Webbed space

Template:Short description 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 Template:Em is a stratified collection of disks satisfying the following absorbency and convergence requirements.Template:Sfn

1. Stratum 1: The first stratum must consist of a sequence $D_1,D_2,D_3,\dots$ of disks in $X$ such that their union $\bigcup _{i\in \mathbb{N}}{D}_i$ absorbs $X.$
2. Stratum 2: For each disk $D_i$ in the first stratum, there must exists a sequence $D_{i1},D_{i2},D_{i3},\dots$ of disks in $X$ such that for every $D_i$: $$D_{ij}\subseteq \left(\frac{1}{2}\right)D_i\text{ for every }j$$ 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.
3. Stratum 3: To each disk $D_{ij}$ in the second stratum, assign another sequence $D_{ij1},D_{ij2},D_{ij3},\dots$ of disks in $X$ satisfying analogously defined properties; explicitly, this means that for every $D_{i,j}$: $$D_{ijk}\subseteq \left(\frac{1}{2}\right)D_{ij}\text{ for every }k$$ 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 $4,5,\dots .$ That is, use induction to define stratum $n+1$ in terms of stratum $n.$

A Template:Em 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 $\sum _{n=1}^{\mathrm{\infty }}{x}_n$ converges.

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

Examples and sufficient conditions

All of the following spaces are webbed:

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

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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References

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• Template:Khaleelulla Counterexamples in Topological Vector Spaces
• Template:Kriegl Michor The Convenient Setting of Global Analysis
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• Template:Narici Beckenstein Topological Vector Spaces
• Template:Schaefer Wolff Topological Vector Spaces