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 and characterizations

A disk in a topological vector space X is called infrabornivorous if it absorbs all Banach disks. If X is locally convex and Hausdorff, then a disk is infrabornivorous if and only if it absorbs all compact disks.

A TVS X is ultrabornological if and only if every bounded linear operator from X into a complete metrizable TVS is necessarily continuous.

If X is a locally convex space then the following are equivalent:

  • X is ultrabornological;
  • Every infrabornivorous disk is a neighborhood of 0;
  • X be the inductive limit of the spaces XD as D varies over all compact disks in X,
  • A seminorm on X that is bounded on each Banach disk is necessarily continuous;
  • 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;
  • For every Banach space Y and every linear map u : X → Y, if u is bounded on each Banach disk then u is continuous.

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.

See also

References
    • 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. (1982). Written at Berlin Heidelberg. Counterexamples in topological vector spaces. GTM. 936. Berlin New York: Springer-Verlag. ISBN 978-3-540-11565-6. OCLC 8588370.CS1 maint: ref=harv (link)
    • H.H. Schaefer (1970). Topological Vector Spaces. GTM. 3. Springer-Verlag. pp. 61–63. ISBN 0-387-05380-8.
    • 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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