Best N-Simultaneous Approximation in Lp(µ, X)

Abstract

Let X be a Banach space. Let1 ≤ p < ∞and denote by Lp(µ, X) the Banach space of all X-valued Bochner p-integrable functions on a certain positive complete σ-finite measure space (Ω, Σ, µ), endowed with the usual p-norm. In this paper, the theory of lifting is used to prove that, for any weakly compact subset W of X, the set Lp(µ, W) is N-simultaneously proximinal in Lp(µ, X) for any arbitrary monotonous norm N in Rn.