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Proceedings of the American Mathematical Society

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

 

 

Dense barrelled subspaces of uncountable codimension


Authors: Stephen A. Saxon and Wendy J. Robertson
Journal: Proc. Amer. Math. Soc. 107 (1989), 1021-1029
MSC: Primary 46A07
DOI: https://doi.org/10.1090/S0002-9939-1989-0990433-2
MathSciNet review: 990433
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Abstract: Let $ E$ be a Hausdorff barrelled space. If there exists a dense barrelled subspace $ M$ such that $ (\operatorname{codim} (M) \geq c)[\operatorname{codim} (M) = \operatorname{dim} (E)]$, we say that ($ M$ is a satisfactory subspace [11]) [$ E$ is barrelledly fit], respectively. Robertson, Tweddle and Yeomans [11] proved that $ E$ has a barrelled countable enlargement (BCE) if it has a satisfactory subspace. (Trivially) $ E$ has a satisfactory subspace if $ \dim (E) \geq c$ and $ E$ is barrelledly fit. We show that $ E$ is barrelledly fit (and $ \dim (E) \geq c$) if $ E \ncong \varphi $ and either (i) $ E$ is an (LF)-space, or (ii) $ E$ is an infinite-dimensional separable space and the continuum hypothesis holds. Conclusion: barrelledly fit spaces and their permanence properties arise from and advance the study of BCE's.


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DOI: https://doi.org/10.1090/S0002-9939-1989-0990433-2
Article copyright: © Copyright 1989 American Mathematical Society