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.

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 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 in ${\displaystyle X,}$ where moreover, ${\displaystyle B_{0}}$ is said to be a(n):

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

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