Quasi-ultrabarrelled space


In functional analysis and related areas of mathematics, a quasi-ultrabarrelled space is a topological vector spaces 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 of closed balanced and bornivorous subsets of X such that Bi+1 + Bi+1 ⊆ Bi for all i = 0, 1,....
In this case, 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

s and ultrabornological spaces are quasi-ultrabarrelled.
Complete and metrizable TVSs are quasi-ultrabarrelled.