Barrelled set
# 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 topological vector space (TVS). A subset of $X$ is called a barrel if it is closed convex balanced and absorbing in $X.$ A subset of $X$ is called bornivorous and a bornivore if it absorbs every bounded subset of $X.$ Every bornivorous subset of $X$ is necessarily an absorbing subset of $X.$

Let $B_{0}\subseteq X$ be a subset of a topological vector space $X.$ If $B_{0}$ is a balanced absorbing subset of $X$ and if there exists a sequence $\left(B_{i}\right)_{i=1}^{\infty }$ of balanced absorbing subsets of $X$ such that $B_{i+1}+B_{i+1}\subseteq B_{i}$ for all $i=0,1,\ldots ,$ then $B_{0}$ is called a suprabarrel in $X,$ where moreover, $B_{0}$ is said to be a(n):

* bornivorous suprabarrel if in addition every $B_{i}$ is a closed and bornivorous subset of $X$ for every $i\geq 0.$
* ultrabarrel if in addition every $B_{i}$ is a closed subset of $X$ for every $i\geq 0.$
* bornivorous ultrabarrel if in addition every $B_{i}$ is a closed and bornivorous subset of $X$ for every $i\geq 0.$

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

## 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.
