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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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Two characterizations of linear Baire spaces
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by Stephen A. Saxon PDF
Proc. Amer. Math. Soc. 45 (1974), 204-208 Request permission

Abstract:

The Wilansky-Klee conjecture is equivalent to the (unproved) conjecture that every dense, $1$-codimensional subspace of an arbitrary Banach space is a Baire space (second category in itself). The following two characterizations may be useful in dealing with this conjecture: (i) A topological vector space is a Baire space if and only if every absorbing, balanced, closed set is a neighborhood of some point, (ii) A topological vector space is a Baire space if and only if it cannot be covered by countably many nowhere dense sets, each of which is a union of lines ($1$-dimensional subspaces). Characterization (i) has a more succinct form, using the definition of Wilansky’s text [8, p. 224]: a topological vector space is a Baire space if and only if it has the $t$ property.
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Additional Information
  • © Copyright 1974 American Mathematical Society
  • Journal: Proc. Amer. Math. Soc. 45 (1974), 204-208
  • MSC: Primary 46A15
  • DOI: https://doi.org/10.1090/S0002-9939-1974-0358274-6
  • MathSciNet review: 0358274