Barrelled set

In functional analysis, a subset of a topological [vector space] 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 be a topological vector space.
A subset of is called a if it is closed convex balanced and absorbing in
A subset of is called and a if it absorbs every bounded subset of Every bornivorous subset of is necessarily an absorbing subset of
Let be a subset of a topological vector space If is a balanced absorbing subset of and if there exists a sequence of balanced absorbing subsets of such that for all then is called a in where moreover, is said to be a:

if in addition every is a closed and bornivorous subset of for every
if in addition every is a closed subset of for every
if in addition every is a closed and bornivorous subset of for every

In this case, is called a for

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.