On lower semi-continuous metric projections onto finite dimensional subspaces of spaces of continuous functions
Let \(T\) be a locally compact Hausdorff topological space and \(G\) a finite dimensional subspace of \(C_0(T)\) (the space of continuous real-valued functions vanishing at infinity, with the sup-norm). The paper is concerned with characterizations of the subspaces \(G\) for which the set-valued metric projection \(P_G\) is lower semi-continuous, completing the characterizations given by \textit{Wu Li} [J. Approximation Theory 57, No. 2, 150--168 (1989; Zbl 0688.41037)] (see also the book [\textit{Wu Li}, Continuous selections for metric projections and interpolating subspaces. Frankfurt a.\,M.: Peter Lang (1991; Zbl 0743.41029)]) and the author [\textit{A. L. Brown}, Suppl. Rend. Circ. Mat. Palermo (2) 76, 17--44 (2005; Zbl 1148.41032)]. The characterizations are given in terms of the continuous mapping \(e:T\to G^*\) given by \(e(t)(g)=g(t)\), \(g\in G\), and a set of five equivalent conditions of Haar-type, each of them being equivalent to the fact that \(G\) is a Chebyshev subspace of \(C_0(T)\), in which case the metric projection \(P_G\) is single-valued and continuous. One introduces a generalized Haar condition (GHC), involving a decomposition of \(T\) into closed subsets, which is necessary for the continuity of \(P_G\), and, for some spaces \(T\), it is also sufficient. It is shown by an example that, in the case of a compact and totally discrete infinite compact space \(T\), the situation is essentially different.
scientific article; zbMATH DE number 6137309