In functional analysis, a topological vector space (TVS) is said to be countably barrelled if every weakly bounded countable union of equicontinuous subsets of its continuous dual space is again equicontinuous. This property is a generalization of barrelled spaces.
A TVS X with continuous dual space is said to be countably barrelled if is a weak-* 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 barrelled if and only if each barrel in X that is equal to the countable intersection of closed convex balanced neighborhoods of 0 is itself a neighborhood of 0.
A TVS with continuous dual space is said to be ÃÂ-barrelled if every weak-* bounded (countable) sequence in is equicontinuous.
A TVS with continuous dual space is said to be sequentially barrelled if every weak-* convergent sequence in is equicontinuous.
Every countably barrelled space is a countably quasibarrelled space, a ÃÂ-barrelled space, a ÃÂ-quasi-barrelled space, and a sequentially barrelled space. An H-space is a TVS whose strong dual space is countably barrelled.
Every countably barrelled space is a ÃÂ-barrelled space and every ÃÂ-barrelled space is sequentially barrelled. Every ÃÂ-barrelled space is a ÃÂ-quasi-barrelled space.
A locally convex quasi-barrelled space that is also a ðÂÂÂ-barrelled space is a barrelled space.
Every barrelled space is countably barrelled. However, there exist semi-reflexive countably barrelled spaces that are not barrelled. The strong dual of a distinguished space and of a metrizable locally convex space is countably barrelled.
There exist ÃÂ-barrelled spaces that are not countably barrelled. There exist normed DF-spaces that are not countably barrelled. There exists a quasi-barrelled space that is not a ðÂÂÂ-barrelled space. There exist ÃÂ-barrelled spaces that are not Mackey spaces. There exist ÃÂ-barrelled spaces that are not countably quasi-barrelled spaces and thus not countably barrelled. There exist sequentially barrelled spaces that are not ÃÂ-quasi-barrelled. There exist quasi-complete locally convex TVSs that are not sequentially barrelled.