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 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 $$\left( B_{i} \right)_{i=1}^{\infty}$$ of closed balanced and bornivorous subsets of X such that Bi+1 + Bi+1 &sube; Bi for all i = 0, 1, .... In this case, $$\left( B_{i} \right)_{i=1}^{\infty}$$ is called a defining sequence for B0. 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.