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Countably quasi-barrelled space
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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 space]]s.
== Definition ==
A [[Hausdorff space|Hausdorff]] [[locally convex]] TVS with continuous dual space <math>X^{\prime}</math> is said to be '''countably quasi-barrelled''' if it satisfies any of the following equivalent conditions:
<ol>
<li>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.} | pp=28-63}}</li>
<li>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.} | pp=28-63}}</li>
</ol>
=== 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.} | pp=28-63}}
=== Sequentially 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]], every [[countably barrelled space]], and every [[quasi-barrelled space]] is countably quasi-barrelled and thus also σ-quasi-barrelled space.} | pp=28-63}}
The [[strong dual]] of a [[distinguished space]] and of a metrizable locally convex space is countably quasi-barrelled.} | pp=28-63}}
Every σ-barrelled space is a σ-quasi-barrelled space.} | pp=28-63}}
Every [[DF-space]] is countably quasi-barrelled.} | pp=28-63}}
== See also ==
* [[Barrelled space]]
* [[Countably barrelled space]]
* [[DF-space]]
* [[H-space]]
* [[Quasibarrelled space]]
== References ==
* } | year=1982 | isbn=978-3-540-11565-6 | oclc=8588370 | ref=harv}} <!-- } | p=}} -->
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<!--- Categories --->
[[Category:Functional analysis]]
This property is a generalization of [[quasibarrelled space]]s.
== Definition ==
A [[Hausdorff space|Hausdorff]] [[locally convex]] TVS with continuous dual space <math>X^{\prime}</math> is said to be '''countably quasi-barrelled''' if it satisfies any of the following equivalent conditions:
<ol>
<li>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.} | pp=28-63}}</li>
<li>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.} | pp=28-63}}</li>
</ol>
=== 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.} | pp=28-63}}
=== Sequentially 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]], every [[countably barrelled space]], and every [[quasi-barrelled space]] is countably quasi-barrelled and thus also σ-quasi-barrelled space.} | pp=28-63}}
The [[strong dual]] of a [[distinguished space]] and of a metrizable locally convex space is countably quasi-barrelled.} | pp=28-63}}
Every σ-barrelled space is a σ-quasi-barrelled space.} | pp=28-63}}
Every [[DF-space]] is countably quasi-barrelled.} | pp=28-63}}
== See also ==
* [[Barrelled space]]
* [[Countably barrelled space]]
* [[DF-space]]
* [[H-space]]
* [[Quasibarrelled space]]
== References ==
* } | year=1982 | isbn=978-3-540-11565-6 | oclc=8588370 | ref=harv}} <!-- } | p=}} -->
* <!-- -->
* <!-- -->
* <!-- -->
<!--- Categories --->
[[Category:Functional analysis]]
June 14, 2020 at 07:38AM
