Ultrabornological space

In functional analysis, a topological vector space (TVS) formula_1 is called ultrabornological if every bounded linear operator from formula_1 into another TVS is necessarily continuous. A general version of the closed graph theorem holds for ultrabornological spaces. 
Ultrabornological spaces were introduced by Alexander Grothendieck (Grothendieck [1955, p. 17] "espace du type (β)"). 
Definitions.
Let formula_1 be a topological vector space (TVS). 
Preliminaries.
A disk is a convex and balanced set. 
A disk in a TVS formula_1 is called bornivorous if it absorbs every bounded subset of formula_5 
A linear map between two TVSs is called infrabounded if it maps Banach disks to bounded disks. 
A disk formula_6 in a TVS formula_1 is called infrabornivorous if it satisfies any of the following equivalent conditions:
while if formula_1 locally convex then we may add to this list:
while if formula_1 locally convex and Hausdorff then we may add to this list:
Ultrabornological space.
A TVS formula_1 is ultrabornological if it satisfies any of the following equivalent conditions:
while if formula_1 is a locally convex space then we may add to this list:
while if formula_1 is a Hausdorff locally convex space then we may add to this list:
Properties.
Every locally convex ultrabornological space is barrelled, quasi-ultrabarrelled space, and a bornological space but there exist bornological spaces that are not ultrabornological.
Examples and sufficient conditions.
The finite product of locally convex ultrabornological spaces is ultrabornological. Inductive limits of ultrabornological spaces are ultrabornological. 
Every Hausdorff sequentially complete bornological space is ultrabornological. Thus every compete Hausdorff bornological space is ultrabornological. In particular, every Fréchet space is ultrabornological. 
The strong dual space of a complete Schwartz space is ultrabornological. 
Every Hausdorff bornological space that is quasi-complete is ultrabornological. 
There exist ultrabarrelled spaces that are not ultrabornological. 
There exist ultrabornological spaces that are not ultrabarrelled.