In functional analysis, a discipline within mathematics, a locally convex topological vector space (TVS) is said to be infrabarrelled (also spelled infrabarreled) if every bounded barrel is a neighborhood 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.
of a topological vector space (TVS)
is called bornivorous if it absorbs all bounded subsets of
is a Hausdorff locally convex space then the canonical injection from
[4] A Hausdorff topological vector space
is quasibarrelled if and only if every bounded closed linear operator from
into a complete metrizable TVS is continuous.
is a metrizable locally convex TVS then the following are equivalent: Every quasi-complete infrabarrelled space is barrelled.
[1] A locally convex Hausdorff quasibarrelled space that is sequentially complete is barrelled.
[6] A locally convex Hausdorff quasibarrelled space is a Mackey space, quasi-M-barrelled, and countably quasibarrelled.
[3] A locally convex space is reflexive if and only if it is semireflexive and quasibarrelled.
Note that there exist quasibarrelled spaces that are neither barrelled nor bornological.
[3] There exist Mackey spaces that are not quasibarrelled.
[3] There exist distinguished spaces, DF-spaces, and
[3] There exists a quasibarrelled DF-space that is not bornological.