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 ' 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 D_{1}, D_{2}, D_{3}, \ldots of disks in X such that their union \bigcup_{i \in \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 \N} D_{ij} absorbs D_i. The sets \left(D_{ij}\right)_{i,j \in \N} will form the second stratum.

1. 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 ' 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 '.

Examples and sufficient conditions

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 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.^[4]
  • So in particular, the strong duals of locally convex metrizable spaces are webbed.^[5]
• If X is a webbed space, then any Hausdorff locally convex topology weaker than this (webbed) topology is also webbed.^[3]

Theorems

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:

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.

References

• De Wilde, Marc (1978). Closed graph theorems and webbed spaces. London: Pitman.
• Kriegl, Andreas & Michor, Peter W. (1997). The Convenient Setting of Global Analysis. Mathematical Surveys and Monographs. American Mathematical Society. pp. 557–578. ISBN 9780821807804.