Best simultaneous approximation in L∞(μ, X)

Abstract

AbstractLet Y be a reflexive subspace of the Banach space X, let (Ω, Σ, μ) be a finite measure space, and let L∞(μ, X) be the Banach space of all essentially bounded μ ‐Bochner integrable functions on Ω with values in X, endowed with its usual norm. Let us suppose that Σ0 is a sub‐σ ‐algebra of Σ, and let μ0 be the restriction of μ to Σ0. Given a natural number n, let N be a monotonous norm in ℝn . We prove that L∞(μ, Y) is N ‐simultaneously proximinal in L∞(μ,X), and that if X is reflexive then L∞(μ0, X) is N ‐simultaneously proximinal in L∞(μ, X) in the sense of Fathi, Hussein, and Khalil [3]. (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)