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. 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. The first stratum must consist of a sequence of disks in X, denoted by ${\displaystyle (D_{i})}$, such that ${\displaystyle X=\cup _{i}D_{i}}$. For each disk ${\displaystyle D_{i}}$ in the first stratum, there must exists a sequence of disks in X, denote by ${\displaystyle (D_{ij})}$ such that

${\displaystyle D_{ij}\subseteq ({\tfrac {1}{2}})D_{i}}$

and

${\displaystyle \cup _{j}D_{ij}}$

absorbs ${\displaystyle D_{i}}$. This sequence of sequences will form the second stratum. To each disk in the second stratum another sequence of disks with analogously defined properties can be assigned. This process continuous for countably many strata.

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 ${\displaystyle \Sigma _{n}x_{n}}$ 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[1] (de Wilde 1978) — A topological vector space X is a Fréchet space if and only if it is 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.[2]
• Countable products of webbed spaces.[2]
• A Hausdorff quotient of a webbed space.[2]
• The image of a webbed space under a sequentially continuous linear map if that image is Hausdorff.[2]
• The bornologification of a webbed space.
• The continuous dual space of a metrizable locally convex space with the strong topology ${\displaystyle \beta (X^{*},X)}$ 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 ${\displaystyle \beta (X^{*},X)}$ is webbed.
  • So in particular, the strong duals of locally convex metrizable spaces are webbed.[3]
• If X is a webbed space, then any Hausdorff locally convex topology that is weaker than this webbed topology is also webbed.[2]

Theorems

Closed Graph Theorem[4] — Let A : X → Y be a linear map between TVSs that is sequentially closed (i.e. its graph is sequentially closed in X × Y). If Y is a webbed space and X is an ultrabornological space (e.g. 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[4] — Any continuous surjective linear map from a webbed locally convex space onto an ultrabornological space is open.

Open Mapping Theorem[4] — If the image of a closed linear operator A : X → Y from locally convex webbed space X into Hausdorff locally convex space Y is nonmeager in Y then A : X → 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.

See also

• Almost open linear map
• Barrelled space
• Closed graph
• Closed graph theorem
• Closed linear operator
• Discontinuous linear map
• F-space – Topological vector space with a complete translation-invariant metric
• Fréchet space
• Kakutani fixed-point theorem
• Metrizable TVS
• Open mapping theorem (functional analysis)

References

1. Narici 2011, p. 472. sfn error: no target: CITEREFNarici2011 (help)
2. Narici 2011, p. 481. sfn error: no target: CITEREFNarici2011 (help)
3. Narici 2011, pp. 459-483. sfn error: no target: CITEREFNarici2011 (help)
4. Narici 2011, pp. 474-476. sfn error: no target: CITEREFNarici2011 (help)

• De Wilde, Marc (1978). Closed graph theorems and webbed spaces. London: Pitman.
• Khaleelulla, S. M. (July 1, 1982). Written at Berlin Heidelberg. Counterexamples in Topological Vector Spaces. Lecture Notes in Mathematics. 936. Berlin New York: Springer-Verlag. ISBN 978-3-540-11565-6. OCLC 8588370.CS1 maint: ref=harv (link) CS1 maint: date and year (link)
• Kriegl, Andreas; Michor, Peter W. (1997). The Convenient Setting of Global Analysis. Mathematical Surveys and Monographs. American Mathematical Society. pp. 557–578. ISBN 9780821807804.
• Narici, Lawrence; Beckenstein, Edward (2011). Topological Vector Spaces. Pure and applied mathematics (Second ed.). Boca Raton, FL: CRC Press. ISBN 978-1584888666. OCLC 144216834.
• Schaefer, Helmut H.; Wolff, Manfred P. (1999). Topological Vector Spaces. GTM. 8 (Second ed.). New York, NY: Springer New York Imprint Springer. ISBN 978-1-4612-7155-0. OCLC 840278135.CS1 maint: ref=harv (link)