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.

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. 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. 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. 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. 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: ==See also== ==Citations== ==References==

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