σ-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 $X^$ is said to be countably barrelled if $B^ \subseteq X^$ is a weak-* bounded subset of $X^$ that is equal to a countable union of equicontinuous subsets of $X^$, then $B^$ 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 $X^$ is said to be σ-barrelled if every weak-* bounded (countable) sequence in $X^$ is equicontinuous.

Sequentially barrelled space

A TVS with continuous dual space $X^$ is said to be sequentially barrelled if every weak-* convergent sequence in $X^$ 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

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{{Topological vector spaces

Functional analysis