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.

Similarly, 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.

A subset ${\displaystyle B}$ of a topological vector space (TVS) ${\displaystyle X}$ is called bornivorous if it absorbs all bounded subsets of ${\displaystyle X}$; that is, if for each bounded subset ${\displaystyle S}$ of ${\displaystyle X,}$ there exists some scalar ${\displaystyle r}$ such that ${\displaystyle S\subseteq rB.}$ 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.

If ${\displaystyle X}$ is a Hausdorff locally convex space then the canonical injection from ${\displaystyle X}$ into its bidual is a topological embedding if and only if ${\displaystyle X}$ is infrabarrelled.

A Hausdorff topological vector space ${\displaystyle X}$ is quasibarrelled if and only if every bounded closed linear operator from ${\displaystyle X}$ into a complete metrizable TVS is continuous. By definition, a linear ${\displaystyle F:X\to Y}$ operator is called closed if its graph is a closed subset of ${\displaystyle X\times Y.}$

For a locally convex space ${\displaystyle X}$ with continuous dual ${\displaystyle X^{\prime }}$ the following are equivalent:

If ${\displaystyle X}$ is a metrizable locally convex TVS then the following are equivalent:

Every quasi-complete infrabarrelled space is barrelled.