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.
Keywords
Citation
Pakhrou, T. (2012), "Some subspaces simultaneously proximinal", Kybernetes, Vol. 41 No. 1/2, pp. 108-115. https://doi.org/10.1108/03684921211213142
Publisher
:Emerald Group Publishing Limited
Copyright © 2012, Emerald Group Publishing Limited