Ultrabornological space

In functional analysis, a topological vector space (TVS) ${\displaystyle X}$ is called ultrabornological if every bounded linear operator from ${\displaystyle X}$ 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]

Let ${\displaystyle X}$ be a topological vector space (TVS).

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

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

A disk ${\displaystyle D}$ in a TVS ${\displaystyle X}$ is called infrabornivorous if it satisfies any of the following equivalent conditions:

1. ${\displaystyle D}$ absorbs every Banach disks in ${\displaystyle X.}$

while if ${\displaystyle X}$ locally convex then we may add to this list:

2. the gauge of ${\displaystyle D}$ is an infrabounded map;^[2]

while if ${\displaystyle X}$ locally convex and Hausdorff then we may add to this list:

3. ${\displaystyle D}$ absorbs all compact disks;^[2] that is, ${\displaystyle D}$ is "compactivorious".

A TVS ${\displaystyle X}$ is ultrabornological if it satisfies any of the following equivalent conditions:

1. every infrabornivorous disk in ${\displaystyle X}$ is a neighborhood of the origin;^[2]

while if ${\displaystyle X}$ is a locally convex space then we may add to this list:

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

while if ${\displaystyle X}$ is a Hausdorff locally convex space then we may add to this list:

8. ${\displaystyle X}$ 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 ${\displaystyle X}$ is the inductive limit of a family of nuclear Fréchet spaces, spanning ${\displaystyle X.}$
• Every ultrabornological space ${\displaystyle X}$ is the inductive limit of a family of nuclear DF-spaces, spanning ${\displaystyle X.}$

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 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

• 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). 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.