Best Simultaneous Approximation in Metric Spaces

Abstract

For a Banach space X and an increasing subadditive continuous function φ on [0,∞) with φ(0) = 0, let us denote by L^φ(I,X), the space of all X-valued φ-integrable functions f : I → X on a certain positive complete σ-finite measure space (equation). The aim of this paper is to prove that for a closed separable subspace G of X, L^φ(I,G) is simultaneously proximinal in L^φ(I,X) if and only if G is simultaneously proximinal in X. Other result on simultaneous approximation of l^φ(G) in l^φ(X) is presented.

Key words and phrases. Simultaneous, Approximation.

1991 Mathematics Subject Classification. Primary: 41A65; Secondary: 41A50