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]\displaystyle{ X }[/math] be a topological vector space (TVS). A subset of [math]\displaystyle{ X }[/math] is called a barrel if it is closed convex balanced and absorbing in [math]\displaystyle{ X. }[/math] A subset of [math]\displaystyle{ X }[/math] is called bornivorous and a bornivore if it absorbs every bounded subset of [math]\displaystyle{ X. }[/math] Every bornivorous subset of [math]\displaystyle{ X }[/math] is necessarily an absorbing subset of [math]\displaystyle{ X. }[/math]

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

• bornivorous suprabarrel if in addition every [math]\displaystyle{ B_i }[/math] is a closed and bornivorous subset of [math]\displaystyle{ X }[/math] for every [math]\displaystyle{ i \geq 0. }[/math]
• ultrabarrel if in addition every [math]\displaystyle{ B_i }[/math] is a closed subset of [math]\displaystyle{ X }[/math] for every [math]\displaystyle{ i \geq 0. }[/math]
• bornivorous ultrabarrel if in addition every [math]\displaystyle{ B_i }[/math] is a closed and bornivorous subset of [math]\displaystyle{ X }[/math] for every [math]\displaystyle{ i \geq 0. }[/math]

In this case, [math]\displaystyle{ \left(B_i\right)_{i=1}^{\infty} }[/math] is called a defining sequence for [math]\displaystyle{ B_0. }[/math]

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

• Barrelled space – Type of topological vector space
• Ultrabarrelled space

Bibliography

• Hogbe-Nlend, Henri (1977). Bornologies and functional analysis. Amsterdam: North-Holland Publishing Co.. pp. xii+144. ISBN 0-7204-0712-5. 
• Khaleelulla, S. M. (July 1, 1982). written at Berlin Heidelberg. Counterexamples in Topological Vector Spaces. Lecture Notes in Mathematics. 936. Berlin New York: Springer-Verlag. ISBN 978-3-540-11565-6. OCLC 8588370. 
• Narici, Lawrence; Beckenstein, Edward (2011). Topological Vector Spaces. Pure and applied mathematics (Second ed.). Boca Raton, FL: CRC Press. ISBN 978-1584888666. OCLC 144216834. 
• H.H. Schaefer (1970). Topological Vector Spaces. GTM. 3. Springer-Verlag. ISBN 0-387-05380-8. 
• Khaleelulla, S.M. (1982). Counterexamples in Topological Vector Spaces. GTM. 936. Berlin Heidelberg: Springer-Verlag. pp. 29–33, 49, 104. ISBN 9783540115656. 
• Kriegl, Andreas; Michor, Peter W. (1997). The Convenient Setting of Global Analysis. Mathematical Surveys and Monographs. American Mathematical Society. ISBN 9780821807804.