Countably barrelled space explained

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

X ^{\prime} is said to be countably barrelled if B ^{\prime} \subseteq X ^{\prime} is a weak-* bounded subset of X ^{\prime} that is equal to a countable union of equicontinuous subsets of X ^{\prime} , then 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.

σ-barrelled space

A TVS with continuous dual space

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

Sequentially barrelled space

A TVS with continuous dual space

X ^{\prime} is said to be sequentially barrelled if every weak-* convergent sequence in X ^{\prime} is equicontinuous.

Properties

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 -barrelled space is a barrelled space.

Examples and sufficient conditions

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.

Counter-examples

There exist σ-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 -barrelled space. There exist σ-barrelled spaces that are not Mackey spaces. There exist σ-barrelled spaces that are not countably quasi-barrelled spaces and thus not countably barrelled.There exist sequentially barrelled spaces that are not σ-quasi-barrelled. There exist quasi-complete locally convex TVSs that are not sequentially barrelled.

See also

• Barrelled space
• H-space
• Quasibarrelled space

References

• • • • • Book: Wong . Schwartz spaces, nuclear spaces, and tensor products . Springer-Verlag . Berlin New York . 1979 . 3-540-09513-6 . 5126158 .