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Webbed space

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In mathematics, particularly in functional analysis, a webbed space is a topological vector space designed with the goal of allowing the results of the open mapping theorem and the closed graph theorem to hold for a wider class of linear maps whose codomains are webbed spaces. A space is called webbed if there exists a collection of sets, called a web that satisfies certain properties. Webs were first investigated by de Wilde.

Web

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Let be a Hausdorff locally convex topological vector space. A web is a stratified collection of disks satisfying the following absorbency and convergence requirements.[1]

  1. Stratum 1: The first stratum must consist of a sequence of disks in such that their union absorbs
  2. Stratum 2: For each disk in the first stratum, there must exists a sequence of disks in such that for every : and absorbs The sets will form the second stratum.
  3. Stratum 3: To each disk in the second stratum, assign another sequence of disks in satisfying analogously defined properties; explicitly, this means that for every : and absorbs The sets form the third stratum.

Continue this process to define strata That is, use induction to define stratum in terms of stratum

A strand is a sequence of disks, with the first disk being selected from the first stratum, say and the second being selected from the sequence that was associated with and so on. We also require that if a sequence of vectors is selected from a strand (with belonging to the first disk in the strand, belonging to the second, and so on) then the series converges.

A Hausdorff locally convex topological vector space on which a web can be defined is called a webbed space.

Examples and sufficient conditions

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Theorem[2] (de Wilde 1978) — A topological vector space is a Fréchet space if and only if it is both a webbed space and a Baire space.

All of the following spaces are webbed:

Theorems

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Closed Graph Theorem[6] — Let be a linear map between TVSs that is sequentially closed (meaning that its graph is a sequentially closed subset of ). If is a webbed space and is an ultrabornological space (such as a Fréchet space or an inductive limit of Fréchet spaces), then is continuous.

Closed Graph Theorem — Any closed linear map from the inductive limit of Baire locally convex spaces into a webbed locally convex space is continuous.

Open Mapping Theorem — Any continuous surjective linear map from a webbed locally convex space onto an inductive limit of Baire locally convex spaces is open.

Open Mapping Theorem[6] — Any continuous surjective linear map from a webbed locally convex space onto an ultrabornological space is open.

Open Mapping Theorem[6] — If the image of a closed linear operator from locally convex webbed space into Hausdorff locally convex space is nonmeager in then is a surjective open map.

If the spaces are not locally convex, then there is a notion of web where the requirement of being a disk is replaced by the requirement of being balanced. For such a notion of web we have the following results:

Closed Graph Theorem — Any closed linear map from the inductive limit of Baire topological vector spaces into a webbed topological vector space is continuous.

See also

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Citations

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References

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  • De Wilde, Marc (1978). Closed graph theorems and webbed spaces. London: Pitman.
  • Khaleelulla, S. M. (1982). Counterexamples in Topological Vector Spaces. Lecture Notes in Mathematics. Vol. 936. Berlin, Heidelberg, New York: Springer-Verlag. ISBN 978-3-540-11565-6. OCLC 8588370.
  • Kriegl, Andreas; Michor, Peter W. (1997). The Convenient Setting of Global Analysis (PDF). Mathematical Surveys and Monographs. Vol. 53. Providence, R.I: American Mathematical Society. ISBN 978-0-8218-0780-4. OCLC 37141279.
  • Kriegl, Andreas; Michor, Peter W. (1997). The Convenient Setting of Global Analysis. Mathematical Surveys and Monographs. American Mathematical Society. pp. 557–578. ISBN 9780821807804.
  • Narici, Lawrence; Beckenstein, Edward (2011). Topological Vector Spaces. Pure and applied mathematics (Second ed.). Boca Raton, FL: CRC Press. ISBN 978-1584888666. OCLC 144216834.
  • Schaefer, Helmut H.; Wolff, Manfred P. (1999). Topological Vector Spaces. GTM. Vol. 8 (Second ed.). New York, NY: Springer New York Imprint Springer. ISBN 978-1-4612-7155-0. OCLC 840278135.