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Countability, Completeness and the Closed Graph Theorem

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Generalized Functions, Convergence Structures, and Their Applications

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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.

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References

  1. R. Beattie, Convergence spaces with webs, Math. Nachr. 116, 159–164 (1984).

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  2. R. Beattie, A convenient category for the closed graph theorem, Categorical Topology, Proc. Conference Toledo, Ohio 1983, Heldermann, Berlin, 29–45 (1984).

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  3. R. Beattie and H. -P. Butzmann, Strongly first countable convergence spaces, Convergence Structures 1984, Proc. Conference on Convergence, Bechyne, Czechoslovakia, Akademie-Verlag, Berlin, 39–46 (1985).

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  4. M. De Wilde, Closed Graph Theorems and Webbed Spaces, Research Notes in Mathematics 19, Pitman, London (1978).

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  5. H. Jarchow, “Locally Convex Spaces”, Teubner, Stuttgart (1981).

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  6. W. Robertson, On the closed graph theorem and spaces with webs, Proc. London Math. Soc. (3) 24, 692–738 (1972).

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Author information

Authors and Affiliations

  1. Dept. of Mathematics and Computer Science, Mount Allison University, Sackville, NB, EOA 3CO, Canada

    R. Beattie

  2. Fakultät für Mathematik und Informatik, Universität Mannheim, 6800, Mannheim, Deutschland

    H.-P. Butzmann

Authors
  1. R. Beattie
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  2. H.-P. Butzmann
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Editor information

Editors and Affiliations

  1. Institute of Mathematics, Novi Sad, Yugoslavia

    Bogoljub Stanković , Endre Pap  & Stevan Pilipović ,  & 

  2. Steklov Institute of Mathematics, Moscow, USSR

    Vasilij S. Vladimirov

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© 1988 Plenum Press, New York

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Beattie, R., Butzmann, HP. (1988). Countability, Completeness and the Closed Graph Theorem. In: Stanković, B., Pap, E., Pilipović, S., Vladimirov, V.S. (eds) Generalized Functions, Convergence Structures, and Their Applications. Springer, Boston, MA. https://doi.org/10.1007/978-1-4613-1055-6_38

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  • DOI: https://doi.org/10.1007/978-1-4613-1055-6_38

  • Publisher Name: Springer, Boston, MA

  • Print ISBN: 978-1-4612-8312-6

  • Online ISBN: 978-1-4613-1055-6

  • eBook Packages: Springer Book Archive

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