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 ${\displaystyle X}$ 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.

Examples and sufficient conditions

All of the following spaces are webbed:

• Fréchet spaces.^[2]
• 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
  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]

Theorems

Closed Graph Theorem^[6] — 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.

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

• Almost open linear map
   – Map that satisfies a condition similar to that of being an open map.Pages displaying short descriptions of redirect targets
• Barrelled space – Type of topological vector space
• Closed graph
   – Graph of a map closed in the product spacePages displaying short descriptions of redirect targets
• Closed graph theorem (functional analysis) – Theorems connecting continuity to closure of graphs
• Closed linear operator
   – Graph of a map closed in the product spacePages displaying short descriptions of redirect targets
• Discontinuous linear map
• F-space – Topological vector space with a complete translation-invariant metric
• Fréchet space – A locally convex topological vector space that is also a complete metric space
• Kakutani fixed-point theorem – Fixed-point theorem for set-valued functions
• Metrizable topological vector space – A topological vector space whose topology can be defined by a metric
• Open mapping theorem (functional analysis) – Condition for a linear operator to be open
• Ursescu theorem – Generalization of closed graph, open mapping, and uniform boundedness theorem

Citations

References

• De Wilde, Marc (1978). Closed graph theorems and webbed spaces. London: Pitman.
• Khaleelulla, S. M. (1982). Counterexamples in Topological Vector Spaces.
  OCLC 8588370
  .
• OCLC 37141279
  .
• Kriegl, Andreas; Michor, Peter W. (1997). The Convenient Setting of Global Analysis. Mathematical Surveys and Monographs.
  ISBN 9780821807804
  .
• Narici, Lawrence; Beckenstein, Edward (2011). Topological Vector Spaces. Pure and applied mathematics (Second ed.). Boca Raton, FL: CRC Press.
  OCLC 144216834
  .
• OCLC 840278135
  .