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. A general version of the closed graph theorem holds for ultrabornological spaces. Ultrabornological spaces were introduced by Alexander Grothendieck (Grothendieck [1955, p. 17] "espace du type (β)").[1]

Definitions

Let X be a topological vector space (TVS).

Preliminaries

A disk is a convex and balanced set. A disk in a TVS X is called bornivorous[2] if it absorbs every bounded subset of X.

A linear map between two TVSs is called infrabounded[2] if it maps Banach disks to bounded disks.

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 gauge of D is an infrabounded map;[2]

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

  1. D absorbs all compact disks;[2] that is, 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;[2]

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;[2]

Properties

Every locally convex ultrabornological space is barrelled,[2] quasi-ultrabarrelled space, and a bornological space but there exist bornological spaces that are not ultrabornological.

Examples and sufficient conditions

The finite product of locally convex ultrabornological spaces is ultrabornological.[2] Inductive limits of ultrabornological spaces are ultrabornological.

Every Hausdorff sequentially complete bornological TVS is ultrabornological.[2] Thus every compete Hausdorff bornological space is ultrabornological. In particular, every Fréchet space is ultrabornological.[2]

The strong dual space of a complete Schwartz space is ultrabornological.

Every Hausdorff bornological space that is quasi-complete is ultrabornological.

Counter-examples

There exist ultrabarrelled spaces that are not ultrabornological. There exist ultrabornological spaces that are not ultrabarrelled.

See also

References
    1. Narici & Beckenstein 2011, p. 441.
    2. Narici & Beckenstein 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.
    • Edwards, Robert E. (1995). Functional Analysis: Theory and Applications. New York: Dover Publications. ISBN 978-0-486-68143-6. OCLC 30593138.
    • Grothendieck, Alexander (1955). "Produits Tensoriels Topologiques et Espaces Nucléaires" [Topological Tensor Products and Nuclear Spaces]. Memoirs of the American Mathematical Society Series (in French). Providence: American Mathematical Society. 16. ISBN 978-0-8218-1216-7. MR 0075539. OCLC 1315788.
    • Grothendieck, Alexander (1973). Topological Vector Spaces. Translated by Chaljub, Orlando. New York: Gordon and Breach Science Publishers. ISBN 978-0-677-30020-7. OCLC 886098.
    • Khaleelulla, S. M. (1982). Counterexamples in Topological Vector Spaces. Lecture Notes in Mathematics. 936. Berlin, Heidelberg, New York: Springer-Verlag. ISBN 978-3-540-11565-6. OCLC 8588370.
    • Kriegl, Andreas; Michor, Peter W. (1997). The Convenient Setting of Global Analysis (PDF). Mathematical Surveys and Monographs. 53. Providence, R.I: American Mathematical Society. ISBN 978-0-8218-0780-4. OCLC 37141279.
    • 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. 8 (Second ed.). New York, NY: Springer New York Imprint Springer. ISBN 978-1-4612-7155-0. OCLC 840278135.
    • Wilansky, Albert (2013). Modern Methods in Topological Vector Spaces. Mineola, New York: Dover Publications, Inc. ISBN 978-0-486-49353-4. OCLC 849801114.
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