Stone–Weierstrass and extension theorems in the nonlocally convex case

Abstract

In the general (nonlocally convex) case we prove a Stone–Weierstrass-type theorem for sets of continuous vector-valued functions on Hausdorff topological spaces whose compact subsets have finite Lebesgue covering dimension (topological dimension). For such topological space T and Hausdorff topological vector space X (real or complex), in the presence of a separating set of multipliers, the theorem characterizes the closure of a subset in: C(T,X) (resp. C0(T,X) and Cb(T,X)), endowed with the compact-open topology (respectively, the uniform convergence topology and the strict topology). Applications include a Stone–Weierstrass theorem for vector subspaces, range-support uniform approximation results (under constraints on both the range and the support of the approximant function), extension theorems for vector-valued functions, and a short proof of a Schauder-type fixed point theorem. Our noncompact version of the Stone–Weierstrass theorem has significant consequences, among which we mention the extension theorem for vector-valued functions defined on closed subsets of paracompact spaces.