Countably barrelled space

In functional analysis, a topological vector space (TVS) is said to be countably barrelled if every weakly bounded countable union of equicontinuous subsets of its continuous dual space is again equicontinuous. This property is a generalization of barrelled spaces.

Definition

A TVS X with continuous dual space [math]\displaystyle{ X^{\prime} }[/math] is said to be countably barrelled if [math]\displaystyle{ B^{\prime} \subseteq X^{\prime} }[/math] is a weak-* 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 barrelled if and only if each barrel in X that is equal to the countable intersection of closed convex balanced neighborhoods of 0 is itself a neighborhood of 0.

σ-barrelled space

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

Sequentially barrelled space

A TVS with continuous dual space [math]\displaystyle{ X^{\prime} }[/math] is said to be sequentially barrelled if every weak-* convergent sequence in [math]\displaystyle{ X^{\prime} }[/math] is equicontinuous.

Properties

Every countably barrelled space is a countably quasibarrelled space, a σ-barrelled space, a σ-quasi-barrelled space, and a sequentially barrelled space. An H-space is a TVS whose strong dual space is countably barrelled.

Every countably barrelled space is a σ-barrelled space and every σ-barrelled space is sequentially barrelled. Every σ-barrelled space is a σ-quasi-barrelled space.

A locally convex quasi-barrelled space that is also a 𝜎-barrelled space is a barrelled space.

Examples and sufficient conditions

Every barrelled space is countably barrelled. However, there exist semi-reflexive countably barrelled spaces that are not barrelled. The strong dual of a distinguished space and of a metrizable locally convex space is countably barrelled.

Counter-examples

There exist σ-barrelled spaces that are not countably barrelled. There exist normed DF-spaces that are not countably barrelled. There exists a quasi-barrelled space that is not a 𝜎-barrelled space. There exist σ-barrelled spaces that are not Mackey spaces. There exist σ-barrelled spaces that are not countably quasi-barrelled spaces and thus not countably 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
• 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.