Stone-Weierstrass theorems in \(C^*(X)\)
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.
scientific article; zbMATH DE number 1552980