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 X be a TVS and let B be a subset of X. Then B is a barrel if it is closed convex balanced and absorbing in X.

A subset B₀ of a TVS X is called an ultrabarrel if it is a closed and balanced subset of X and if there exists a sequence ${\displaystyle \left(B_{i}\right)_{i=1}^{\infty }}$ of closed balanced and absorbing subsets of X such that B_i+1 + B_i+1 ⊆ B_i for all i = 0, 1, .... In this case, ${\displaystyle \left(B_{i}\right)_{i=1}^{\infty }}$ is called a defining sequence for B₀.[1]

A subset B₀ of a TVS X is called a bornivorous ultrabarrel if it is a closed balanced and bornivorous subset of X and if there exists a sequence ${\displaystyle \left(B_{i}\right)_{i=1}^{\infty }}$ of closed balanced and bornivorous subsets of X such that B_i+1 + B_i+1 ⊆ B_i for all i = 0, 1, ....[1]

A subset B₀ of a TVS X is called an suprabarrel if it is a balanced subset of X and if there exists a sequence ${\displaystyle \left(B_{i}\right)_{i=1}^{\infty }}$ of balanced and absorbing subsets of X such that B_i+1 + B_i+1 ⊆ B_i for all i = 0, 1, .... In this case, ${\displaystyle \left(B_{i}\right)_{i=1}^{\infty }}$ is called a defining sequence for B₀.[1]

A subset B₀ of a TVS X is called a bornivorous suprabarrel if it is a balanced and bornivorous subset of X and if there exists a sequence ${\displaystyle \left(B_{i}\right)_{i=1}^{\infty }}$ of balanced and bornivorous subsets of X such that B_i+1 + B_i+1 ⊆ B_i for all i = 0, 1, ....[1]

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
• Space of linear maps
• Ultrabarrelled space

References

1. Khaleelulla 1982, p. 65.

Sources

• 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. 936. Berlin, Heidelberg, New York: Springer-Verlag. ISBN 978-3-540-11565-6. OCLC 8588370.
• 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.