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

Author/Creator:
Lindenstrauss, Joram, 1936-
Publication:
Princeton : Princeton University Press, 2012.
Format/Description:
Book
1 online resource (436 p.)
Edition:
Course Book
Series:
Annals of mathematics studies ; no. 179.
Annals of mathematics studies ; no. 179
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Subjects:
Banach spaces.
Calculus of variations.
Functional analysis.
Form/Genre:
Electronic books.
Language:
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.
Contributor:
Preiss, David.
Tišer, Jaroslav, 1957-
ISBN:
1-283-37995-3
9786613379955
1-4008-4269-7
OCLC:
769343169
Publisher Number:
10.1515/9781400842698 doi