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

Theorem ^[2]  (de Wilde 1978) — A topological vector space ${\displaystyle X}$ 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. ^[2]
• Projective limits and 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 strict inductive limit of a denumerable family of locally convex metrizable spaces, then 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.

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 ^[6]  — Any continuous surjective linear map from a webbed locally convex space onto an ultrabornological space is open.

Open Mapping Theorem ^[6]  — If the image of a closed linear operator ${\displaystyle A:X\to Y}$ from locally convex webbed space ${\displaystyle X}$ into Hausdorff locally convex space ${\displaystyle Y}$ is nonmeager in ${\displaystyle Y}$ then ${\displaystyle A:X\to Y}$ 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
• Barrelled space  – Type of topological vector space
• Closed graph
• Closed graph theorem (functional analysis)  – Theorems connecting continuity to closure of graphs
• Closed linear operator
• 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  – On when a function f: S→Pow(S) on a compact nonempty convex subset S⊂ℝⁿ has a fixed point
• 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

1. Narici & Beckenstein 2011, p. 470−471.
2. Narici & Beckenstein 2011, p. 472.
3. Narici & Beckenstein 2011, p. 481.
4. Narici & Beckenstein 2011, p. 473.
5. Narici & Beckenstein 2011, pp. 459–483.
6. Narici & Beckenstein 2011, pp. 474–476.

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.