The embedding of proximinal sets (Q1083634)

Let M be a proximinal subset of a normed space X (i.e. \(\inf \{\| x- m\|\), \(m\in M\}\) is attained for each \(x\in X)\). This paper deals with the following general question: if X is embedded (isometrically) in another normed space Z, is M proximinal also as a subset of Z? Various results are given in the case when \(X=C(S)\) and \(Z=C(S\times T)\), where S and T are compact Hausdorff spaces. In particular, the proximinal subspaces of C(S) which are proximinal in C(S\(\times T)\) for every T are characterized. A generalization of Mazur's proximinality theorem is also proved; this generalization gives a condition under which a subspace of functions \(v\circ f\) is proximinal, when f is held fixed and v ranges over a proximinal subspace.