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 ${\displaystyle X^{\prime }}$  is said to be countably quasi-barrelled if ${\displaystyle B^{\prime }\subseteq X^{\prime }}$  is a strongly bounded subset of ${\displaystyle X^{\prime }}$  that is equal to a countable union of equicontinuous subsets of ${\displaystyle X^{\prime }}$ , then ${\displaystyle B^{\prime }}$  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 ${\displaystyle X^{\prime }}$  is said to be σ-quasi-barrelled if every strongly bounded (countable) sequence in ${\displaystyle X^{\prime }}$  is equicontinuous.^[1]

Sequentially quasi-barrelled space

A TVS with continuous dual space ${\displaystyle X^{\prime }}$  is said to be sequentially quasi-barrelled if every strongly convergent sequence in ${\displaystyle X^{\prime }}$  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

• Barrelled space
• Countably barrelled space
• DF-space
• H-space
• Quasibarrelled space

References

1. Khaleelulla 1982, pp. 28–63.

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• 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.
• Trèves, François (2006) [1967]. Topological Vector Spaces, Distributions and Kernels. Mineola, N.Y.: Dover Publications. ISBN 978-0-486-45352-1. OCLC 853623322.
• Wong (1979). Schwartz spaces, nuclear spaces, and tensor products. Berlin New York: Springer-Verlag. ISBN 3-540-09513-6. OCLC 5126158.