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.
be a topological vector space (TVS).
is called a barrel if it is closed convex balanced and absorbing in
is called bornivorous[1] and a bornivore if it absorbs every bounded subset of
Every bornivorous subset of
is necessarily an absorbing subset of
be a subset of a topological vector space
is a balanced absorbing subset of
and if there exists a sequence
of balanced absorbing subsets of
is called a suprabarrel[2] in
is called a defining sequence for
[2] Note that every bornivorous ultrabarrel is an ultrabarrel and that every bornivorous suprabarrel is a suprabarrel.