Webbed space
# 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 web is a stratified collection of disks satisfying the following absorbency and convergence requirements.

* Stratum 1: The first stratum must consist of a sequence $D_{1},D_{2},D_{3},\ldots$ of disks in $X$ such that their union $\bigcup _{i\in \mathbb {N} }D_{i}$ absorbs $X.$
* Stratum 2: For each disk $D_{i}$ in the first stratum, there must exists a sequence $D_{i1},D_{i2},D_{i3},\ldots$ of disks in $X$ such that for every $D_{i}$: $D_{ij}\subseteq \left({\tfrac {1}{2}}\right)D_{i}\quad {\text{ for every }}j$ and $\cup _{j\in \mathbb {N} }D_{ij}$ absorbs $D_{i}.$ The sets $\left(D_{ij}\right)_{i,j\in \mathbb {N} }$ will form the second stratum.
* Stratum 3: To each disk $D_{ij}$ in the second stratum, assign another sequence $D_{ij1},D_{ij2},D_{ij3},\ldots$ of disks in $X$ satisfying analogously defined properties; explicitly, this means that for every $D_{i,j}$: $D_{ijk}\subseteq \left({\tfrac {1}{2}}\right)D_{ij}\quad {\text{ for every }}k$ and $\cup _{k\in \mathbb {N} }D_{ijk}$ absorbs $D_{ij}.$ The sets $\left(D_{ijk}\right)_{i,j,k\in \mathbb {N} }$ 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 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 $\sum _{n=1}^{\infty }x_{n}$ converges.

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

## Examples and sufficient conditions

Theorem (de Wilde 1978) — A topological vector space $X$ 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:

* 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

Closed Graph Theorem — Let $A:X\to Y$ be a linear map between TVSs that is sequentially closed (meaning that its graph is a sequentially closed subset of $X\times Y$). If $Y$ is a webbed space and $X$ is an ultrabornological space (such as a Fréchet space or an inductive limit of Fréchet spaces), then $A$ 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 — Any continuous surjective linear map from a webbed locally convex space onto an ultrabornological space is open.

Open Mapping Theorem — If the image of a closed linear operator $A:X\to Y$ from locally convex webbed space $X$ into Hausdorff locally convex space $Y$ is nonmeager in $Y$ then $A:X\to Y$ 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.

