In functional analysis and related areas of mathematics, a quasi-ultrabarrelled space is a topological vector spaces (TVS) for which every bornivorous ultrabarrel is a neighbourhood of the origin.
A subset B0 of a TVS X is called a bornivorous ultrabarrel if it is a closed, balanced, and bornivorous subset of X and if there exists a sequence
of closed balanced and bornivorous subsets of X such that Bi+1 + Bi+1 ⊆ Bi for all i = 0, 1, ....
A TVS X is called quasi-ultrabarrelled if every bornivorous ultrabarrel in X is a neighbourhood of the origin.
[1] A locally convex quasi-ultrabarrelled space is quasi-barrelled.