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

Sequentially barrelled space
A TVS with continuous dual space $$X^{\prime}$$ is said to be sequentially barrelled if every weak-* convergent sequence in $$X^{\prime}$$ 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.