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 $$\left(B_i\right)_{i=1}^{\infty}$$ of closed balanced and absorbing subsets of $$X$$ such that $$B_{i+1} + B_{i+1} \subseteq B_i$$ for all $$i = 0, 1, \ldots.$$ In this case, $$\left(B_i\right)_{i=1}^{\infty}$$ 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 a barrelled space. 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 $$B_0$$ is a closed balanced and bounded neighborhood of the origin, then $$B_0$$ 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.