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 if it is closed convex balanced and absorbing in X. A subset of X is called and a if it absorbs every bounded subset of X. Every bornivorous subset of X is necessarily an absorbing subset of X. Let 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 of balanced absorbing subsets of X such that for all then B_0 is called a in X, where moreover, B_0 is said to be a(n): In this case, is called a for B_0.

Properties

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

Examples