Best simultaneous approximation in L ∞ ( μ, X )

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)