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 $X$ be a Hausdorff locally convex topological vector space. A web is a stratified collection of disks satisfying the following absorbency and convergence requirements.

1. 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 $\bigcup _{i\in \mathbb {N} }D_{i}$ absorbs $X.$
2. Stratum 2: For each disk $D_{i}$ in the first stratum, there must exists a sequence $D_{i1},D_{i2},D_{i3},\ldots$ 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 $\cup _{j\in \mathbb {N} }D_{ij}$ 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 $D_{ij1},D_{ij2},D_{ij3},\ldots$ 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 $\cup _{k\in \mathbb {N} }D_{ijk}$ 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 $4,5,\ldots .$ That is, use induction to define stratum $n+1$ in terms of stratum $n.$

A 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 webbed space.

Examples and sufficient conditions

Theorem (de Wilde 1978) — A topological vector space $X$ 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 $X$ 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.
  • So in particular, the strong duals of locally convex metrizable spaces are webbed.
• If $X$ is a webbed space, then any Hausdorff locally convex topology weaker than this (webbed) topology is also webbed.

Theorems

Closed Graph Theorem — 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.

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 $A:X\to Y$ 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.

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 – 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

References

• De Wilde, Marc (1978). Closed graph theorems and webbed spaces. London: Pitman.
• Khaleelulla, S. M. (1982). Counterexamples in Topological Vector Spaces. Lecture Notes in Mathematics. Vol. 936. Berlin, Heidelberg, New York: Springer-Verlag. ISBN 978-3-540-11565-6. OCLC 8588370.
• Kriegl, Andreas; Michor, Peter W. (1997). The Convenient Setting of Global Analysis (PDF). Mathematical Surveys and Monographs. Vol. 53. Providence, R.I: American Mathematical Society. ISBN 978-0-8218-0780-4. OCLC 37141279.
• 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. Vol. 8 (Second ed.). New York, NY: Springer New York Imprint Springer. ISBN 978-1-4612-7155-0. OCLC 840278135.