A new proof of Fréchet differentiability of Lipschitz functions
type: Item
instance of: scholarly article
title: A new proof of Fréchet differentiability of Lipschitz functions
zbMATH Open document ID: 0968.58006
published in: Journal of the European Mathematical Society (JEMS)
publication date: 2001-09-18T00:00:00Z
review text: As well known, if \(f\) is a real valued Lipschitz function from an open set \(G\) (in a separable Banach space), then \(f\) is Gateaux differentiable in a subset \(D\) whose complement in \(G\) is a Gauss null set. Let \(A=\{f'(x)\); \(x\in D\}\). Assuming that the \(w^*\)-closure of \(A\) is norm separable, it is proved that \(G\) contains a point of Fréchet differentiability of \(f\). Moreover, for every segment \([u,v]\subset G\) and every \(m<f(v) -f(u)\), there is a point \(x\) of Fréchet differentiability of \(f\) such that \(f'(x)(v-u) >m\). The argument is self-contained and exhibits an algorithm for producing a sequence of points \((x_n)_n\) which converge to a point of Fréchet differentiability.
Mathematics Subject Classification ID: 58C20
Mathematics Subject Classification ID: 46G05
zbMATH DE Number: 1529495
zbMATH Keywords: Lipschitz function
zbMATH Keywords: Fréchet differentiability
author: J. Lindenstrauss
author: David Preiss
MaRDI profile type: Publication
full work available at URL: s100970000019
OpenAlex ID: W2027087150
DOI: 10.1007/S100970000019
description: scientific article; zbMATH DE number 1529495