Barrelled set

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

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

== 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 balanced and absorbing in <math>X.</math> A subset of <math>X</math> is called {{em|bornivorous}}{{sfn|Narici|Beckenstein|2011|pp=441-457}} and a {{em|bornivore}} if it absorbs every 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 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 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 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 ==

== 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}}  * {{Narici Beckenstein Topological Vector Spaces|edition=2}} * {{cite book|author=H.H. Schaefer|title=Topological Vector Spaces|publisher=Springer-Verlag|series=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=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}}

Category:Topological vector spaces