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 be a topological vector space (TVS). A subset of is called a barrel if it is closed convex balanced and absorbing in A subset of is called bornivorous and a bornivore 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 suprabarrel in where moreover, is said to be a(n):

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

In this case, is called a defining sequence 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.

See also

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