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

1. Stratum 1: The first stratum must consist of a sequence of disks in such that their union absorbs
2. Stratum 2: For each disk in the first stratum, there must exists a sequence of disks in such that for every : and absorbs The sets will form the second stratum.
3. Stratum 3: To each disk in the second stratum, assign another sequence of disks in satisfying analogously defined properties; explicitly, this means that for every : and absorbs The sets form the third stratum.

Continue this process to define strata That is, use induction to define stratum in terms of stratum

A is a sequence of disks, with the first disk being selected from the first stratum, say and the second being selected from the sequence that was associated with and so on. We also require that if a sequence of vectors is selected from a strand (with belonging to the first disk in the strand, belonging to the second, and so on) then the series 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.
• 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 is the strict inductive limit of a denumerable family of locally convex metrizable spaces, then the continuous dual space of with the strong topology is webbed.
• So in particular, the strong duals of locally convex metrizable spaces are webbed.
• If is a webbed space, then any Hausdorff locally convex topology weaker than this (webbed) topology is also webbed.

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:

See also

Citations

References