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.^[2][3]

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.^[4]

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.^[5] 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:

1. ${\displaystyle X}$  is quasibarrelled.
2. Every bounded lower semi-continuous semi-norm on ${\displaystyle X}$  is continuous.
3. Every ${\displaystyle \beta (X',X)}$ -bounded subset of the continuous dual space ${\displaystyle X^{\prime }}$  is equicontinuous.

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

1. The strong dual of ${\displaystyle X}$  is quasibarrelled.
2. The strong dual of ${\displaystyle X}$  is barrelled.
3. The strong dual of ${\displaystyle X}$  is bornological.

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.^[7]

A locally convex quasibarrelled space that is also a σ-barrelled space is necessarily a barrelled space.^[3]

A locally convex space is reflexive if and only if it is semireflexive and quasibarrelled.^[3]

Every barrelled space is infrabarrelled.^[1] A closed vector subspace of an infrabarrelled space is, however, not necessarily infrabarrelled.^[8]

Every product and locally convex direct sum of any family of infrabarrelled spaces is infrabarrelled.^[8] Every separated quotient of an infrabarrelled space is infrabarrelled.^[8]

Every Hausdorff barrelled space and every Hausdorff bornological space is quasibarrelled.^[9] Thus, every metrizable TVS is 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 ${\displaystyle \sigma }$ -barrelled spaces that are not quasibarrelled.^[3]

The strong dual space ${\displaystyle X_{b}^{\prime }}$  of a Fréchet space ${\displaystyle X}$  is distinguished if and only if ${\displaystyle X}$  is quasibarrelled.^[10]

Counter-examples edit

There exists a DF-space that is not quasibarrelled.^[3]

There exists a quasibarrelled DF-space that is not bornological.^[3]

There exists a quasibarrelled space that is not a σ-barrelled space.^[3]

• Barrelled space – Type of topological vector space
• Reflexive space – Locally convex topological vector space
• Semi-reflexive space

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