View source for Infrabarrelled space

{{Multiple issues|{{one source|date=June 2020}}{{technical|date=June 2020}}}}

In [[functional analysis]], a discipline within mathematics, a [[Locally convex topological vector space|locally convex]] [[topological vector space]] (TVS) is said to be '''infrabarrelled''' (also spelled '''infra[[Wiktionary:barreled|barreled]]''') if every [[Bounded set (topological vector space)|bounded]] [[Barreled space|barrel]] is a neighborhood of the origin.{{sfn|Schaefer|Wolff|1999|p=142}}

Similarly, '''quasibarrelled spaces''' are [[topological vector spaces]] (TVS) for which every [[bornivorous]] barrelled set in the space is a [[Neighbourhood (topology)|neighbourhood]] of the origin. 
Quasibarrelled spaces are studied because they are a weakening of the defining condition of [[barrelled space]]s, for which a form of the [[Banach–Steinhaus theorem]] holds.

==Definition==
A subset <math>B</math> of a [[topological vector space]] (TVS) <math>X</math> is called '''bornivorous''' if it absorbs all bounded subsets of <math>X</math>; 
that is, if for each bounded subset <math>S</math> of <math>X,</math> there exists some scalar <math>r</math> such that <math>S \subseteq r B.</math> 
A '''barrelled set''' or a '''barrel''' in a TVS is a [[Set (mathematics)|set]] which is [[Convex set|convex]], [[Balanced set|balanced]], [[Absorbing set|absorbing]] and [[Closed set|closed]].  
A '''quasibarrelled space''' is a TVS for which every bornivorous barrelled set in the space is a [[Neighbourhood (topology)|neighbourhood]] of the origin.{{sfn|Jarchow|1981|p=222}}{{sfn|Khaleelulla|1982|pp=28-63}} 

==Characterizations==
If <math>X</math> is a Hausdorff locally convex space then [[Semi-reflexive space|the canonical injection]] from <math>X</math> into its bidual is a topological embedding if and only if <math>X</math> is infrabarrelled.{{sfn|Narici|Beckenstein|2011|pp=488–491}}

A Hausdorff topological vector space <math>X</math> is '''quasibarrelled''' if and only if every bounded closed linear operator from <math>X</math> into a [[F-space|complete metrizable TVS]] is continuous.{{sfn|Adasch|Ernst|Keim|1978|p=43}}  
By definition, a linear <math>F : X \to Y</math> operator is called '''closed''' if its graph is a closed subset of <math>X \times Y.</math> 

For a [[locally convex space]] <math>X</math> with continuous dual <math>X^{\prime}</math> the following are equivalent:
# <math>X</math> is quasibarrelled.
# Every bounded lower semi-continuous semi-norm on <math>X</math> is continuous.
# Every <math>\beta(X', X)</math>-bounded subset of the continuous dual space <math>X^{\prime}</math> is equicontinuous.

If <math>X</math> is a metrizable locally convex TVS then the following are equivalent:
# The [[strong dual]] of <math>X</math> is quasibarrelled.
# The [[strong dual]] of <math>X</math> is barrelled.
# The [[strong dual]] of <math>X</math> is [[Bornological space|bornological]].

==Properties==
Every [[Quasi-complete space|quasi-complete]] infrabarrelled space is barrelled.{{sfn|Schaefer|Wolff|1999|p=142}} 

A locally convex Hausdorff quasibarrelled space that is [[sequentially complete]] is barrelled.{{sfn|Khaleelulla|1982|p=28}}  

A locally convex Hausdorff quasibarrelled space is a [[Mackey space]], [[quasi-M-barrelled]], and countably quasibarrelled.{{sfn|Khaleelulla|1982|pp=35}}  

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

A locally convex space is [[reflexive space|reflexive]] if and only if it is [[Semireflexive space|semireflexive]] and quasibarrelled.{{sfn|Khaleelulla|1982|pp=28-63}}

==Examples==
Every [[barrelled space]] is infrabarrelled.{{sfn|Schaefer|Wolff|1999|p=142}}  
A closed vector subspace of an infrabarrelled space is, however, not necessarily infrabarrelled.{{sfn|Schaefer|Wolff|1999|p=194}} 

Every product and locally convex direct sum of any family of infrabarrelled spaces is infrabarrelled.{{sfn|Schaefer|Wolff|1999|p=194}}  
Every [[Hausdorff space|separated]] quotient of an infrabarrelled space is infrabarrelled.{{sfn|Schaefer|Wolff|1999|p=194}} 

Every Hausdorff [[barrelled space]] and every Hausdorff [[bornological space]] is quasibarrelled.{{sfn|Adasch|Ernst|Keim|1978|pp=70-73}}  
Thus, every [[metrizable TVS]] is quasibarrelled.  

Note that there exist quasibarrelled spaces that are neither barrelled nor bornological.{{sfn|Khaleelulla|1982|pp=28-63}}  
There exist [[Mackey space]]s that are not quasibarrelled.{{sfn|Khaleelulla|1982|pp=28-63}}  
There exist [[distinguished space]]s, [[DF-space]]s, and <math>\sigma</math>-barrelled spaces that are not quasibarrelled.{{sfn|Khaleelulla|1982|pp=28-63}}  

The [[strong dual space]] <math>X_b^{\prime}</math> of a [[Fréchet space]] <math>X</math> is [[Distinguished space|distinguished]] if and only if <math>X</math> is quasibarrelled.<ref name="Gabriyelyan 2014">Gabriyelyan, S.S. [https://arxiv.org/pdf/1412.1497.pdf "On topological spaces and topological groups with certain local countable networks] (2014)</ref>

===Counter-examples===
There exists a [[DF-space]] that is not quasibarrelled.{{sfn|Khaleelulla|1982|pp=28-63}}  

There exists a quasibarrelled [[DF-space]] that is not [[Bornological space|bornological]].{{sfn|Khaleelulla|1982|pp=28-63}}  

There exists a quasibarrelled space that is not a [[σ-barrelled space]].{{sfn|Khaleelulla|1982|pp=28-63}}

==See also==

* {{annotated link|Barrelled space}}
* {{annotated link|Reflexive space}}
* {{annotated link|Semi-reflexive space}}

==References==
{{reflist}}

==Bibliography==
* {{Adasch Topological Vector Spaces|edition=2}} <!-- {{sfn|Adasch|Ernst|Keim|1978|p=}} -->
* {{Berberian Lectures in Functional Analysis and Operator Theory}} <!-- {{sfn|Berberian|1974|p=}} -->
* {{Bourbaki Topological Vector Spaces Part 1 Chapters 1–5}} <!-- {{sfn|Bourbaki|1987|p=}} -->
* {{Conway A Course in Functional Analysis|edition=2}} <!-- {{sfn|Conway|1990|p=}} -->
* {{Edwards Functional Analysis Theory and Applications}} <!-- {{sfn|Edwards|1995|p=}} -->
* {{Grothendieck Topological Vector Spaces}} <!-- {{sfn|Grothendieck|1973|p=}} -->
* {{Hogbe-Nlend Bornologies and Functional Analysis}} <!-- {{sfn|Hogbe-Nlend|1977|p=}} -->
* {{Husain Khaleelulla Barrelledness in Topological and Ordered Vector Spaces}} <!-- {{sfn|Husain|Khaleelulla|1978|p=}} -->
* {{Jarchow Locally Convex Spaces}} <!-- {{sfn|Jarchow|1981|p=}} -->
* {{Khaleelulla Counterexamples in Topological Vector Spaces}} <!-- {{sfn|Khaleelulla|1982|p=}} -->
* {{Köthe Topological Vector Spaces I}} <!-- {{sfn|Köthe|1969|p=}} -->
* {{Köthe Topological Vector Spaces II}} <!-- {{sfn|Köthe|1979|p=}} -->
* {{Narici Beckenstein Topological Vector Spaces|edition=2}} <!-- {{sfn|Narici|Beckenstein|2011|p=}} -->
* {{Schaefer Wolff Topological Vector Spaces|edition=2}} <!-- {{sfn|Schaefer|Wolff|1999|p=}} -->
* {{Swartz An Introduction to Functional Analysis}} <!-- {{sfn|Swartz|1992|p=}} -->
* {{Trèves François Topological vector spaces, distributions and kernels}} <!-- {{sfn|Trèves|2006|p=}} -->
* {{Wilansky Modern Methods in Topological Vector Spaces|edition=1}} <!-- {{sfn|Wilansky|2013|p=}} -->
* {{Wong Schwartz Spaces, Nuclear Spaces, and Tensor Products}} <!-- {{sfn|Wong|1979|p=}} -->

{{Topological vector spaces}}
{{Boundedness and bornology}}
{{Functional analysis}}

[[Category:Functional analysis]]
[[Category:Topological vector spaces]]