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 <math>X^{\prime}</math> is said to be '''countably barrelled''' if <math>B^{\prime} \subseteq X^{\prime}</math> is a weak-* bounded subset of <math>X^{\prime}</math> that is equal to a countable union of equicontinuous subsets of <math>X^{\prime}</math>, then <math>B^{\prime}</math> is itself equicontinuous.{{sfn | Khaleelulla | 1982 | pp=28-63}} 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.{{sfn | Khaleelulla | 1982 | pp=28-63}}

=== σ-barrelled space ===

A TVS with continuous dual space <math>X^{\prime}</math> is said to be '''σ-barrelled''' if every weak-* bounded (countable) sequence in <math>X^{\prime}</math> is equicontinuous.{{sfn | Khaleelulla | 1982 | pp=28-63}}

=== Sequentially barrelled space ===

A TVS with continuous dual space <math>X^{\prime}</math> is said to be '''sequentially barrelled''' if every weak-* convergent sequence in <math>X^{\prime}</math> is equicontinuous.{{sfn | Khaleelulla | 1982 | pp=28-63}}

== Properties ==

Every countably barrelled space is a countably quasibarrelled space, a σ-barrelled space, a σ-quasi-barrelled space, and a sequentially barrelled space.{{sfn | Khaleelulla | 1982 | pp=28-63}} An H-space is a TVS whose strong dual space is countably barrelled.{{sfn | Khaleelulla | 1982 | pp=28-63}}

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

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

== Examples and sufficient conditions ==

Every barrelled space is countably barrelled.{{sfn | Khaleelulla | 1982 | pp=28-63}} However, there exist semi-reflexive countably barrelled spaces that are not barrelled.{{sfn | Khaleelulla | 1982 | pp=28-63}} The strong dual of a distinguished space and of a metrizable locally convex space is countably barrelled.{{sfn | Khaleelulla | 1982 | pp=28-63}}

=== Counter-examples ===

There exist σ-barrelled spaces that are not countably barrelled.{{sfn | Khaleelulla | 1982 | pp=28-63}} There exist normed DF-spaces that are not countably barrelled.{{sfn | Khaleelulla | 1982 | pp=28-63}} There exists a quasi-barrelled space that is not 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 that are not countably quasi-barrelled spaces and thus not countably 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}}

==References== {{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 }}

Category:Functional analysis