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On the proximinality of the unit ball of proximinal subspaces in Banach spaces: A counterexample
Authors:Fathi B Saidi
Institution:Mathematics Division, University of Sharjah, P. O. Box 27272, Sharjah, United Arab Emirates
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.

Keywords:Proximinality  best approximation  approximation  Banach spaces  unit ball  counterexample  example  renorming
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