Approximation with monotone norms in tensor product spaces (Q1185940)

A monotone norm on \(C(S)\), \(S\) a compact Hausdorff space, is a norm \(\|\cdot\|_ \alpha\) satisfying \(\| x\|_ \alpha\leq\| y\|_ \alpha\) for \(0\leq x\leq y\) in \(C(S)\). A typical example of a monotone norm is an \(L^ p\)-norm on \(C(S)\). If \(Y\) is a normed space then \(\|\cdot\|_ \alpha\) can be lifted to \(C(S,Y)\) by \(\| f\|_ \alpha=\| f(s)\|\), for \(s\in S\) and \(f\in C(S,Y)\). A first problem considered in this paper is that of proximinality of the subspaces of \(C(S,Y)\). The strict convexity of \(C(S,Y)\) is also studied - -- \(C(S,Y)\) is strictly convex if and only if \(\|\cdot\|_ \alpha\) and \(Y\) are strictly convex. A monotone norm \(\|\cdot\|_ \alpha\) on \(C(S)\) induces in a natural way a norm on the tensor product \(C(S)\otimes Y\). Following the ideas developed in \textit{W. A. Light} and \textit{E. W. Cheney} [Approximation Theory in Tensor Product Spaces, Lect. Notes Math. 1169, (1985; Zbl 0575.41001)] the authors study the proximinality of the subspaces of \(C(S)\otimes C(T)\), for \(C(S)\) and \(C(T)\) equipped with monotone norms.