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 "B"0 of a TVS "X" is called an ultrabarrel if it is a closed and balanced subset of "X" and if there exists a sequence formula_1 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, formula_1 is called a defining sequence for "B"0. 
A subset "B"0 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 formula_1 of closed balanced and bornivorous subsets of "X" such that "B""i"+1 + "B""i"+1 ⊆ "B""i" for all "i" = 0, 1, ... 
A subset "B"0 of a TVS "X" is called an suprabarrel if it is a balanced subset of "X" and if there exists a sequence formula_1 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, formula_1 is called a defining sequence for "B"0.
A subset "B"0 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 formula_1 of balanced and bornivorous subsets of "X" such that "B""i"+1 + "B""i"+1 ⊆ "B""i" for all "i" = 0, 1, ...
Properties.
Note that every bornivorous ultrabarrel is an ultrabarrel and that every bornivorous suprabarrel is a suprabarrel. 