Countably quasi-barrelled space

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]

Every countably quasi-barrelled space is a σ-quasi-barrelled space.

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]

• Barrelled space
• Countably barrelled space
• DF-space
• H-space
• Quasibarrelled space

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