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Two characterizations of linear Baire spaces


Author: Stephen A. Saxon
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
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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

DOI: https://doi.org/10.1090/S0002-9939-1974-0358274-6
Keywords: Wilansky-Klee conjecture, topological vector spaces, Baire spaces, absorbing, balanced, closed set, unordered Baire-like spaces
Article copyright: © Copyright 1974 American Mathematical Society

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