On the Weierstrass-Stone theorem
Let \(S\) be a compact Hausdorff space, let \(E\) be a normed space, and let \(C(S,E)\) be the space of continuous functions from \(S\) to \(E\) with the usual supremum norm. A collection \(M\) of \([0,1]\)-valued continuous functions on \(S\) is said to have property \(V\) if (1) \(\varphi\in M\Leftrightarrow 1-\varphi\in M\) and (2) \(\varphi,\psi\in M \Leftrightarrow \varphi\psi\in M\). For a subset \(W\) of \(C(S,E)\), a continuous function \(\varphi: S\to [0,1]\) is said to be a multiplier of \(W\) if \(f,g\in W\to\varphi f+ (1-\varphi) g\in W\). It is easy to see that the set \(M\) of multipliers of \(W\) has property V; also, it is a non- trivial result that \(\overline{M}\) is a lattice see \textit{R. I. Jewett}, Proc. Am. Math. Soc. 14, 690-693 (1963; Zbl 0113.148)]. Using these ideas, the author proves the following main theorem, which is a generalization of \textit{S. Machado's} Stone-Weierstrass theorem. (see e.g., \textit{S. Machado} [Indag. Math. 39, 218-224 (1977; Zbl 0412.46029)] or \textit{J. B. Prolla} [Approximation of vector valued functions (1977; Zbl 0373.46048)]. See also \textit{T. J. Ransford} [Math. Proc. Camb. Philos. Soc. 96, 309-311 (1984; Zbl 0537.41018)] for a very short proof which uses Zorn's Lemma.) Theorem. Let \(M\) be a set of multipliers of \(W\) which separates the points of \(S\), let \(f\in C(S,E)\), and let \(\varepsilon>0\). Then there is a function \(g\in W\) such that \(| f-g|< \varepsilon \Leftrightarrow\) for all \(x\in S\), there is a function \(g_ x\in W\) such that \(| f(x)- g_ x(x)|< \varepsilon\). Among several interesting examples and applications, the author shows the following: Let \(A\subset C(S,E)\) such that for all finite subsets \(F\subset S\) and all \(f\in C(S,E)\), there is a function \(g\in A\) with \(g|_ F= f_ F\). If, in addition, the set of multipliers of \(A\) separates the points of \(S\), then simultaneous approximation and interpolation is possible; i.e., for all finite subsets \(F\subset S\), for all \(f\in C(S,E),\) and for all \(\varepsilon\), there is \(g\in A\) such that \(g|_ F= f|_ F\) and \(| f-g|< \varepsilon\).
scientific article; zbMATH DE number 644989
scientific article