KNOWPIA
WELCOME TO KNOWPIA

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]}

Let be a topological vector space (TVS).

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:

- absorbs every Banach disks in

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

- the gauge of is an infrabounded map;
^{[2]}

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

- absorbs all compact disks;
^{[2]}that is, is "compactivorious".

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

- 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:

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

- is an inductive limit of Banach spaces;
^{[2]}

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

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.

- Bounded linear operator – Linear transformation between topological vector spaces
- 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

- Some characterizations of ultrabornological spaces

- 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. 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.