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 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

Let $X$ be a Hausdorff locally convex topological vector space. A is a stratified collection of disks satisfying the following absorbency and convergence requirements.
# Stratum 1: The first stratum must consist of a sequence $D_, D_, D_, \ldots$ of disks in $X$ such that their union $\bigcup_ D_i$ absorbs $X.$
# Stratum 2: For each disk $D_i$ in the first stratum, there must exists a sequence $D_, D_, D_, \ldots$ of disks in $X$ such that for every $D_i$: $D_ \subseteq \left(\tfrac\right) D_i \quad \text j$ and $\cup_ D_$ absorbs $D_i.$ The sets $\left(D_\right)_$ will form the second stratum.
# Stratum 3: To each disk $D_$ in the second stratum, assign another sequence $D_, D_, D_, \ldots$ of disks in $X$ satisfying analogously defined properties; explicitly, this means that for every $D_$: $D_ \subseteq \left(\tfrac\right) D_ \quad \text k$ and $\cup_ D_$ absorbs $D_.$ The sets $\left(D_\right)_$ form the third stratum.

Continue this process to define strata $4, 5, \ldots.$ That is, use induction to define stratum $n + 1$ in terms of stratum $n.$

A 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 $\sum_^ x_n$ converges.

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

Examples and sufficient conditions

All of the following spaces are webbed:
* Fréchet spaces.
* Projective limits and inductive limits of sequences of webbed spaces.
* A sequentially closed vector subspace of a webbed space.
* Countable products of webbed spaces.
* A Hausdorff quotient of a webbed space.
* The image of a webbed space under a sequentially continuous linear map if that image is Hausdorff.
* The bornologification of a webbed space.
* The continuous dual space of a metrizable locally convex space endowed with the strong dual topology is webbed.
* If $X$ is the strict inductive limit of a denumerable family of locally convex metrizable spaces, then the continuous dual space of $X$ with the strong topology is webbed.
** So in particular, the strong duals of locally convex metrizable spaces are webbed.
* If $X$ is a webbed space, then any Hausdorff locally convex topology weaker than this (webbed) topology is also webbed.

Theorems

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:

See also

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Citations

References

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{{Topological vector spaces

Functional analysis
Topological vector spaces