Quasibarrelled space

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.[1][2]

A locally convex Hausdorff quasibarrelled space that is sequentially complete is barrelled.[3] A locally convex Hausdorff quasibarrelled space is a Mackey space, quasi-M-barrelled, and countably quasibarrelled.[4] A locally convex quasibarrelled space that is also a σ-barrelled space is necessarily a barrelled space.[2]

A locally convex space is reflexive if and only if it is semireflexive and quasibarrelled.[2]

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 Hausdorff barrelled space and every Hausdorff bornological space is quasibarrelled.[6] Thus, every metrizable TVS is quasibarrelled.

Note that there exist quasibarrelled spaces that are neither barrelled nor bornological.[2] There exist Mackey spaces that are not quasibarrelled.[2] There exist distinguished spaces, DF-spaces, and ${\displaystyle \sigma }$-barrelled spaces that are not quasibarrelled.[2]

The strong dual space ${\displaystyle X_{b}^{\prime }}$ of a Fréchet space ${\displaystyle X}$ is distinguished if and only if ${\displaystyle X}$ is quasibarrelled.[7]

• Barrelled space – Type of topological vector space
• Countably barrelled space
• Countably quasibarrelled space
• Infrabarrelled space
• Uniform boundedness principle#Generalisations – A theorem stating that pointwise boundedness implies uniform boundedness

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