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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

Extending continuous linear functionals in convergence vector spaces


Authors: S. K. Kranzler and T. S. McDermott
Journal: Trans. Amer. Math. Soc. 200 (1974), 149-168
MSC: Primary 46A15
MathSciNet review: 0407557
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Abstract: Let $ (E,\tau )$ be a convergence vector space, $ M$ a subspace of $ E$, and $ \varphi $ a linear functional on $ M$ continuous in the induced convergence structure. Sufficient and sometimes necessary conditions are given that (1) $ \varphi $ has a continuous linear extension to the $ \tau $-adherence $ \bar M$ of$ M$; (2) $ \varphi $ has a continuous linear extension to $ E$; (3) $ \bar M$ is $ \tau $-closed; (4) every $ \tau $-closed convex subset of $ E$ is $ \sigma (E,E')$-closed. Several examples are included illustrating the extent and limitations of the theory presented.


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DOI: http://dx.doi.org/10.1090/S0002-9947-1974-0407557-4
PII: S 0002-9947(1974)0407557-4
Article copyright: © Copyright 1974 American Mathematical Society