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 [math]\displaystyle{ X^{\prime} }[/math] is said to be countably quasi-barrelled if [math]\displaystyle{ B^{\prime} \subseteq X^{\prime} }[/math] is a strongly bounded subset of [math]\displaystyle{ X^{\prime} }[/math] that is equal to a countable union of equicontinuous subsets of [math]\displaystyle{ X^{\prime} }[/math], then [math]\displaystyle{ B^{\prime} }[/math] is itself equicontinuous. 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.

σ-quasi-barrelled space

A TVS with continuous dual space [math]\displaystyle{ X^{\prime} }[/math] is said to be σ-quasi-barrelled if every strongly bounded (countable) sequence in [math]\displaystyle{ X^{\prime} }[/math] is equicontinuous.

Sequentially quasi-barrelled space

A TVS with continuous dual space [math]\displaystyle{ X^{\prime} }[/math] is said to be sequentially quasi-barrelled if every strongly convergent sequence in [math]\displaystyle{ X^{\prime} }[/math] 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. The strong dual of a distinguished space and of a metrizable locally convex space is countably quasi-barrelled.

Every σ-barrelled space is a σ-quasi-barrelled space. Every DF-space is countably quasi-barrelled. A σ-quasi-barrelled space that is sequentially complete is a σ-barrelled space.

There exist σ-barrelled spaces that are not Mackey spaces. There exist σ-barrelled spaces (which are consequently σ-quasi-barrelled spaces) that are not countably quasi-barrelled spaces. There exist sequentially complete Mackey spaces that are not σ-quasi-barrelled. There exist sequentially barrelled spaces that are not σ-quasi-barrelled. There exist quasi-complete locally convex TVSs that are not sequentially barrelled.

See also

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

References

• 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. 
• 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. 
• Trèves, François (August 6, 2006). 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.