Fréchet differentiability of Lipschitz functions and porous sets in Banach spaces

Author(s)

• Lindenstrauss, Joram
• Preiss, David
• Tišer, Jaroslav

Bibliographic Information

Fréchet differentiability of Lipschitz functions and porous sets in Banach spaces

Joram Lindenstrauss, David Preiss, Jaroslav Tišer

（Annals of mathematics studies, no. 179）

Princeton University Press, 2012

• : hbk
• : pbk

Note

Includes bibliography (p. [415]-418) and indexes

Description and Table of Contents

Description

This book makes a significant inroad into the unexpectedly difficult question of existence of Frechet 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 Frechet 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 Frechet 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.

Table of Contents

*FrontMatter, pg. i*Contents, pg. vii*Chapter One: Introduction, pg. 1*Chapter Two: Gateaux differentiability of Lipschitz functions, pg. 12*Chapter Three: Smoothness, convexity, porosity, and separable determination, pg. 23*Chapter Four: epsilon-Frechet differentiability, pg. 46*Chapter Five: GAMMA-null and GAMMAn-null sets, pg. 72*Chapter Six: Ferchet differentiability except for GAMMA-null sets, pg. 96*Chapter Seven: Variational principles, pg. 120*Chapter Eight: Smoothness and asymptotic smoothness, pg. 133*Chapter Nine: Preliminaries to main results, pg. 156*Chapter Ten: Porosity, GAMMAn- and GAMMA-null sets, pg. 169*Chapter Eleven: Porosity and epsilon-Frechet differentiability, pg. 202*Chapter Twelve: Frechet differentiability of real-valued functions, pg. 222*Chapter Thirteen: Frechet differentiability of vector-valued functions, pg. 262*Chapter Fourteen: Unavoidable porous sets and nondifferentiable maps, pg. 319*Chapter Fifteen: Asymptotic Frechet differentiability, pg. 355*Chapter Sixteen: Differentiability of Lipschitz maps on Hilbert spaces, pg. 392*Bibliography, pg. 415*Index, pg. 419*Index of Notation, pg. 423

by "Nielsen BookData"

Details

• NCID: BB08253033
• ISBN:
  • 9780691153551
  • 9780691153568
• LCCN: 2011037616
• Country Code: us
• Title Language Code: eng
• Text Language Code: eng
• Place of Publication: Princeton
• Pages/Volumes: ix, 425 p.
• Size: 25 cm
• Classification:
  • LCC : QA322.2
  • DC23 : 515/.88
• Subject Headings:
  • LCSH : Banach spaces
  • LCSH : Calculus of variations
  • LCSH : Functional analysis
• Parent Bibliography ID:
  • BA00300336