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.
be a Hausdorff locally convex topological vector space.
A web is a stratified collection of disks satisfying the following absorbency and convergence requirements.
[1] Continue this process to define strata
A Hausdorff locally convex topological vector space on which a web can be defined is called a webbed space.
Theorem[2] (de Wilde 1978) — A topological vector space
All of the following spaces are webbed: Closed Graph Theorem[6] — Let
be a linear map between TVSs that is sequentially closed (meaning that its graph is a sequentially closed subset of
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
from locally convex webbed space
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.