Quasi-ultrabarrelled space

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. 
Definition.
A subset "B"0 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 formula_1 of closed balanced and bornivorous subsets of "X" such that "B""i"+1 + "B""i"+1 ⊆ "B""i" for all "i" = 0, 1, ... 
In this case, formula_1 is called a defining sequence for "B"0. 
A TVS "X" is called quasi-ultrabarrelled if every bornivorous ultrabarrel in "X" is a neighbourhood of the origin. 
Properties.
A locally convex quasi-ultrabarrelled space is quasi-barrelled. 
Examples and sufficient conditions.
Ultrabarrelled spaces and ultrabornological spaces are quasi-ultrabarrelled. 
Complete and metrizable TVSs are quasi-ultrabarrelled. 