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.¹

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

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.

All of the following spaces are webbed:

• Fréchet spaces.²
• Projective limits and inductive limits of sequences of webbed spaces.
• A sequentially closed vector subspace of a webbed space.³
• Countable products of webbed spaces.³
• A Hausdorff quotient of a webbed space.³
• The image of a webbed space under a sequentially continuous linear map if that image is Hausdorff.³
• The bornologification of a webbed space.
• The continuous dual space of a metrizable locally convex space endowed with the strong dual topology is webbed.²
• If ${\displaystyle X}$ is the strict inductive limit of a denumerable family of locally convex metrizable spaces, then the continuous dual space of ${\displaystyle X}$ with the strong topology is webbed.⁴
  • So in particular, the strong duals of locally convex metrizable spaces are webbed.⁵
• If ${\displaystyle X}$ is a webbed space, then any Hausdorff locally convex topology weaker than this (webbed) topology is also webbed.³

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: