Approximation by \(C^p\)-smooth, Lipschitz functions on Banach spaces
type: Item
instance of: scholarly article
title: Approximation by \(C^p\)-smooth, Lipschitz functions on Banach spaces
zbMATH Open document ID: 1111.46061
author: R. Fry
published in: Journal of Mathematical Analysis and Applications
publication date: 2006-03-29T00:00:00Z
review text: The problem of uniformly approximating real continuous functions on Banach spaces by smooth functions has a long history. If the Banach space has a \(C^p\)-smooth bump function, \(C^p\)-smooth approximating functions are sought; the answer is positive, for example, in the class of weakly compactly generated Banach spaces [see, for example, \textit{R.\,Deville, G.\,Godefroy} and \textit{V.\,Zizler}, ``Smoothness and renormings in Banach spaces'' (Pitman Monographs and Surveys in Pure and Applied Mathematics 64, Longman, Harlow and John Wiley \& Sons, New York) (1993; Zbl 0782.46019)]. If the bump function is, moreover, Lipschitz and the Banach space is separable, bounded uniformly continuous functions can be uniformly approximated by Lipschitz \(C^p\)-smooth functions [\textit{D.\,Azagra, R.\,Fry} and \textit{A.\,Montesinos}, Proc.\ Am.\ Math.\ Soc.\ 133, No.\,3, 727--734 (2005; Zbl 1060.46060)]. In this interesting note, answering in the positive a question in the aforesaid paper, independently posed by J.\,A.\,Jaramillo, the boundedness assumption is dropped when \(X\) has an unconditional Schauder basis. Precisely, the main theorem reads: Let \(X\) be a Banach space with an unconditional basis \(\{e_j,e^*_j\}_{j=1}^{\infty}\) which admits a \(C^p\)-smooth Lipschitz bump function. Let \(Y\subset X\) be a subset and \(f:Y\rightarrow \mathbb R\) a uniformly continuous map. Then for each \(\varepsilon>0\) there exists a \(C^p\)-smooth Lipschitz function \(g:X\rightarrow \mathbb R\) such that for all \(y\in Y\), \(| f(y)-g(y)| <\varepsilon\). Moreover, if \(X\) admits a \(C^p\)-smooth norm with unconditional constant \(C\), and if \(f\) is Lipschitz with constant \(\eta\), then we can choose \(g\) to have Lipschitz constant \(12C(1+C)\eta\). As a consequence, in a Banach space \(X\) with unconditional Schauder basis, it is equivalent to have a \(C^p\)-smooth Lipschitz bump function and the fact that for every subset \(Y\subset X\), uniformly continuous map \(f:Y\righta…
reviewed by: V. Montesinos
Mathematics Subject Classification ID: 46T20
Mathematics Subject Classification ID: 46B15
Mathematics Subject Classification ID: 46B20
Mathematics Subject Classification ID: 41A65
Mathematics Subject Classification ID: 46E50
Mathematics Subject Classification ID: 46G05
Mathematics Subject Classification ID: 58C25
zbMATH DE Number: 5016151
zbMATH Keywords: smooth approximation
zbMATH Keywords: Banach spaces
zbMATH Keywords: unconditional Schauder bases
MaRDI profile type: Publication
full work available at URL: j.jmaa.2005.07.009
OpenAlex ID: W1992110643
cites work: On the topological classification of starlike bodies in Banach spaces
cites work: C1-fine approximation of functions on Banach spaces with unconditional basis
cites work: Perturbed smooth Lipschitz extensions of uniformly continuous functions on Banach spaces
cites work: Approximation of Lipschitz functions by \(\Delta\)-convex functions in Banach spaces
cites work: Q4938152
cites work: Q5519417
cites work: Q3134552
cites work: Functional analysis and infinite-dimensional geometry
cites work: Approximation by functions with bounded derivative on Banach spaces
cites work: Q5588008
cites work: Approximation de fonctions différentiables sur certains espaces de Banach
DOI: 10.1016/J.JMAA.2005.07.009
description: scientific article; zbMATH DE number 5016151