Quasibarrelled space
In functional analysis and related areas of mathematics, 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
A subset B of a TVS X is called bornivorous if it absorbs all bounded subsets of X; that is, if for each bounded subset S of X, there exists some scalar r such that S ⊆ rB. 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.[1][2]
Properties
A locally convex Hausdorff quasibarrelled space that is sequentially complete is barrelled.[3] A locally convex Hausdorff quasibarrelled space is a Mackey space, quasi-M-barrelled, and countably quasibarrelled.[4]
A locally convex space is reflexive if and only if it is semireflexive and quasibarrelled.[2]
A locally convex quasi-barreled space that is also a 𝜎-barrelled space is a barrelled space.[2]
Characterizations
For a locally convex space X with continuous dual the following are equivalent:
- X is quasi-barrelled,
- every bounded lower semi-continuous semi-norm on X is continuous,
- every -bounded subset of the continuous dual space is equicontinuous.
If X is a metrizable locally convex TVS then the following are equivalent:
- The strong dual of X is quasibarrelled.
- The strong dual of X is barrelled.
- The strong dual of X is bornological.
Examples and sufficient conditions
Every locally convex Hausdorff barrelled space is quasibarreled.[3] Every locally convex Hausdorff bornological space is quasibarreled.[3] Thus, every locally convex metrizable TVS is quasibarreled.[3]
Note that there exist quasibarrelled spaces that are neither barrelled nor bornological.[2] There exist Mackey spaces that are not quasibarrelled.[2] There exist distinguished spaces, DF-spaces, and -barrelled spaces that are not quasibarrelled.[2]
Counter-examples
There exists a DF-space that is not quasibarrelled.[2] There exists a quasibarrelled DF-space that is not bornological.[2] There exists a quasi-barreled space that is not a 𝜎-barrelled space.[2]
See also
- Barrelled space
- Countably barrelled space
- Countably quasi-barrelled space
- Infrabarreled space
- Uniform boundedness principle#Generalisations
References
- ^ Jarhow 1981, p. 222.
- ^ a b c d e f g h i Khaleelulla 1982, pp. 28–63.
- ^ a b c d Khaleelulla 1982, p. 28.
- ^ Khaleelulla 1982, pp. 35.
- Bourbaki, Nicolas (1950). "Sur certains espaces vectoriels topologiques". Annales de l'Institut Fourier (in French). 2: 5–16 (1951). MR 0042609.
- Robertson, Alex P.; Robertson, Wendy J. (1964). Topological vector spaces. Cambridge Tracts in Mathematics. Vol. 53. Cambridge University Press. pp. 65–75.
- Husain, Taqdir (1978). Barrelledness in topological and ordered vector spaces. Berlin New York: Springer-Verlag. ISBN 3-540-09096-7. OCLC 4493665.
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(help) - Jarhow, Hans (1981). Locally convex spaces. Teubner. ISBN 978-3-322-90561-1.
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(help) - Khaleelulla, S. M. (1982). Written at Berlin Heidelberg. Counterexamples in topological vector spaces. GTM. Vol. 936. Berlin New York: Springer-Verlag. ISBN 978-3-540-11565-6. OCLC 8588370.
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(help) - Schaefer, Helmut H. (1971). Topological vector spaces. GTM. Vol. 3. New York: Springer-Verlag. p. 60. ISBN 0-387-98726-6.
- Schaefer, Helmut H. (1999). Topological Vector Spaces. GTM. Vol. 3. New York, NY: Springer New York Imprint Springer. ISBN 978-1-4612-7155-0. OCLC 840278135.
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(help) - Treves, Francois (2006). Topological vector spaces, distributions and kernels. Mineola, N.Y: Dover Publications. ISBN 978-0-486-45352-1. OCLC 853623322.
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