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

Nonfiction, Science & Nature, Mathematics, Functional Analysis, Calculus
Cover of the book Fréchet Differentiability of Lipschitz Functions and Porous Sets in Banach Spaces (AM-179) by Joram Lindenstrauss, David Preiss, Jaroslav Tišer, Princeton University Press
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Author: Joram Lindenstrauss, David Preiss, Jaroslav Tišer ISBN: 9781400842698
Publisher: Princeton University Press Publication: February 26, 2012
Imprint: Princeton University Press Language: English
Author: Joram Lindenstrauss, David Preiss, Jaroslav Tišer
ISBN: 9781400842698
Publisher: Princeton University Press
Publication: February 26, 2012
Imprint: Princeton University Press
Language: English

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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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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