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Best simultaneous approximation in L ∞ ( μ, X )
Author(s) -
Pakhrou Tijani
Publication year - 2008
Publication title -
mathematische nachrichten
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.913
H-Index - 50
eISSN - 1522-2616
pISSN - 0025-584X
DOI - 10.1002/mana.200510610
Subject(s) - mathematics , banach space , subspace topology , bounded function , norm (philosophy) , combinatorics , space (punctuation) , discrete mathematics , mathematical analysis , linguistics , philosophy , political science , law
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)