Ultrabornological space
In functional analysis, a topological vector space (TVS) [math]\displaystyle{ X }[/math] is called ultrabornological if every bounded linear operator from [math]\displaystyle{ X }[/math] 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
Let [math]\displaystyle{ X }[/math] be a topological vector space (TVS).
Preliminaries
A disk is a convex and balanced set. A disk in a TVS [math]\displaystyle{ X }[/math] is called bornivorous[2] if it absorbs every bounded subset of [math]\displaystyle{ X. }[/math]
A linear map between two TVSs is called infrabounded[2] if it maps Banach disks to bounded disks.
A disk [math]\displaystyle{ D }[/math] in a TVS [math]\displaystyle{ X }[/math] is called infrabornivorous if it satisfies any of the following equivalent conditions:
- [math]\displaystyle{ D }[/math] absorbs every Banach disks in [math]\displaystyle{ X. }[/math]
while if [math]\displaystyle{ X }[/math] locally convex then we may add to this list:
while if [math]\displaystyle{ X }[/math] locally convex and Hausdorff then we may add to this list:
- [math]\displaystyle{ D }[/math] absorbs all compact disks;[2] that is, [math]\displaystyle{ D }[/math] is "compactivorious".
Ultrabornological space
A TVS [math]\displaystyle{ X }[/math] is ultrabornological if it satisfies any of the following equivalent conditions:
- every infrabornivorous disk in [math]\displaystyle{ X }[/math] is a neighborhood of the origin;[2]
while if [math]\displaystyle{ X }[/math] is a locally convex space then we may add to this list:
- every bounded linear operator from [math]\displaystyle{ X }[/math] into a complete metrizable TVS is necessarily continuous;
- every infrabornivorous disk is a neighborhood of 0;
- [math]\displaystyle{ X }[/math] be the inductive limit of the spaces [math]\displaystyle{ X_D }[/math] as D varies over all compact disks in [math]\displaystyle{ X }[/math];
- a seminorm on [math]\displaystyle{ X }[/math] that is bounded on each Banach disk is necessarily continuous;
- for every locally convex space [math]\displaystyle{ Y }[/math] and every linear map [math]\displaystyle{ u : X \to Y, }[/math] if [math]\displaystyle{ u }[/math] is bounded on each Banach disk then [math]\displaystyle{ u }[/math] is continuous;
- for every Banach space [math]\displaystyle{ Y }[/math] and every linear map [math]\displaystyle{ u : X \to Y, }[/math] if [math]\displaystyle{ u }[/math] is bounded on each Banach disk then [math]\displaystyle{ u }[/math] is continuous.
while if [math]\displaystyle{ X }[/math] is a Hausdorff locally convex space then we may add to this list:
- [math]\displaystyle{ X }[/math] is an inductive limit of Banach spaces;[2]
Properties
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 [math]\displaystyle{ X }[/math] is the inductive limit of a family of nuclear Fréchet spaces, spanning [math]\displaystyle{ X. }[/math]
- Every ultrabornological space [math]\displaystyle{ X }[/math] is the inductive limit of a family of nuclear DF-spaces, spanning [math]\displaystyle{ X. }[/math]
Examples and sufficient conditions
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
- 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
- Topological vector space – Vector space with a notion of nearness
- Vector bornology
External links
References
- Hogbe-Nlend, Henri (1977). Bornologies and functional analysis. Amsterdam: North-Holland Publishing Co.. pp. xii+144. ISBN 0-7204-0712-5.
- Edwards, Robert E. (Jan 1, 1995). Functional Analysis: Theory and Applications. New York: Dover Publications. ISBN 978-0-486-68143-6. OCLC 30593138.
- Template:Grothendieck Produits Tensoriels Topologiques et Espaces Nucléaires
- Grothendieck, Alexander (January 1, 1973). Topological Vector Spaces. New York: Gordon and Breach Science Publishers. ISBN 978-0-677-30020-7. OCLC 886098. https://archive.org/details/topologicalvecto0000grot.
- Khaleelulla, S. M. (July 1, 1982). written at Berlin Heidelberg. Counterexamples in Topological Vector Spaces. Lecture Notes in Mathematics. 936. Berlin New York: Springer-Verlag. ISBN 978-3-540-11565-6. OCLC 8588370.
- Template:Kriegl Michor The Convenient Setting of Global Analysis
- 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. 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.
Original source: https://en.wikipedia.org/wiki/Ultrabornological space.
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