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.
that is equal to a countable union of equicontinuous subsets of
[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.
is said to be σ-quasi-barrelled if every strongly bounded (countable) sequence in
is said to be sequentially quasi-barrelled if every strongly convergent sequence in
[1] The strong dual of a distinguished space and of a metrizable locally convex space is countably quasi-barrelled.
[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.