Best simultaneous approximation on metric spaces via monotonous norms

Abstract

For a Banach space X, L?(T,X) denotes the metric space of all X-valued ?-integrable functions f : T ? X, where the measure space (T,?,?) is a complete positive ?-finite and ? is an increasing subadditive continuous function on [0,?) with ?(0) = 0. In this paper we discuss the proximinality problem for the monotonous norm on best simultaneous approximation from the closed subspace Y?X to a finite number of elements in X.