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Infunctional analysis, a discipline within mathematics, alocally convextopological vector space (TVS) is said to beinfrabarrelled (also spelledinfrabarreled) if everyboundedbarrel is a neighborhood of the origin.^[1]

Similarly,quasibarrelled spaces aretopological vector spaces (TVS) for which everybornivorous barrelled set in the space is aneighbourhood of the origin. Quasibarrelled spaces are studied because they are a weakening of the defining condition ofbarrelled spaces, for which a form of theBanach–Steinhaus theorem holds.

A subset${\displaystyle B}$ of atopological vector space (TVS)${\displaystyle X}$ is calledbornivorous 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${\displaystyle S\subseteq rB.}$ Abarrelled set or abarrel in a TVS is aset which isconvex,balanced,absorbing andclosed. Aquasibarrelled space is a TVS for which every bornivorous barrelled set in the space is aneighbourhood of the origin.^[2][3]

If${\displaystyle X}$ is a Hausdorff locally convex space thenthe 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${\displaystyle X}$ isquasibarrelled if and only if every bounded closed linear operator from${\displaystyle X}$ into acomplete metrizable TVS is continuous.^[5] By definition, a linear${\displaystyle F:X\to Y}$ operator is calledclosed if its graph is a closed subset of${\displaystyle X\times Y.}$

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

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

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

1. Thestrong dual of${\displaystyle X}$ is quasibarrelled.
2. Thestrong dual of${\displaystyle X}$ is barrelled.
3. Thestrong dual of${\displaystyle X}$ isbornological.

Everyquasi-complete infrabarrelled space is barrelled.^[1]

A locally convex Hausdorff quasibarrelled space that issequentially complete is barrelled.^[6]

A locally convex Hausdorff quasibarrelled space is aMackey space,quasi-M-barrelled, and countably quasibarrelled.^[7]

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

A locally convex space isreflexive if and only if it issemireflexive and quasibarrelled.^[3]

Everybarrelled 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] Everyseparated quotient of an infrabarrelled space is infrabarrelled.^[8]

Every Hausdorffbarrelled space and every Hausdorffbornological space is quasibarrelled.^[9] Thus, everymetrizable TVS is quasibarrelled.

Note that there exist quasibarrelled spaces that are neither barrelled nor bornological.^[3] There existMackey spaces that are not quasibarrelled.^[3] There existdistinguished spaces,DF-spaces, and${\displaystyle \sigma }$-barrelled spaces that are not quasibarrelled.^[3]

Thestrong dual space${\displaystyle X_b^{\mathrm{\prime }}}$ of aFréchet space${\displaystyle X}$ isdistinguished if and only if${\displaystyle X}$ is quasibarrelled.^[10]

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

There exists a quasibarrelledDF-space that is notbornological.^[3]

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

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

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• Functional analysis
• Topological vector spaces

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