Countably quasi-barrelled space

A TVS X with continuous dual space $X^\{\backslash prime\}$ is said to be countably quasi-barrelled if $B^\{\backslash prime\}\; \backslash subseteq\; X^\{\backslash prime\}$ is a strongly bounded subset of $X^\{\backslash prime\}$ that is equal to a countable union of equicontinuous subsets of $X^\{\backslash prime\}$, then $B^\{\backslash prime\}$ is itself equicontinuous.{{sfn | Khaleelulla | 1982 | pp=28-63}}

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.{{sfn | Khaleelulla | 1982 | pp=28-63}}

A TVS with continuous dual space $X^\{\backslash prime\}$ is said to be σ-quasi-barrelled if every strongly bounded (countable) sequence in $X^\{\backslash prime\}$ is equicontinuous.{{sfn | Khaleelulla | 1982 | pp=28-63}}

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

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

Every barrelled space, every countably barrelled space, and every quasi-barrelled space is countably quasi-barrelled and thus also σ-quasi-barrelled space.{{sfn | Khaleelulla | 1982 | pp=28-63}}

The strong dual of a distinguished space and of a metrizable locally convex space is countably quasi-barrelled.{{sfn | Khaleelulla | 1982 | pp=28-63}}

Every σ-barrelled space is a σ-quasi-barrelled space.{{sfn | Khaleelulla | 1982 | pp=28-63}}

Every DF-space is countably quasi-barrelled.{{sfn | Khaleelulla | 1982 | pp=28-63}}

A σ-quasi-barrelled space that is sequentially complete is a σ-barrelled space.{{sfn | Khaleelulla | 1982 | pp=28-63}}

There exist σ-barrelled spaces that are not Mackey spaces.{{sfn | Khaleelulla | 1982 | pp=28-63}}

There exist σ-barrelled spaces (which are consequently σ-quasi-barrelled spaces) that are not countably quasi-barrelled spaces.{{sfn | Khaleelulla | 1982 | pp=28-63}}

There exist sequentially complete Mackey spaces that are not σ-quasi-barrelled.{{sfn | Khaleelulla | 1982 | pp=28-63}}

There exist sequentially barrelled spaces that are not σ-quasi-barrelled.{{sfn | Khaleelulla | 1982 | pp=28-63}}

There exist quasi-complete locally convex TVSs that are not sequentially barrelled.{{sfn | Khaleelulla | 1982 | pp=28-63}}

• Barrelled space
• Countably barrelled space
• DF-space
• H-space
• Quasibarrelled space

{{Reflist}}

• {{Khaleelulla Counterexamples in Topological Vector Spaces}}
• {{Narici Beckenstein Topological Vector Spaces|edition=2}}
• {{Schaefer Wolff Topological Vector Spaces|edition=2}}
• {{Trèves François Topological vector spaces, distributions and kernels}}
• {{cite book | author=Wong | title=Schwartz spaces, nuclear spaces, and tensor products | publisher=Springer-Verlag | location=Berlin New York | year=1979 | isbn=3-540-09513-6 | oclc=5126158 }}

{{Topological vector spaces}}

Category:Functional analysis