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Countability, Completeness and the Closed Graph Theorem

In: Generalized Functions, Convergence Structures, and Their Applications

Author

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  • R. Beattie

    (Mount Allison University, Dept. of Mathematics and Computer Science)

  • H.-P. Butzmann

    (Universität Mannheim, Fakultät für Mathematik und Informatik)

Abstract

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.

Suggested Citation

  • R. Beattie & H.-P. Butzmann, 1988. "Countability, Completeness and the Closed Graph Theorem," Springer Books, in: Bogoljub Stanković & Endre Pap & Stevan Pilipović & Vasilij S. Vladimirov (ed.), Generalized Functions, Convergence Structures, and Their Applications, pages 375-381, Springer.
    • Handle: RePEc:spr:sprchp:978-1-4613-1055-6_38
    DOI: 10.1007/978-1-4613-1055-6_38
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