Barrelled set

Let ${\displaystyle X}$ be a topological vector space (TVS). A subset of ${\displaystyle X}$ is called a barrel if it is closed convex balanced and absorbing in ${\displaystyle X.}$ A subset of ${\displaystyle X}$ is called bornivorous[1] and a bornivore if it absorbs every bounded subset of ${\displaystyle X.}$ Every bornivorous subset of ${\displaystyle X}$ is necessarily an absorbing subset of ${\displaystyle X.}$

Let ${\displaystyle B_{0}\subseteq X}$ be a subset of a topological vector space ${\displaystyle X.}$ If ${\displaystyle B_{0}}$ is a balanced absorbing subset of ${\displaystyle X}$ and if there exists a sequence ${\displaystyle \left(B_{i}\right)_{i=1}^{\infty }}$ of balanced absorbing subsets of ${\displaystyle X}$ such that ${\displaystyle B_{i+1}+B_{i+1}\subseteq B_{i}}$ for all ${\displaystyle i=0,1,\ldots ,}$ then ${\displaystyle B_{0}}$ is called a suprabarrel[2] in ${\displaystyle X,}$ where moreover, ${\displaystyle B_{0}}$ is said to be a(n):

• bornivorous suprabarrel if in addition every ${\displaystyle B_{i}}$ is a closed and bornivorous subset of ${\displaystyle X}$ for every ${\displaystyle i\geq 0.}$[2]
• ultrabarrel if in addition every ${\displaystyle B_{i}}$ is a closed subset of ${\displaystyle X}$ for every ${\displaystyle i\geq 0.}$[2]
• bornivorous ultrabarrel if in addition every ${\displaystyle B_{i}}$ is a closed and bornivorous subset of ${\displaystyle X}$ for every ${\displaystyle i\geq 0.}$[2]

In this case, ${\displaystyle \left(B_{i}\right)_{i=1}^{\infty }}$ is called a defining sequence for ${\displaystyle B_{0}.}$[2]

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

• 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.

• Barrelled space – Type of topological vector space
• Space of linear maps
• Ultrabarrelled space

• Hogbe-Nlend, Henri (1977). Bornologies and functional analysis. Amsterdam: North-Holland Publishing Co. pp. xii+144. ISBN 0-7204-0712-5. MR 0500064.
• Khaleelulla, S. M. (1982). Counterexamples in Topological Vector Spaces. Lecture Notes in Mathematics. Vol. 936. Berlin, Heidelberg, 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. Vol. 3. Springer-Verlag. ISBN 0-387-05380-8.
• Khaleelulla, S.M. (1982). Counterexamples in Topological Vector Spaces. GTM. Vol. 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.