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 B<sub>0</sub> 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 B<sub>i+1</sub> + B<sub>i+1</sub> ⊆ B<sub>i</sub> for all i = 0, 1, .... In this case, is called a defining sequence for B<sub>0</sub>. A TVS X is called quasi-ultrabarrelled if every bornivorous ultrabarrel in X is a neighbourhood of the origin.
A locally convex quasi-ultrabarrelled space is quasi-barrelled.
Ultrabarrelled spaces and ultrabornological spaces are quasi-ultrabarrelled. Complete and metrizable TVSs are quasi-ultrabarrelled.