Stone-Weierstrass theorems in \(C^*(X)\)
type: Item
instance of: scholarly article
title: Stone-Weierstrass theorems in \(C^*(X)\)
zbMATH Open document ID: 0971.41021
author: Francisco Montalvo
author: Jorge Bustamante
published in: Journal of Approximation Theory
publication date: 2001-10-28T00:00:00Z
review text: Let \(X\) be a completely regular Hausdorff topological space and let \(C^{\ast }( X) \) be the Banach space of all real-valued bounded continuous functions on \(X\) endowed with the sup-norm. The authors obtain characterizations of the uniform closure of a family \(W\subset C^{\ast}( X) \). The main result is the following: Let \(W\subset C^{\ast}( X) \) and \(f\in C^{\ast}( X) \). Then \(f\in cl( W) \) if and only if for every finite set of real numbers \(\alpha _{1},\alpha_{2},\dots ,\alpha_{n}\) and every \(\varepsilon>0\) there exists \(g\in W\) such that \(|f( x) -\alpha_{i}|<\varepsilon\) implies \(|g( x) -\alpha_{i}|<2\varepsilon\), \((i=1,2,\dots ,n),\)\ \(x\in X\). Imposing some supplementary conditions on the family \(W\) one obtains new characterizations of the \(cl( W) \) as well as simpler proofs for some known characterizations.
reviewed by: Costică Mustăţa
Mathematics Subject Classification ID: 41A65
zbMATH DE Number: 1552980
zbMATH Keywords: Stone-Weierstrass theorems
MaRDI profile type: Publication
full work available at URL: jath.2000.3507
OpenAlex ID: W1978938825
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DOI: 10.1006/JATH.2000.3507
description: scientific article; zbMATH DE number 1552980