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^[1] 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^[2] in ${\displaystyle X,}$  where moreover, ${\displaystyle B_{0}}$  is said to be a(n):

• bornivorous suprabarrel if in addition every ${\displaystyle B_{i}}$  is a closed and bornivorous subset of ${\displaystyle X}$  for every ${\displaystyle i\geq 0.}$ ^[2]
• ultrabarrel if in addition every ${\displaystyle B_{i}}$  is a closed subset of ${\displaystyle X}$  for every ${\displaystyle i\geq 0.}$ ^[2]
• bornivorous ultrabarrel if in addition every ${\displaystyle B_{i}}$  is a closed and bornivorous subset of ${\displaystyle X}$  for every ${\displaystyle i\geq 0.}$ ^[2]

In this case, ${\displaystyle \left(B_{i}\right)_{i=1}^{\infty }}$  is called a defining sequence for ${\displaystyle B_{0}.}$ ^[2]

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

• 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.

• Barrelled space – Type of topological vector space
• Space of linear maps
• Ultrabarrelled space

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