymptotic Fréchet differentiability
Joram Lindenstrauss and others
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Sixteen
Differentiability of Lipschitz maps on Hilbert spaces
Joram Lindenstrauss and others
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End Matter
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