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[edit]

Let be a topological vector space (TVS).

Preliminaries[edit]

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:

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

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

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

Ultrabornological space[edit]

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:

2. every bounded linear operator from into a complete metrizable TVS is necessarily continuous;
3. every infrabornivorous disk is a neighborhood of 0;
4. be the inductive limit of the spaces as D varies over all compact disks in ;
5. a seminorm on that is bounded on each Banach disk is necessarily continuous;
6. for every locally convex space and every linear map if is bounded on each Banach disk then is continuous;
7. 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:

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

Properties[edit]

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[edit]

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[edit]

• Bounded linear operator – Linear transformation between topological vector spacesPages displaying short descriptions of redirect targets
• Bounded set (topological vector space) – Generalization of boundedness
• Bornological space – Space where bounded operators are continuous
• Bornology – Mathematical generalization of boundedness
• Locally convex topological vector space – A vector space with a topology defined by convex open sets
• Space of linear maps
• Topological vector space – Vector space with a notion of nearness
• Vector bornology

External links[edit]

• Some characterizations of ultrabornological spaces

References[edit]

• 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.