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.
is called an ultrabarrel if it is a closed and balanced subset of
of closed balanced and absorbing subsets of
is called a defining sequence for
[1] A locally convex ultrabarrelled space is a barrelled space.
[1] Complete and metrizable TVSs are ultrabarrelled.
is a complete locally bounded non-locally convex TVS and if
is a closed balanced and bounded neighborhood of the origin, then
is an ultrabarrel that is not convex and has a defining sequence consisting of non-convex sets.
[1] There exist barrelled spaces that are not ultrabarrelled.
[1] There exist TVSs that are complete and metrizable (and thus ultrabarrelled) but not barrelled.