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 web is a stratified collection of disks satisfying the following absorbency and convergence requirements. The first stratum must consist of a sequence of disks in X, denoted by ${\displaystyle D_{\bullet }=(D_{i})_{i=1}^{\infty },}$ such that ${\displaystyle X=\bigcup _{i=1}^{\infty }D_{i}}$. For each disk ${\displaystyle D_{i}}$ in the first stratum, there must exists a sequence of disks in X, denote by ${\displaystyle D_{i\bullet }=(D_{ij})_{j=1}^{\infty }}$ such that

${\displaystyle D_{ij}\subseteq \left({\tfrac {1}{2}}\right)D_{i}}$ for every ${\displaystyle j}$

and ${\displaystyle \cup _{j=1}^{\infty }D_{ij}}$ absorbs ${\displaystyle D_{i}.}$ This sequence of sequences will form the second stratum. To each disk in the second stratum another sequence of disks with analogously defined properties can be assigned. This process continuous for countably many strata.

A strand is a sequence of disks, with the first disk being selected from the first stratum, say ${\displaystyle D_{i}}$, and the second being selected from the sequence that was associated with ${\displaystyle D_{i}}$, and so on. We also require that if a sequence of vectors ${\displaystyle (x_{n})}$ is selected from a strand (with ${\displaystyle x_{1}}$ belonging to the first disk in the strand, ${\displaystyle x_{2}}$ belonging to the second, and so on) then the series ${\displaystyle \sum _{n=1}^{\infty }x_{n}}$ converges.

A Hausdorff locally convex topological vector space on which a web can be defined is called a webbed space.

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.^[2]
• Countable products of webbed spaces.^[2]
• A Hausdorff quotient of a webbed space.^[2]
• The image of a webbed space under a sequentially continuous linear map if that image is Hausdorff.^[2]
• The bornologification of a webbed space.
• The continuous dual space of a metrizable locally convex space with the strong topology ${\displaystyle \beta (X^{*},X)}$ 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 ${\displaystyle \beta (X^{*},X)}$ is webbed.
  • So in particular, the strong duals of locally convex metrizable spaces are webbed.^[3]
• If X is a webbed space, then any Hausdorff locally convex topology that is weaker than this webbed topology is also webbed.^[2]

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[]

• Almost open linear map
• Barrelled space – Topological vector space
• Closed graph – Graph of a map closed in the product space
• Closed graph theorem (functional analysis) – Theorems for deducing continuity
• Closed linear operator
• Discontinuous linear map
• F-space – Topological vector space with a complete translation-invariant metric
• Fréchet space – A locally convex topological vector space that is also a complete metric space
• Kakutani fixed-point theorem – On when a function f: S→Pow(S) on a compact nonempty convex subset S⊂ℝⁿ has a fixed point
• Metrizable topological vector space – A topological vector space whose topology can be defined by a metric
• Open mapping theorem (functional analysis) – Condition for a linear operator to be open
• Ursescu theorem – Generalization of closed graph, open mapping, and uniform boundedness theorem

Citations[]

References[]

• De Wilde, Marc (1978). Closed graph theorems and webbed spaces. London: Pitman.
• Khaleelulla, S. M. (1982). Counterexamples in Topological Vector Spaces. Lecture Notes in Mathematics. Vol. 936. Berlin, Heidelberg, New York: Springer-Verlag. ISBN 978-3-540-11565-6. OCLC 8588370.
• ; (1997). The Convenient Setting of Global Analysis (PDF). Mathematical Surveys and Monographs. Vol. 53. Providence, R.I: American Mathematical Society. ISBN 978-0-8218-0780-4. OCLC 37141279.
• Kriegl, Andreas; Michor, Peter W. (1997). The Convenient Setting of Global Analysis. Mathematical Surveys and Monographs. American Mathematical Society. pp. 557–578. ISBN 9780821807804.
• Narici, Lawrence; Beckenstein, Edward (2011). Topological Vector Spaces. Pure and applied mathematics (Second ed.). Boca Raton, FL: CRC Press. ISBN 978-1584888666. OCLC 144216834.
• Schaefer, Helmut H.; Wolff, Manfred P. (1999). Topological Vector Spaces. GTM. Vol. 8 (Second ed.). New York, NY: Springer New York Imprint Springer. ISBN 978-1-4612-7155-0. OCLC 840278135.