Infrabarrelled space

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.Template:Sfn

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.

Definition

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

Characterizations

If $X$ is a Hausdorff locally convex space then the canonical injection from $X$ into its bidual is a topological embedding if and only if $X$ is infrabarrelled.Template:Sfn

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

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

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

If $X$ is a metrizable locally convex TVS then the following are equivalent:

1. The strong dual of $X$ is quasibarrelled.
2. The strong dual of $X$ is barrelled.
3. The strong dual of $X$ is bornological.

Properties

Every quasi-complete infrabarrelled space is barrelled.Template:Sfn

A locally convex Hausdorff quasibarrelled space that is sequentially complete is barrelled.Template:Sfn

A locally convex Hausdorff quasibarrelled space is a Mackey space, quasi-M-barrelled, and countably quasibarrelled.Template:Sfn

A locally convex quasibarrelled space that is also a σ-barrelled space is necessarily a barrelled space.Template:Sfn

A locally convex space is reflexive if and only if it is semireflexive and quasibarrelled.Template:Sfn

Examples

Every barrelled space is infrabarrelled.Template:Sfn A closed vector subspace of an infrabarrelled space is, however, not necessarily infrabarrelled.Template:Sfn

Every product and locally convex direct sum of any family of infrabarrelled spaces is infrabarrelled.Template:Sfn Every separated quotient of an infrabarrelled space is infrabarrelled.Template:Sfn

Every Hausdorff barrelled space and every Hausdorff bornological space is quasibarrelled.Template:Sfn Thus, every metrizable TVS is quasibarrelled.

Note that there exist quasibarrelled spaces that are neither barrelled nor bornological.Template:Sfn There exist Mackey spaces that are not quasibarrelled.Template:Sfn There exist distinguished spaces, DF-spaces, and $\sigma$-barrelled spaces that are not quasibarrelled.Template:Sfn

The strong dual space $X_b^{\prime }$ of a Fréchet space $X$ is distinguished if and only if $X$ is quasibarrelled.

Counter-examples

There exists a DF-space that is not quasibarrelled.Template:Sfn

There exists a quasibarrelled DF-space that is not bornological.Template:Sfn

There exists a quasibarrelled space that is not a σ-barrelled space.Template:Sfn

See also

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Bibliography

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