Ultrabarrelled space

In functional analysis and related areas of mathematics, an ultrabarrelled space is a topological vector spaces (TVS) for which every ultrabarrel is a neighbourhood of the origin. 
Definition.
A subset "B"0 of a TVS "X" is called an ultrabarrel if it is a closed and balanced subset of "X" and if there exists a sequence formula_1 of closed balanced and absorbing 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 ultrabarrelled if every ultrabarrel in "X" is a neighbourhood of the origin. 
Properties.
A locally convex ultrabarrelled space is barrelled. 
Every ultrabarrelled space is a quasi-ultrabarrelled space.
Examples and sufficient conditions.
Complete and metrizable TVSs are ultrabarrelled. 
If "X" is a complete locally bounded non-locally convex TVS and if "B" is a closed balanced and bounded neighborhood of the origin, then "B" is an ultrabarrel that is not convex and has a defining sequence consisting of non-convex sets. 
Counter-examples.
There exist barrelled spaces that are not ultrabarrelled. 
There exist TVSs that are complete and metrizable (and thus ultrabarrelled) but not barrelled. 