Countably quasi-barrelled space
In functional analysis, a topological vector space (TVS) is said to be countably quasi-barrelled if every strongly bounded countable union of equicontinuous subsets of its continuous dual space is again equicontinuous. This property is a generalization of quasibarrelled spaces.
Definition
A TVS X with continuous dual space is said to be countably quasi-barrelled if is a strongly bounded subset of that is equal to a countable union of equicontinuous subsets of , then is itself equicontinuous.[1] A Hausdorff locally convex TVS is countably quasi-barrelled if and only if each bornivorous barrel in X that is equal to the countable intersection of closed convex balanced neighborhoods of 0 is itself a neighborhood of 0.[1]
σ-quasi-barrelled space
A TVS with continuous dual space is said to be σ-quasi-barrelled if every strongly bounded (countable) sequence in is equicontinuous.[1]
Sequentially quasi-barrelled space
A TVS with continuous dual space is said to be sequentially quasi-barrelled if every strongly convergent sequence in is equicontinuous.
Properties
Every countably quasi-barrelled space is a σ-quasi-barrelled space.
Examples and sufficient conditions
Every barrelled space, every countably barrelled space, and every quasi-barrelled space is countably quasi-barrelled and thus also σ-quasi-barrelled space.[1] The strong dual of a distinguished space and of a metrizable locally convex space is countably quasi-barrelled.[1]
Every σ-barrelled space is a σ-quasi-barrelled space.[1] Every DF-space is countably quasi-barrelled.[1] A σ-quasi-barrelled space that is sequentially complete is a σ-barrelled space.[1]
There exist σ-barrelled spaces that are not Mackey spaces.[1] There exist σ-barrelled spaces (which are consequently σ-quasi-barrelled spaces) that are not countably quasi-barrelled spaces.[1] There exist sequentially complete Mackey spaces that are not σ-quasi-barrelled.[1] There exist sequentially barrelled spaces that are not σ-quasi-barrelled.[1] There exist quasi-complete locally convex TVSs that are not sequentially barrelled.[1]
See also
References
Sources
- 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. (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, François (2006). Topological vector spaces, distributions and kernels. Mineola, N.Y: Dover Publications. ISBN 978-0-486-45352-1. OCLC 853623322.
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(help) - Wong (1979). Schwartz spaces, nuclear spaces, and tensor products. Berlin New York: Springer-Verlag. ISBN 3-540-09513-6. OCLC 5126158.
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(help)