Countably barrelled space

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 ${\displaystyle X^{\prime }}$ is said to be countably barrelled if ${\displaystyle B^{\prime }\subseteq X^{\prime }}$ is a weak-* 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. 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 ${\displaystyle X^{\prime }}$ is said to be σ-barrelled if every weak-* bounded (countable) sequence in ${\displaystyle X^{\prime }}$ is equicontinuous.

A TVS with continuous dual space ${\displaystyle X^{\prime }}$ is said to be sequentially barrelled if every weak-* convergent sequence in ${\displaystyle X^{\prime }}$ is equicontinuous.

Every countably barrelled space is a countably quasibarrelled space, a σ-barrelled space, a σ-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 σ-barrelled space and every σ-barrelled space is sequentially barrelled. Every σ-barrelled space is a σ-quasi-barrelled space.

A locally convex quasi-barrelled space that is also a 𝜎-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.