Webbed space explained

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. 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 c up _{i} D _{i} absorbs X.

1. 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} : $$D\_\; \backslash subseteq\; \backslash left(\backslash tfrac\backslash right)\; D\_i\; \backslash quad\; \backslash text\; j$$ and \cup _{j} D _{ij} absorbs D _{i.} The sets \left(D _{ij} \right) _{i,j} 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} : $$D\_\; \backslash subseteq\; \backslash left(\backslash tfrac\backslash right)\; D\_\; \backslash quad\; \backslash text\; k$$ and \cup _{k} D _{ijk} absorbs D _{ij} . The sets \left(D _{ijk} \right) _{i,j,k} 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 ^{inf ty} _{n =1} 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.
• 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

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:

References

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