Quasibarrelled space

In functional analysis and related areas of mathematics, quasibarrelled spaces are topological vector spaces (TVS) for which every bornivorous barrelled set in the space is a neighbourhood of the origin. 
Quasibarrelled spaces are studied because they are a weakening of the defining condition of barrelled spaces, for which a form of the Banach–Steinhaus theorem holds.
Definition.
A subset formula_1 of a topological vector space (TVS) formula_2 is called bornivorous if it absorbs all bounded subsets of formula_2; 
that is, if for each bounded subset formula_4 of formula_5 there exists some scalar formula_6 such that formula_7 
A barrelled set or a barrel in a TVS is a set which is convex, balanced, absorbing and closed. 
A quasibarrelled space is a TVS for which every bornivorous barrelled set in the space is a neighbourhood of the origin. 
Properties.
A locally convex Hausdorff quasibarrelled space that is sequentially complete is barrelled. 
A locally convex Hausdorff quasibarrelled space is a Mackey space, quasi-M-barrelled, and countably quasibarrelled. 
A locally convex quasi-barreled space that is also a 𝜎-barrelled space is necessarily a barrelled space. 
A locally convex space is reflexive if and only if it is semireflexive and quasibarrelled. 
Characterizations.
A Hausdorff topological vector space formula_2 is quasibarrelled if and only if every bounded closed linear operator from formula_2 into a complete metrizable TVS is continuous. 
By definition, a linear formula_10 operator is called closed if its graph is a closed subset of formula_11 
For a locally convex space formula_2 with continuous dual formula_13 the following are equivalent:
If formula_2 is a metrizable locally convex TVS then the following are equivalent:
Examples and sufficient conditions.
Every Hausdorff barrelled space and every Hausdorff bornological space is quasibarrelled. 
Thus, every metrizable TVS is quasibarrelled. 
Note that there exist quasibarrelled spaces that are neither barrelled nor bornological. 
There exist Mackey spaces that are not quasibarrelled. 
There exist distinguished spaces, DF-spaces, and formula_15-barrelled spaces that are not quasibarrelled. 
The strong dual space formula_16 of a Fréchet space formula_2 is distinguished if and only if formula_2 is quasibarrelled.
Counter-examples.
There exists a DF-space that is not quasibarrelled. 
There exists a quasibarrelled DF-space that is not bornological. 
There exists a quasi-barreled space that is not a 𝜎-barrelled space.