# Barrelled set

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In [functional analysis](/source/functional_analysis), a subset of a [topological vector space](/source/topological_vector_space) (TVS) is called a '''barrel''' or a '''barrelled set''' if it is closed, [convex](/source/Convex_set), [balanced](/source/Balanced_set), and [absorbing](/source/Absorbing_set). 

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

== Definitions ==

Let <math>X</math> be a [topological vector space](/source/topological_vector_space) (TVS). 
A subset of <math>X</math> is called a {{em|barrel}} if it is closed [convex](/source/Convex_set) [balanced](/source/Balanced_set) and [absorbing](/source/Absorbing_set) in <math>X.</math> 
A subset of <math>X</math> is called {{em|[bornivorous](/source/Bornivorous_set)}}{{sfn|Narici|Beckenstein|2011|pp=441-457}} and a {{em|bornivore}} if it [absorbs](/source/Absorbing_set) every [bounded subset](/source/Bounded_set_(topological_vector_space)) 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](/source/Balanced_set) [absorbing subset](/source/Absorbing_set) 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 subset](/source/Bornivorous_set) 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 subset](/source/Closed_set) 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](/source/semi_normed_vector_space) the closed [unit ball](/source/unit_ball) is a barrel.
* Every [locally convex topological vector space](/source/locally_convex_topological_vector_space) has a [neighbourhood basis](/source/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](/source/Springer-Verlag)|series=[GTM](/source/Graduate_Texts_in_Mathematics)|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](/source/Springer-Verlag)|location=Berlin Heidelberg|series=[GTM](/source/Graduate_Texts_in_Mathematics)|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](/source/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}}
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Category:Topological vector spaces

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Adapted from the Wikipedia article [Barrelled set](https://en.wikipedia.org/wiki/Barrelled_set) by Wikipedia contributors ([contributor history](https://en.wikipedia.org/wiki/Barrelled_set?action=history)). Available under [Creative Commons Attribution-ShareAlike 4.0 International](https://creativecommons.org/licenses/by-sa/4.0/). Changes may have been made.