Webbed space

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

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

$D_{ij} \subseteq (\tfrac{1}{2})D_i$

and

$\cup_{j} D_{ij}$

absorbs $D_i$. This sequence of sequences will form the second stratum. To each disk in the second stratum another sequence of disks with analogously defined properties can be assigned. 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.

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

Examples

• 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

• 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

• Closed graph theorem