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.
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.