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Proceedings of the American Mathematical Society

Published by the American Mathematical Society, the Proceedings of the American Mathematical Society (PROC) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6826 (online) ISSN 0002-9939 (print)

The 2020 MCQ for Proceedings of the American Mathematical Society is 0.85.

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On the proximinality of the unit ball of proximinal subspaces in Banach spaces: A counterexample
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by Fathi B. Saidi PDF
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.
References
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Additional Information
  • Fathi B. Saidi
  • Affiliation: Mathematics Division, University of Sharjah, P. O. Box 27272, Sharjah, United Arab Emirates
  • Email: fsaidi@sharjah.ac.ae
  • Received by editor(s): April 28, 2004
  • Published electronically: April 25, 2005
  • Communicated by: Jonathan M. Borwein
  • © Copyright 2005 American Mathematical Society
    The copyright for this article reverts to public domain 28 years after publication.
  • Journal: Proc. Amer. Math. Soc. 133 (2005), 2697-2703
  • MSC (2000): Primary 41A65, 41A50; Secondary 41A52, 41A30
  • DOI: https://doi.org/10.1090/S0002-9939-05-08152-9
  • MathSciNet review: 2146216