Infrabarrelled space

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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.[1]

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

Characterizations

If [math]\displaystyle{ X }[/math] is a Hausdorff locally convex space then the canonical injection from [math]\displaystyle{ X }[/math] into its bidual is a topological embedding if and only if [math]\displaystyle{ X }[/math] is infrabarrelled.[4]

A Hausdorff topological vector space [math]\displaystyle{ X }[/math] is quasibarrelled if and only if every bounded closed linear operator from [math]\displaystyle{ X }[/math] into a complete metrizable TVS is continuous.[5] By definition, a linear [math]\displaystyle{ F : X \to Y }[/math] operator is called closed if its graph is a closed subset of [math]\displaystyle{ X \times Y. }[/math]

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

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

If [math]\displaystyle{ X }[/math] is a metrizable locally convex TVS then the following are equivalent:

  1. The strong dual of [math]\displaystyle{ X }[/math] is quasibarrelled.
  2. The strong dual of [math]\displaystyle{ X }[/math] is barrelled.
  3. The strong dual of [math]\displaystyle{ X }[/math] is bornological.

Properties

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]

Examples

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 [math]\displaystyle{ \sigma }[/math]-barrelled spaces that are not quasibarrelled.[3]

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

Counter-examples

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]

See also

References

Bibliography