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 $$X^{\prime}$$ is said to be countably quasi-barrelled if $$B^{\prime} \subseteq X^{\prime}$$ is a strongly 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 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 $$X^{\prime}$$ is said to be σ-quasi-barrelled if every strongly bounded (countable) sequence in $$X^{\prime}$$ is equicontinuous.

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