On lower semi-continuous metric projections onto finite dimensional subspaces of spaces of continuous functions

Abstract

The context of the paper is: a locally compact Hausdorff space T; the space C0(T), equipped with the uniform norm, of real continuous functions on T which vanish at infinity; a linear subspace G, of finite dimension n, of the space C0(T); and the set-valued metric projection PG of C0(T) into the family of non-empty compact convex subsets of G, defined by PG(f)= the set of best uniform approximations to f from G. Those G which are Chebyshev (that is the metric projection is single point valued) were characterised by Haar (1918). Those G for which the metric projection PG is lower semi-continuous have been characterised by Wu Li (1989) and A.L. Brown (2005); the paper interprets and exploits the characterisation. A ‘Generalised Haar Condition’ which is necessary for PG to be lower semi-continuous is identified; in some circumstances it is also sufficient. The condition has a calculable determinantal form. The results of the paper include an essentially complete determination, in case T is a compact space in which no net of components is convergent to a single point set, of those subspaces G of C(T) for which PG is lower semi-continuous. Simple examples show that if the space T is compact and totally disconnected, but not finite, the situation is essentially different.