Various notions of best approximation property in spaces of Bochner integrable functions

Paul, Tanmoy (2017) Various notions of best approximation property in spaces of Bochner integrable functions. Advances in Operator Theory, 2 (1). pp. 59-77. ISSN 2538-225X

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Abstract

We show that a separable proximinal subspace of X , say Y is strongly proximinal (strongly ball proximinal) if and only if L p ( I, Y ) is strongly proximinal (strongly ball proximinal) in L p ( I, X ), for 1 ≤ p < ∞ . The p = ∞ case requires a stronger assumption, that of ’uniform proximinality’. Further, we show that Y is ball proximinal in X if and only if L p ( I, Y ) is ball proximinal in L p ( I, X ) for 1 ≤ p ≤ ∞ . We develop the notion of ’uniform proximinality’ of a closed convex set in a Banach space, rectifying one that was defined in a recent paper by P.-K Lin et al. [J. Approx. Theory 183 (2014), 72–81]. We also provide several examples viz. any U -subspace of a Banach space has this property. Recall the notion of 3 . 2 .I.P. by Joram Lindenstrauss, a Banach space X is said to have 3 . 2 .I.P. if any three closed balls which are pairwise intersecting actually intersect in X . It is proved the closed unit ball B X of a space with 3 . 2 .I.P and closed unit ball of any M-ideal of a space with 3 . 2 .I.P. are uniformly proximinal. A new class of examples are given having this property.

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IITH Creators:
IITH CreatorsORCiD
Paul, Tanmoyhttp://orcid.org/0000-0002-2043-3888
Item Type: Article
Subjects: Mathematics
Divisions: Department of Mathematics
Depositing User: Team Library
Date Deposited: 25 Jan 2017 06:58
Last Modified: 14 Jan 2019 09:38
URI: http://raiith.iith.ac.in/id/eprint/3015
Publisher URL: https://doi.org/10.22034/aot.1611-1052
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