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Proceedings of the American Mathematical Society

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

 

 

Almost Chebyshev subspaces, lower semicontinuity, and Hahn-Banach extensions


Author: Edward Rozema
Journal: Proc. Amer. Math. Soc. 39 (1973), 117-121
MSC: Primary 41A65; Secondary 46B05
DOI: https://doi.org/10.1090/S0002-9939-1973-0312131-9
MathSciNet review: 0312131
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Abstract: A subset $ M$ of a Banach space is called almost Chebyshev iff the set of elements with more than one best approximation from $ M$ is the first category. It is first shown that if the metric projection onto a proximinal almost Chebyshev subset $ M$ is lower semicontinuous, then $ M$ is Chebyshev. Next, let $ M$ be a subspace of a separable Banach space. Then $ {M^ \bot }$ is almost Chebyshev iff the set of elements in $ {M^\ast }$ which fail to have a unique Hahn-Banach extension is the first category.


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Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1973-0312131-9
Keywords: Almost Chebyshev sets, lower semicontinuity, Hahn-Banach extensions, selections, Brown property $ ($P$ )$
Article copyright: © Copyright 1973 American Mathematical Society