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Fréchet Differentiability of Lipschitz Functions and Porous Sets in Banach Spaces (AM-179)$
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Joram Lindenstrauss, David Preiss, and Jaroslav Tier

Print publication date: 2012

Print ISBN-13: 9780691153551

Published to Princeton Scholarship Online: October 2017

DOI: 10.23943/princeton/9780691153551.001.0001

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Fr ´Echet Differentiability Except For Γ‎-Null Sets

Fr ´Echet Differentiability Except For Γ‎-Null Sets

Chapter:
(p.96) Chapter Six Fr ´Echet Differentiability Except For Γ‎-Null Sets
Source:
Fréchet Differentiability of Lipschitz Functions and Porous Sets in Banach Spaces (AM-179)
Author(s):

Joram Lindenstrauss

David Preiss

Tiˇser Jaroslav

Publisher:
Princeton University Press
DOI:10.23943/princeton/9780691153551.003.0006

This chapter gives an account of the known genuinely infinite dimensional results proving Fréchet differentiability almost everywhere except for Γ‎-null sets. Γ‎-null sets provide the only notion of negligible sets with which a Fréchet differentiability result is known. Porous sets appear as sets at which Gâteaux derivatives can behave irregularly, and they turn out to be the only obstacle to validity of a Fréchet differentiability result Γ‎-almost everywhere. Furthermore, geometry of the space may (or may not) guarantee that porous sets are Γ‎-null. The chapter also shows that on some infinite dimensional Banach spaces countable collections of real-valued Lipschitz functions, and even of fairly general Lipschitz maps to infinite dimensional spaces, have a common point of Fréchet differentiability.

Keywords:   infinite dimensional space, Fréchet differentiability, Γ‎-null sets, porous sets, Gâteaux derivative, Banach space, Lipschitz function, Lipschitz map

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