Ultrabornological space

Summary

In functional analysis, a topological vector space (TVS) is called ultrabornological if every bounded linear operator from 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

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Let   be a topological vector space (TVS).

Preliminaries

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

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

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

  1.   absorbs every Banach disks in  

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

  1. the gauge of   is an infrabounded map;[2]

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

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

Ultrabornological space

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

  1. every infrabornivorous disk in   is a neighborhood of the origin;[2]

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

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

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

  1.   is an inductive limit of Banach spaces;[2]

Properties

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

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

Examples and sufficient conditions

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

Every Hausdorff sequentially complete bornological space is ultrabornological.[2] Thus every complete 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.[citation needed]

Counter-examples

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

See also

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  • Some characterizations of ultrabornological spaces

References

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  1. ^ Narici & Beckenstein 2011, p. 441.
  2. ^ a b c d e f g h i j 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). 16. Providence: American Mathematical Society. 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. Vol. 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. Vol. 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. Vol. 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.