Lindenstrauss
Joram
Joram Lindenstrauss
1936
2012
Mathematics
MATHEMATICS--Calculus
Frechet Differentiability of Lipschitz Functions and Porous Sets in Banach Spaces.
Print version:
electronic bk.
9781400842698
1400842697
Preiss
David
David Preiss
MATHEMATICS--Mathematical Analysis
Banach spaces
Calculus of variations
nju
2015-02-27
Annals of mathematics studies ;
Functional analysis
Princeton University Press
Princeton
2012
MATHEMATICS--Set Theory
Tišer
Jaroslav
Jaroslav Tišer
1957
2012
Frechet Differentiability of Lipschitz Functions and Porous Sets in Banach Spaces
Electronic books
769343169
en
769343169
Cover; Title Page; Copyright Page; Table of Contents; Chapter 1. Introduction; 1.1 Key notions and notation; Chapter 2. Gâteaux Dfferentiability of Lipschitz Functions; 2.1 Radon-Nikodým Property; 2.2 Haar and Aronszajn-Gauss Null Sets; 2.3 Existence Results for Gâteaux Derivatives; 2.4 Mean Value Estimates; Chapter 3. Smoothness, Convexity, Porosity, and Separable Determination; 3.1 A criterion of Differentiability of Convex Functions; 3.2 Fréchet Smooth and Nonsmooth Renormings; 3.3 Fréchet Differentiability of Convex Functions; 3.4 Porosity and Nondifferentiability.
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.