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, -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 (-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 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