Barrelled set


In functional analysis, a subset of a topological vector space 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 of closed balanced and absorbing subsets of X such that Bi+1 + Bi+1 ⊆ Bi for all i = 0, 1,....
In this case, is called a defining sequence for B0.
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 of closed balanced and bornivorous subsets of X such that Bi+1 + Bi+1 ⊆ Bi for all i = 0, 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 of balanced and absorbing subsets of X such that Bi+1 + Bi+1 ⊆ Bi for all i = 0, 1,....
In this case, is called a defining sequence for B0.
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 of balanced and bornivorous subsets of X such that Bi+1 + Bi+1 ⊆ Bi for all i = 0, 1,....

Properties

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

Examples