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 $X$ be a Hausdorff locally convex topological vector space. A {{em|{{visible anchor|web}}}} is a stratified collection of disks satisfying the following absorbency and convergence requirements.{{sfn|Narici|Beckenstein|2011|p=470−471}}

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

Continue this process to define strata $4, 5, \ldots.$ That is, use induction to define stratum $n + 1$ in terms of stratum $n.$

A {{em|{{visible anchor|strand}}}} 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 $\sum_{n = 1}^{\infty} x_n$ converges.

A Hausdorff locally convex topological vector space on which a web can be defined is called a {{em|{{visible anchor|webbed space}}}}.

Examples and sufficient conditions

{{Math theorem|name=Theorem{{sfn|Narici|Beckenstein|2011|p=472}}|note=de Wilde 1978|math_statement=

A topological vector space $X$ is a Fréchet space if and only if it is both a webbed space and a Baire space.

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All of the following spaces are webbed:

Fréchet spaces.{{sfn|Narici|Beckenstein|2011|p=472}}
Projective limits and inductive limits of sequences of webbed spaces.
A sequentially closed vector subspace of a webbed space.{{sfn|Narici|Beckenstein|2011|p=481}}
Countable products of webbed spaces.{{sfn|Narici|Beckenstein|2011|p=481}}
A Hausdorff quotient of a webbed space.{{sfn|Narici|Beckenstein|2011|p=481}}
The image of a webbed space under a sequentially continuous linear map if that image is Hausdorff.{{sfn|Narici|Beckenstein|2011|p=481}}
The bornologification of a webbed space.
The continuous dual space of a metrizable locally convex space endowed with the strong dual topology is webbed.{{sfn|Narici|Beckenstein|2011|p=472}}
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.{{sfn|Narici|Beckenstein|2011|p=473}}
So in particular, the strong duals of locally convex metrizable spaces are webbed.{{sfn|Narici|Beckenstein|2011|pp=459-483}}
If $X$ is a webbed space, then any Hausdorff locally convex topology weaker than this (webbed) topology is also webbed.{{sfn|Narici|Beckenstein|2011|p=481}}

Theorems

{{Math theorem|name=Closed Graph Theorem{{sfn|Narici|Beckenstein|2011|pp=474-476}}|math_statement=

Let $A : X \to Y$ be a linear map between TVSs that is sequentially closed (meaning that its graph is a sequentially closed subset of $X \times Y$ ).

If $Y$ is a webbed space and $X$ is an ultrabornological space (such as a Fréchet space or an inductive limit of Fréchet spaces), then $A$ is continuous.

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{{Math theorem|name=Closed Graph Theorem|math_statement=

Any closed linear map from the inductive limit of Baire locally convex spaces into a webbed locally convex space is continuous.

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{{Math theorem|name=Open Mapping Theorem|math_statement=

Any continuous surjective linear map from a webbed locally convex space onto an inductive limit of Baire locally convex spaces is open.

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{{Math theorem|name=Open Mapping Theorem{{sfn|Narici|Beckenstein|2011|pp=474-476}}|math_statement=

Any continuous surjective linear map from a webbed locally convex space onto an ultrabornological space is open.

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{{Math theorem|name=Open Mapping Theorem{{sfn|Narici|Beckenstein|2011|pp=474-476}}|math_statement=

If the image of a closed linear operator $A : X \to Y$ from locally convex webbed space $X$ into Hausdorff locally convex space $Y$ is nonmeager in $Y$ then $A : X \to Y$ is a surjective open map.

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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:

{{Math theorem|name=Closed Graph Theorem|math_statement=

Any closed linear map from the inductive limit of Baire topological vector spaces into a webbed topological vector space is continuous.

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Citations

References

{{cite book|last=De Wilde|first= Marc|date=1978|title=Closed graph theorems and webbed spaces|publisher=Pitman|location=London}}
{{Khaleelulla Counterexamples in Topological Vector Spaces}}
{{Kriegl Michor The Convenient Setting of Global Analysis}}
{{Cite book|isbn=9780821807804|title=The Convenient Setting of Global Analysis|last1=Kriegl|first1=Andreas|year=1997|publisher=American Mathematical Society|last2=Michor|first2=Peter W.|series=Mathematical Surveys and Monographs|pages=557–578}}
{{Narici Beckenstein Topological Vector Spaces|edition=2}}
{{Schaefer Wolff Topological Vector Spaces|edition=2}}

Category:Functional analysis

Category:Topological vector spaces

webbed space

Web

Examples and sufficient conditions

Theorems

See also

Citations

References