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 $L(E, F)$ from a barreled space $E$ to a locally convex TVS boundedness in the topology of pointwise convergence implies boundedness in the topology of bounded convergence.
A subset $T \subset E$ of a TVS E is a barrel if it is
absorbing
balanced
closed
convex
A TVS $E$ 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).
A locally convex TVS which is a Baire space is barreled.
A locally convex TVS is barreled iff its topology is the strong topology.
Since all locally convex TVSes that are Baire spaces are barreled, the examples naturally include Fréchet spaces, Banach spaces and Hilbert spaces.
See the functional analysis bibliography.
The definition of quasibarreled is from
It is called infrabarreled in