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Existence of best simultaneous approximations in L p (S,Σ,X) without the RNP assumption

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Abstract

Let (S,Σ, µ) be a complete positive σ-finite measure space and let X be a Banach space. We consider the simultaneous proximinality problem in L p (S,Σ,X) for 1 ⩽ p < +∞. We establish some N-simultaneous proximinality results of L p (S,Σ0, Y) in L p (S,Σ,X) without the Radon-Nikodým property (RNP) assumptions on the space \(\overline {spanY}\) and its dual \(\overline {spanY} ^*\), where Σ0 is a sub-σ-algebra of Σ and Y a nonempty locally weakly compact closed convex subset of X. In particular, we completely solve one open problem and partially solve another one in Luo et al. (2011).

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Authors and Affiliations

  1. Department of Mathematics, China Jiliang University, Hangzhou, 310018, China

    XianFa Luo

  2. Department of Mathematics, Zhejiang University, Hangzhou, 310027, China

    Chong Li

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  1. XianFa Luo
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  2. Chong Li
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Correspondence to XianFa Luo.

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Luo, X., Li, C. Existence of best simultaneous approximations in L p (S,Σ,X) without the RNP assumption. Sci. China Math. 58, 813–820 (2015). https://doi.org/10.1007/s11425-014-4884-1

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  • DOI: https://doi.org/10.1007/s11425-014-4884-1

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