On the proximinality of the unit ball of proximinal subspaces in Banach spaces: A counterexample

Saidi, Fathi

doi:10.1090/S0002-9939-05-08152-9

On the proximinality of the unit ball of proximinal subspaces in Banach spaces: A counterexample
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Proc. Amer. Math. Soc. 133 (2005), 2697-2703 Request permission

Abstract:

A known, and easy to establish, fact in Best Approximation Theory is that, if the unit ball of a subspace $G$ of a Banach space $X$ is proximinal in $X$, then $G$ itself is proximinal in $X$. We are concerned in this article with the reverse implication, as the knowledge of whether the unit ball is proximinal or not is useful in obtaining information about other problems. We show, by constructing a counterexample, that the answer is negative in general.

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