In functional analysis, a discipline within mathematics, a locally convex topological vector space (TVS) is said to be infrabarrelled (also spelled infrabarreled) if every bounded barrel is a neighborhood of the origin.[1]
Similarly, quasibarrelled spaces are topological vector spaces (TVS) for which every bornivorous barrelled set in the space is a neighbourhood of the origin. Quasibarrelled spaces are studied because they are a weakening of the defining condition of barrelled spaces, for which a form of the Banach–Steinhaus theorem holds.
Definition
[edit]A subset of a topological vector space (TVS) is called bornivorous if it absorbs all bounded subsets of ; that is, if for each bounded subset of there exists some scalar such that A barrelled set or a barrel in a TVS is a set which is convex, balanced, absorbing and closed. A quasibarrelled space is a TVS for which every bornivorous barrelled set in the space is a neighbourhood of the origin.[2][3]
Characterizations
[edit]If is a Hausdorff locally convex space then the canonical injection from into its bidual is a topological embedding if and only if is infrabarrelled.[4]
A Hausdorff topological vector space is quasibarrelled if and only if every bounded closed linear operator from into a complete metrizable TVS is continuous.[5] By definition, a linear operator is called closed if its graph is a closed subset of
For a locally convex space with continuous dual the following are equivalent:
- is quasibarrelled.
- Every bounded lower semi-continuous semi-norm on is continuous.
- Every -bounded subset of the continuous dual space is equicontinuous.
If is a metrizable locally convex TVS then the following are equivalent:
- The strong dual of is quasibarrelled.
- The strong dual of is barrelled.
- The strong dual of is bornological.
Properties
[edit]Every quasi-complete infrabarrelled space is barrelled.[1]
A locally convex Hausdorff quasibarrelled space that is sequentially complete is barrelled.[6]
A locally convex Hausdorff quasibarrelled space is a Mackey space, quasi-M-barrelled, and countably quasibarrelled.[7]
A locally convex quasibarrelled space that is also a σ-barrelled space is necessarily a barrelled space.[3]
A locally convex space is reflexive if and only if it is semireflexive and quasibarrelled.[3]
Examples
[edit]Every barrelled space is infrabarrelled.[1] A closed vector subspace of an infrabarrelled space is, however, not necessarily infrabarrelled.[8]
Every product and locally convex direct sum of any family of infrabarrelled spaces is infrabarrelled.[8] Every separated quotient of an infrabarrelled space is infrabarrelled.[8]
Every Hausdorff barrelled space and every Hausdorff bornological space is quasibarrelled.[9] Thus, every metrizable TVS is quasibarrelled.
Note that there exist quasibarrelled spaces that are neither barrelled nor bornological.[3] There exist Mackey spaces that are not quasibarrelled.[3] There exist distinguished spaces, DF-spaces, and -barrelled spaces that are not quasibarrelled.[3]
The strong dual space of a Fréchet space is distinguished if and only if is quasibarrelled.[10]
Counter-examples
[edit]There exists a DF-space that is not quasibarrelled.[3]
There exists a quasibarrelled DF-space that is not bornological.[3]
There exists a quasibarrelled space that is not a σ-barrelled space.[3]
See also
[edit]- Barrelled space – Type of topological vector space
- Reflexive space – Locally convex topological vector space
- Semi-reflexive space
References
[edit]- ^ a b c Schaefer & Wolff 1999, p. 142.
- ^ Jarchow 1981, p. 222.
- ^ a b c d e f g h i Khaleelulla 1982, pp. 28–63.
- ^ Narici & Beckenstein 2011, pp. 488–491.
- ^ Adasch, Ernst & Keim 1978, p. 43.
- ^ Khaleelulla 1982, p. 28.
- ^ Khaleelulla 1982, pp. 35.
- ^ a b c Schaefer & Wolff 1999, p. 194.
- ^ Adasch, Ernst & Keim 1978, pp. 70–73.
- ^ Gabriyelyan, S.S. "On topological spaces and topological groups with certain local countable networks (2014)
Bibliography
[edit]- Adasch, Norbert; Ernst, Bruno; Keim, Dieter (1978). Topological Vector Spaces: The Theory Without Convexity Conditions. Lecture Notes in Mathematics. Vol. 639. Berlin New York: Springer-Verlag. ISBN 978-3-540-08662-8. OCLC 297140003.
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- 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.
- Köthe, Gottfried (1983) [1969]. Topological Vector Spaces I. Grundlehren der mathematischen Wissenschaften. Vol. 159. Translated by Garling, D.J.H. New York: Springer Science & Business Media. ISBN 978-3-642-64988-2. MR 0248498. OCLC 840293704.
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