Barreled spaces are topological vector spaces for which the theorem of Banach–Steinhaus is valid. This theorem says, roughly, that for a set of continuous linear maps from a barreled space to a locally convex TVS boundedness in the topology of pointwise convergence implies boundedness in the topology of bounded convergence.
A subset of a TVS E is a barrel if it is
A TVS is barreled (or barrelled) if every barrel is a neighborhood of zero.
Sometimes locally convex is included in the definition, this is not implied by barreled as defined above, i.e. there are barreled spaces that are not locally convex.
In the definition of quasibarreled or infrabarreled the barrels are replaced by sets that are barrels and which absorb all bounded sets (sets with the latter property are also called bornivorous).
See the functional analysis bibliography.
The definition of quasibarreled is from
It is called infrabarreled in