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Proceedings of the American Mathematical Society

Published by the American Mathematical Society since 1950, Proceedings of the American Mathematical Society is devoted to shorter research articles 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


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.
  • W. A. Light, Proximinality in $L_p(S,Y)$, Rocky Mountain J. Math. 19 (1989), no.Β 1, 251–259. Constructive Function Theoryβ€”86 Conference (Edmonton, AB, 1986). MR 1016178, DOI 10.1216/RMJ-1989-19-1-251
  • W. A. Light and E. W. Cheney, Approximation theory in tensor product spaces, Lecture Notes in Mathematics, vol. 1169, Springer-Verlag, Berlin, 1985. MR 817984, DOI 10.1007/BFb0075391
  • Fathi B. Saidi, On the smoothness of the metric projection and its application to proximinality in $L^p(S,X)$, J. Approx. Theory 83 (1995), no.Β 2, 205–219. MR 1357587, DOI 10.1006/jath.1995.1117
  • Fathi B. Saidi, Subdifferentials and approximation theory, Numer. Funct. Anal. Optim. 16 (1995), no.Β 5-6, 765–783. MR 1341111, DOI 10.1080/01630569508816644
  • Ivan Singer, Best approximation in normed linear spaces by elements of linear subspaces, Die Grundlehren der mathematischen Wissenschaften, Band 171, Publishing House of the Academy of the Socialist Republic of Romania, Bucharest; Springer-Verlag, New York-Berlin, 1970. Translated from the Romanian by Radu Georgescu. MR 0270044
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Additional Information
  • Fathi B. Saidi
  • Affiliation: Mathematics Division, University of Sharjah, P. O. Box 27272, Sharjah, United Arab Emirates
  • Email:
  • 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:
  • MathSciNet review: 2146216