Ultrabornological space

In functional analysis, a topological vector space (TVS) X is called ultrabornological if every bounded linear operator from X into another TVS is necessarily continuous.

Definitions

Let X be a topological vector space (TVS).

Preliminaries
Definition: A disk is a convex and balanced set.
Definition:[1] A linear map between two TVSs is called infrabounded if it maps Banach disks to bounded disks.
Definition:[1] A disk in a TVS X is called bornivorous if it absorbs every bounded subset of X.

A disk D in a TVS X is called infrabornivorous if it satisfies any of the following equivalent conditions:

  1. D absorbs every Banach disks in X.

while if X locally convex then we may add to this list:

  1. the guage of D is an infrabounded map;[1]

while if X locally convex and Hausdorff then we may add to this list:

  1. D absorbs all compact disks.[1]
    • i.e. D is "compactivorious".

Ultrabornological space

A TVS X is ultrabornological if it satisfies any of the following equivalent conditions:

  1. every infrabornivorous disk in X is a neighborhood of the origin;[1]

while if X is a locally convex space then we may add to this list:

  1. every bounded linear operator from X into a complete metrizable TVS is necessarily continuous;
  2. every infrabornivorous disk is a neighborhood of 0;
  3. X be the inductive limit of the spaces XD as D varies over all compact disks in X;
  4. a seminorm on X that is bounded on each Banach disk is necessarily continuous;
  5. for every locally convex space Y and every linear map u : X → Y, if u is bounded on each Banach disk then u is continuous;
  6. for every Banach space Y and every linear map u : X → Y, if u is bounded on each Banach disk then u is continuous.

while if X is a Hausdorff locally convex space then we may add to this list:

  1. X is an inductive limit of Banach spaces;[1]

Properties

Every ultrabornological space X is the inductive limit of a family of nuclear Fréchet spaces, spanning X.

Every ultrabornological space X is the inductive limit of a family of nuclear DF-spaces, spanning X.

Every ultrabornological space is a quasi-ultrabarrelled space. Every locally convex ultrabornological space is a bornological space but there exist bornological spaces that are not ultrabornological.

Examples and sufficient conditions
  • Every bornological space that is quasi-complete is ultrabornological.
  • Every metrizable TVS is ultrabornological.
  • The finite product of ultrabornological spaces is ultrabornological.
  • Inductive limits of ultrabornological spaces are ultrabornological.

Counter-exmples
  • There exist ultrabarrelled spaces that are not ultrabornological.
  • There exist ultrabornological spaces that are not ultrabarrelled.
gollark: Well, that seems to partly be for vaccines and testing, which don't seem to have much of a possible counting issue going on. The other bit is for treatment, and I don't think they would particularly want to go to the hassle of treating people who don't actually have it.
gollark: Why?
gollark: For who to do that, exactly?
gollark: I don't really see what the "principle" is here.
gollark: or come up with some way to split the result based on *how* close each person was.

See also

References
  1. Narici 2011, pp. 441-457.
  • Hogbe-Nlend, Henri (1977). Bornologies and functional analysis. Amsterdam: North-Holland Publishing Co. pp. xii+144. ISBN 0-7204-0712-5. MR 0500064.
  • Khaleelulla, S. M. (July 1, 1982). Written at Berlin Heidelberg. Counterexamples in Topological Vector Spaces. Lecture Notes in Mathematics. 936. Berlin New York: Springer-Verlag. ISBN 978-3-540-11565-6. OCLC 8588370.CS1 maint: ref=harv (link) CS1 maint: date and year (link)
  • Schaefer, Helmut H.; Wolff, Manfred P. (1999). Topological Vector Spaces. GTM. 8 (Second ed.). New York, NY: Springer New York Imprint Springer. ISBN 978-1-4612-7155-0. OCLC 840278135.CS1 maint: ref=harv (link)
  • Khaleelulla, S.M. (1982). Counterexamples in Topological Vector Spaces. GTM. 936. Berlin Heidelberg: Springer-Verlag. pp. 29–33, 49, 104. ISBN 9783540115656.
  • Kriegl, Andreas; Michor, Peter W. (1997). The Convenient Setting of Global Analysis. Mathematical Surveys and Monographs. American Mathematical Society. ISBN 9780821807804.
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