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Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

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


Author: Fathi B. Saidi
Journal: Proc. Amer. Math. Soc. 133 (2005), 2697-2703
MSC (2000): Primary 41A65, 41A50; Secondary 41A52, 41A30
Published electronically: April 25, 2005
MathSciNet review: 2146216
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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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Additional Information

Fathi B. Saidi
Affiliation: Mathematics Division, University of Sharjah, P. O. Box 27272, Sharjah, United Arab Emirates
Email: fsaidi@sharjah.ac.ae

DOI: http://dx.doi.org/10.1090/S0002-9939-05-08152-9
PII: S 0002-9939(05)08152-9
Keywords: Proximinality, best approximation, approximation, Banach spaces, unit ball, counterexample, example, renorming
Received by editor(s): April 28, 2004
Published electronically: April 25, 2005
Communicated by: Jonathan M. Borwein
Article copyright: © Copyright 2005 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.