Ultrabornological space
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 (
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:
- absorbs every Banach disks in
while if locally convex then we may add to this list:
while if locally convex and Hausdorff then we may add to this list:
- 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:
- 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]
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 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
External links
[edit]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.