title: Barrelled set
text: # 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.
  