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In [[functional analysis]], a subset of a [[topological vector space]] (TVS) is called a '''barrel''' or a '''barrelled set''' if it is closed [[Convex set|convex]] [[Balanced set|balanced]] and [[Absorbing set|absorbing]]. 

Barrelled sets play an important role in the definitions of several classes of topological vector spaces, such as [[barrelled space]]s. 

== Definitions ==

Let <math>X</math> be a [[topological vector space]] (TVS). 
A subset of <math>X</math> is called a {{em|barrel}} if it is closed [[Convex set|convex]] [[Balanced set|balanced]] and [[Absorbing set|absorbing]] in <math>X.</math> 
A subset of <math>X</math> is called {{em|[[Bornivorous set|bornivorous]]}}{{sfn|Narici|Beckenstein|2011|pp=441-457}} and a {{em|bornivore}} if it [[Absorbing set|absorbs]] every [[Bounded set (topological vector space)|bounded subset]] of <math>X.</math> Every bornivorous subset of <math>X</math> is necessarily an absorbing subset of <math>X.</math> 

Let <math>B_0 \subseteq X</math> be a subset of a topological vector space <math>X.</math> If <math>B_0</math> is a [[Balanced set|balanced]] [[Absorbing set|absorbing subset]] of <math>X</math> and if there exists a sequence <math>\left(B_i\right)_{i=1}^{\infty}</math> of balanced absorbing subsets of <math>X</math> such that <math>B_{i+1} + B_{i+1} \subseteq B_i</math> for all <math>i = 0, 1, \ldots,</math> then <math>B_0</math> is called a {{em|suprabarrel}}{{sfn|Khaleelulla|1982|p=65}} in <math>X,</math> where moreover, <math>B_0</math> is said to be a(n):

*{{em|bornivorous suprabarrel}} if in addition every <math>B_i</math> is a closed and [[Bornivorous set|bornivorous subset]] of <math>X</math> for every <math>i \geq 0.</math>{{sfn|Khaleelulla|1982|p=65}}
*{{em|ultrabarrel}} if in addition every <math>B_i</math> is a [[Closed set|closed subset]] of <math>X</math> for every <math>i \geq 0.</math>{{sfn|Khaleelulla|1982|p=65}}
*{{em|bornivorous ultrabarrel}} if in addition every <math>B_i</math> is a closed and bornivorous subset of <math>X</math> for every <math>i \geq 0.</math>{{sfn|Khaleelulla|1982|p=65}}

In this case, <math>\left(B_i\right)_{i=1}^{\infty}</math> is called a {{em|defining sequence}} for <math>B_0.</math>{{sfn|Khaleelulla|1982|p=65}}

== Properties ==

Note that every bornivorous ultrabarrel is an ultrabarrel and that every bornivorous suprabarrel is a suprabarrel. 

== Examples ==

* In a [[semi normed vector space]] the closed [[unit ball]] is a barrel.
* Every [[locally convex topological vector space]] has a [[neighbourhood basis]] consisting of barrelled sets, although the space itself need not be a barreled space.

== See also ==

* {{annotated link|Barrelled space}}
* {{annotated link|Space of linear maps}}
* {{annotated link|Ultrabarrelled space}}

== References ==

{{reflist}}

== Bibliography ==

* {{cite book|last=Hogbe-Nlend|first=Henri|title=Bornologies and functional analysis|publisher=North-Holland Publishing Co.|location=Amsterdam|year=1977|pages=xii+144|isbn=0-7204-0712-5|mr=0500064}}
* {{Khaleelulla Counterexamples in Topological Vector Spaces}} <!-- {{sfn|Khaleelulla|1982|p=}} --> 
* {{Narici Beckenstein Topological Vector Spaces|edition=2}}
* {{cite book|author=H.H. Schaefer|title=Topological Vector Spaces|publisher=[[Springer-Verlag]]|series=[[Graduate Texts in Mathematics|GTM]]|volume=3|year=1970|isbn=0-387-05380-8}}
* {{Cite book|isbn=9783540115656|title=Counterexamples in Topological Vector Spaces|last1=Khaleelulla|first1=S.M.|year=1982|publisher=[[Springer-Verlag]]|location=Berlin Heidelberg|series=[[Graduate Texts in Mathematics|GTM]]|volume=936 |pages=29–33, 49, 104}}
* {{Cite book|isbn=9780821807804|title=The Convenient Setting of Global Analysis|last1=Kriegl|first1=Andreas|year=1997|publisher=[[American Mathematical Society]]|last2=Michor|first2=Peter W.|series=Mathematical Surveys and Monographs}} <!-- Kriegl and Michor's The Convenient Setting of Global Analysis -->

{{Functional Analysis}}
{{BoundednessAndBornology}}
{{TopologicalVectorSpaces}}

[[Category:Topological vector spaces]]