Advances in Operator Theory

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

Tanmoy Paul

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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 \leq p \lt \infty $. The $p = \infty$ case requires a stronger assumption, that of ’uniform proximinality’. Further, we show that a separable subspace $Y$ is ball proximinal in $X$ if and only if $L_{p}(I,Y)$ is ball proximinal in $L_{p}(I,X)$ for $1 \leq p \leq \infty$. 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 having this property; 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.

Article information

Adv. Oper. Theory, Volume 2, Number 1 (2017), 59-77.

Received: 6 November 2016
Accepted: 21 January 2017
First available in Project Euclid: 4 December 2017

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 41A50: Best approximation, Chebyshev systems
Secondary: 46B20: Geometry and structure of normed linear spaces 46E40: Spaces of vector- and operator-valued functions 46E15: Banach spaces of continuous, differentiable or analytic functions

$L_{p}(I,X)$ proximinality strong proximinality ball proximinality


Paul, Tanmoy. Various notions of best approximation property in spaces of Bochner integrable functions. Adv. Oper. Theory 2 (2017), no. 1, 59--77. doi:10.22034/aot.1611-1052.

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