Countably quasi-barrelled space


In functional analysis, a topological vector space 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 is said to be countably quasi-barrelled if is a strongly bounded subset of that is equal to a countable union of equicontinuous subsets of, then 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 is said to be σ-quasi-barrelled if every strongly bounded sequence in is equicontinuous.

Sequentially quasi-barrelled space

A TVS with continuous dual space is said to be sequentially quasi-barrelled if every strongly convergent sequence in 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 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.