View source for Countably quasi-barrelled space

In [[Functional analysis|functional analysis]], a [[Topological vector space|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 space|quasibarrelled space]]s. 

== Definition ==

A TVS ''X'' with continuous dual space <math>X^{\prime}</math> is said to be '''countably quasi-barrelled''' if <math>B^{\prime} \subseteq X^{\prime}</math> is a [[Strong dual space|strongly 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 space|Hausdorff]] locally convex TVS is countably quasi-barrelled if and only if each [[Bornivorous set|bornivorous]] [[Barrelled set|barrel]] in ''X'' that is equal to the countable intersection of closed [[Convex set|convex]] [[Balanced set|balanced]] neighborhoods of 0 is itself a neighborhood of 0.{{sfn | Khaleelulla | 1982 | pp=28-63}}

=== σ-quasi-barrelled space ===

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

=== Sequentially quasi-barrelled space ===

A TVS with continuous dual space <math>X^{\prime}</math> is said to be '''sequentially quasi-barrelled''' if every [[Strong dual space|strongly]] convergent sequence in <math>X^{\prime}</math> is equicontinuous.

== Properties ==

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

== Examples and sufficient conditions ==

Every [[Barrelled space|barrelled space]], every [[Countably barrelled space|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|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|sequentially complete]] is a σ-barrelled space.{{sfn | Khaleelulla | 1982 | pp=28-63}} 

There exist σ-barrelled spaces that are not [[Mackey space]]s.{{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|sequentially complete]] [[Mackey space]]s 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}}

== See also ==

* [[Barrelled space]]
* [[Countably barrelled space]]
* [[DF-space]]
* [[H-space]]
* [[Quasibarrelled space]]

==References==
{{Reflist}}

* {{Khaleelulla Counterexamples in Topological Vector Spaces}} <!-- {{sfn | Khaleelulla | 1982 | p=}} -->
* {{Narici Beckenstein Topological Vector Spaces|edition=2}} <!-- {{sfn | Narici | 2011 | p=}} -->
* {{Schaefer Wolff Topological Vector Spaces|edition=2}} <!-- {{sfn | Schaefer | 1999 | p=}} -->
* {{Trèves François Topological vector spaces, distributions and kernels}} <!-- {{sfn | Treves | 2006 | p=}} -->
* {{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]]

{{Sourceattribution|Countably quasi-barrelled space}}