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Best simultaneous approximation in L∞(μ, X)

✍ Scribed by Tijani Pakhrou

Publisher
John Wiley and Sons
Year
2008
Tongue
English
Weight
106 KB
Volume
281
Category
Article
ISSN
0025-584X

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✦ Synopsis

Abstract

Let 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 Σ~0~ is a sub‐σ ‐algebra of Σ, and let μ~0~ be the restriction of μ to Σ~0~. Given a natural number n, let N be a monotonous norm in ℝ^n^ . We prove that L~∞~(μ, Y) is N ‐simultaneously proximinal in L~∞~(μ,X), and that if X is reflexive then L~∞~(μ~0~, 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)

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