Countably quasi-barrelled space

In functional analysis, a topological vector space (TVS) is said to be countably quasi-barrelled if every strongly bounded countable union of equicontinuous subsets of its continuous dual space is again equicontinuous. This property is a generalization of quasibarrelled spaces.

Definition

A TVS X with continuous dual space $X^{\prime }$ is said to be countably quasi-barrelled if $B^{\prime }\subseteq X^{\prime }$ is a strongly 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 quasi-barrelled if and only if each bornivorous 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-quasi-barrelled space

A TVS with continuous dual space $X^{\prime }$ is said to be σ-quasi-barrelled if every strongly bounded (countable) sequence in $X^{\prime }$ is equicontinuous.Template:Sfn

Sequentially quasi-barrelled space

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

Properties

Every countably quasi-barrelled space is a σ-quasi-barrelled space.

Examples and sufficient conditions

Every barrelled space, every countably barrelled space, and every quasi-barrelled space is countably quasi-barrelled and thus also σ-quasi-barrelled space.Template:Sfn The strong dual of a distinguished space and of a metrizable locally convex space is countably quasi-barrelled.Template:Sfn

Every σ-barrelled space is a σ-quasi-barrelled space.Template:Sfn Every DF-space is countably quasi-barrelled.Template:Sfn A σ-quasi-barrelled space that is sequentially complete is a σ-barrelled space.Template:Sfn

There exist σ-barrelled spaces that are not Mackey spaces.Template:Sfn There exist σ-barrelled spaces (which are consequently σ-quasi-barrelled spaces) that are not countably quasi-barrelled spaces.Template:Sfn There exist sequentially complete Mackey spaces that are not σ-quasi-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
• Countably barrelled space
• DF-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