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 $X^{\prime }$ 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.Template:Sfn 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.Template:Sfn

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.Template:Sfn

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.Template:Sfn

Properties

Every countably barrelled space is a countably quasibarrelled space, a σ-barrelled space, a σ-quasi-barrelled space, and a sequentially barrelled space.Template:Sfn An H-space is a TVS whose strong dual space is countably barrelled.Template:Sfn

Every countably barrelled space is a σ-barrelled space and every σ-barrelled space is sequentially barrelled.Template:Sfn Every σ-barrelled space is a σ-quasi-barrelled space.Template:Sfn

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

Examples and sufficient conditions

Every barrelled space is countably barrelled.Template:Sfn However, there exist semi-reflexive countably barrelled spaces that are not barrelled.Template:Sfn The strong dual of a distinguished space and of a metrizable locally convex space is countably barrelled.Template:Sfn

Counter-examples

There exist σ-barrelled spaces that are not countably barrelled.Template:Sfn There exist normed DF-spaces that are not countably barrelled.Template:Sfn There exists a quasi-barrelled space that is not a 𝜎-barrelled space.Template:Sfn There exist σ-barrelled spaces that are not Mackey spaces.Template:Sfn There exist σ-barrelled spaces that are not countably quasi-barrelled spaces and thus not countably barrelled.Template:Sfn There exist sequentially barrelled spaces that are not σ-quasi-barrelled.Template:Sfn There exist quasi-complete locally convex TVSs that are not sequentially barrelled.Template:Sfn

See also

• Barrelled space
• H-space
• Quasibarrelled space

References

• Template:Khaleelulla Counterexamples in Topological Vector Spaces
• Template:Narici Beckenstein Topological Vector Spaces
• Template:Schaefer Wolff Topological Vector Spaces
• Template:Trèves François Topological vector spaces, distributions and kernels
• Template:Cite book