Best Simultaneous Approximation on Metric Spaces via Monotonous Norms

Abstract

For a Banach space X, L^{Φ}(T,X) denotes the metric space of all X-valued Φ-integrable functions f:T→X , where the measure space (T,∑,μ) is a complete positive σ-finite and Φ is an increasing subadditive continuous function on [0,∞) with Φ(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.