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 ${\displaystyle X^{\prime ))$ is said to be countably quasi-barrelled if ${\displaystyle B^{\prime }\subseteq X^{\prime ))$ is a strongly bounded subset of ${\displaystyle X^{\prime ))$ that is equal to a countable union of equicontinuous subsets of ${\displaystyle X^{\prime ))$, then ${\displaystyle B^{\prime ))$ is itself equicontinuous.^[1] 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.^[1]

σ-quasi-barrelled space

A TVS with continuous dual space ${\displaystyle X^{\prime ))$ is said to be σ-quasi-barrelled if every strongly bounded (countable) sequence in ${\displaystyle X^{\prime ))$ is equicontinuous.^[1]

Sequentially quasi-barrelled space

A TVS with continuous dual space ${\displaystyle X^{\prime ))$ is said to be sequentially quasi-barrelled if every strongly convergent sequence in ${\displaystyle X^{\prime ))$ 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.^[1] The strong dual of a distinguished space and of a metrizable locally convex space is countably quasi-barrelled.^[1]

Every σ-barrelled space is a σ-quasi-barrelled space.^[1] Every DF-space is countably quasi-barrelled.^[1] A σ-quasi-barrelled space that is sequentially complete is a σ-barrelled space.^[1]

There exist σ-barrelled spaces that are not Mackey spaces.^[1] There exist σ-barrelled spaces (which are consequently σ-quasi-barrelled spaces) that are not countably quasi-barrelled spaces.^[1] There exist sequentially complete Mackey spaces that are not σ-quasi-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]