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.

Let ${\displaystyle X}$ be a Hausdorff locally convex topological vector space. A web is a stratified collection of disks satisfying the following absorbency and convergence requirements.

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

A strand is a sequence of disks, with the first disk being selected from the first stratum, say ${\displaystyle D_{i},}$ and the second being selected from the sequence that was associated with ${\displaystyle D_{i},}$ and so on. We also require that if a sequence of vectors ${\displaystyle (x_{n})}$ is selected from a strand (with ${\displaystyle x_{1}}$ belonging to the first disk in the strand, ${\displaystyle x_{2}}$ belonging to the second, and so on) then the series ${\displaystyle \sum _{n=1}^{\infty }x_{n}}$ converges.

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

Theorem (de Wilde 1978)—A topological vector space ${\displaystyle X}$ 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:

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