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. 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[edit]

Let X be a Hausdorff locally convex topological vector space. A web is a stratified collection of disks satisfying the following absorbency and convergence requirements. The first stratum must consist of a sequence of disks in X, denoted by (D_i), such that X = \cup_{i} D_i. For each disk D_i in the first stratum, there must exists a sequence of disks in X, denote by (D_{ij}) such that D_{ij} \subseteq (\frac{1}{2})D_i and \cup_{j} D_{ij} absorbs D_i. This sequence of sequences will form the second stratum. To each disk in the second stratum we assign another sequence of disks with analogously defined properties. This process continuous for countably many strata.

A strand is a sequence of disks, with the first disk being selected from the first stratum, say D_i, and the second being selected from the sequence that was associated with D_i, and so on. We also require that if a sequence of vectors (x_n) is selected from a strand (with x_1 belonging to the first disk in the strand, x_2 belonging to the second, and so on) then the series \Sigma_{n} x_n converges.

Webbed space[edit]

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

Examples[edit]

  • Frechet spaces are exactly the webbed spaces with the Baire property.
  • Projective limits and inductive limits of sequences of webbed spaces are webbed.
  • The bornologification of a webbed space is webbed.
  • If X is a metrizable locally convex space then the continuous dual space of X with the strong topology \beta(X^*, X) is webbed.
  • If X is the strict inductive limit of a denumerable family of metrizable locally convex spaces, then the continuous dual space of X with the strong topology \beta(X^*, X) is webbed.
    • So in particular, the strong duals of metrizable locally convex spaces are webbed.

Theorems[edit]

  • 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 into an inductive limit of Baire locally convex spaces is open.

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[edit]

References[edit]

  • Lawrence Narici, Edward Beckenstein (1985). Topological Vector Spaces, Second Edition (Chapman & Hall/CRC Pure and Applied Mathematics). Amsterdam: CRC Press. pp. 320–325. ISBN 0824773152.