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.

Definition

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. ^[1] 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. ^[1]

σ-barrelled space

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. ^[1]

Sequentially barrelled space

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. ^[1]

Properties

Every countably barrelled space is a countably quasibarrelled space, a σ-barrelled space, a σ-quasi-barrelled space, and a sequentially barrelled space. ^[1] An H-space is a TVS whose strong dual space is countably barrelled. ^[1]

Every countably barrelled space is a σ-barrelled space and every σ-barrelled space is sequentially barrelled. ^[1] Every σ-barrelled space is a σ-quasi-barrelled space. ^[1]

A locally convex quasi-barrelled space that is also a 𝜎-barrelled space is a barrelled space. ^[1]

Examples and sufficient conditions

Every barrelled space is countably barrelled. ^[1] However, there exist semi-reflexive countably barrelled spaces that are not barrelled. ^[1] The strong dual of a distinguished space and of a metrizable locally convex space is countably barrelled. ^[1]

Counter-examples

There exist σ-barrelled spaces that are not countably barrelled. ^[1] There exist normed DF-spaces that are not countably barrelled. ^[1] There exists a quasi-barrelled space that is not a 𝜎-barrelled space. ^[1] There exist σ-barrelled spaces that are not Mackey spaces. ^[1] There exist σ-barrelled spaces that are not countably quasi-barrelled spaces and thus not countably 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]

See also

• Barrelled space
• H-space
• Quasibarrelled space

References

1. Khaleelulla 1982, pp. 28–63.

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