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.

Citation

Pakhrou, T. (2012), "Some subspaces simultaneously proximinal", Kybernetes, Vol. 41 No. 1/2, pp. 108-115. https://doi.org/10.1108/03684921211213142