# 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.^{[1]}
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.^{[1]}

### 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.^{[1]}

### 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.^{[1]}

## Properties

Every countably barrelled space is a countably quasibarrelled space, a σ-barrelled space, a σ-quasi-barrelled space, and a sequentially barrelled space.^{[1]}
An H-space is a TVS whose strong dual space is countably barrelled.^{[1]}

Every countably barrelled space is a σ-barrelled space and every σ-barrelled space is sequentially barrelled.^{[1]}
Every σ-barrelled space is a σ-quasi-barrelled space.^{[1]}

A locally convex quasi-barrelled space that is also a 𝜎-barrelled space is a barrelled space.^{[1]}

## Examples and sufficient conditions

Every barrelled space is countably barrelled.^{[1]}
However, there exist semi-reflexive countably barrelled spaces that are not barrelled.^{[1]}
The strong dual of a distinguished space and of a metrizable locally convex space is countably barrelled.^{[1]}

### Counter-examples

There exist σ-barrelled spaces that are not countably barrelled.^{[1]}
There exist normed DF-spaces that are not countably barrelled.^{[1]}
There exists a quasi-barrelled space that is not a 𝜎-barrelled space.^{[1]}
There exist σ-barrelled spaces that are not Mackey spaces.^{[1]}
There exist σ-barrelled spaces that are not countably quasi-barrelled spaces and thus not countably 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

- ↑
^{1.00}^{1.01}^{1.02}^{1.03}^{1.04}^{1.05}^{1.06}^{1.07}^{1.08}^{1.09}^{1.10}^{1.11}^{1.12}^{1.13}^{1.14}^{1.15}^{1.16}^{1.17}^{1.18}Khaleelulla 1982, pp. 28-63.

- 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.

Original source: https://en.wikipedia.org/wiki/Countably barrelled space.
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