1988 | OriginalPaper | Buchkapitel
Countability, Completeness and the Closed Graph Theorem
verfasst von : R. Beattie, H.-P. Butzmann
Erschienen in: Generalized Functions, Convergence Structures, and Their Applications
Verlag: Springer US
Enthalten in: Professional Book Archive
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The webs of M. De Wilde [4] have made an enormous contribution to the closed graph theorems in locally convex spaces(lcs). Although webs have a very intricate layered construction, two properties in particular have contributed to the closed graph theorem. First of all, webs possess a strong countability condition in the range space which suitably matches the Baire property of Fréchet spaces in the domain space; as a result the zero neighbourhood filter is mapped to a p-Cauchy filter, a filter attempting to settle down. Secondly webs provide a completeness condition which allow p-Cauchy filters to converge.