Some subspaces simultaneously proximinal

Abstract

Purpose – In this paper the aim is to present some subspace simultaneously proximinal in the Banach space L1(μ, X) of X‐valued Bochner μ‐integrable functions.Design/methodology/approach – By lower semicontinuity and compactness the existence of best simultaneous approximation is obtained.Findings – If Y is a reflexive subspace of a Banach space X, then L1(μ, Y) is simultaneously proximinal in L1(μ, X). Furthermore, if X is reflexive and μ0 is the restriction of μ to a sub‐σ‐algebra, then L1(μ0, X) is simultaneously proximinal in L1(μ, X).Practical implications – Given a finite number of points in the Banach space X, is about finding a point in the subspace Y⊂X that comes close to all this points.Originality/value – By the property of reflexivity two types subspaces simultaneously proximinal in L1(μ, X) are obtained.