Webbed space

Short description: Space where open mapping and closed graph theorems hold

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

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

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

A strand is a sequence of disks, with the first disk being selected from the first stratum, say [math]\displaystyle{ D_i, }[/math] and the second being selected from the sequence that was associated with [math]\displaystyle{ D_i, }[/math] and so on. We also require that if a sequence of vectors [math]\displaystyle{ (x_n) }[/math] is selected from a strand (with [math]\displaystyle{ x_1 }[/math] belonging to the first disk in the strand, [math]\displaystyle{ x_2 }[/math] belonging to the second, and so on) then the series [math]\displaystyle{ \sum_{n = 1}^{\infty} x_n }[/math] 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 (de Wilde 1978) — A topological vector space [math]\displaystyle{ X }[/math] 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:

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

Theorems

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

Open Mapping Theorem — If the image of a closed linear operator [math]\displaystyle{ A : X \to Y }[/math] from locally convex webbed space [math]\displaystyle{ X }[/math] into Hausdorff locally convex space [math]\displaystyle{ Y }[/math] is nonmeager in [math]\displaystyle{ Y }[/math] then [math]\displaystyle{ A : X \to Y }[/math] 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 – Type of topological vector space
• Closed graph – Graph of a map closed in the product space
• Closed graph theorem (functional analysis) – Theorems connecting continuity to closure of graphs
• 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

References

• 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. 
• Template:Kriegl Michor The Convenient Setting of Global Analysis
• 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.