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 ${\displaystyle X}$ 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]

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  – Topological vector space
• Space of linear maps
• Ultrabarrelled space

References

1. Narici & Beckenstein 2011, pp. 441–457.
2. Khaleelulla 1982, p. 65.

Bibliography

• 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.