Quasibarrelled space

In functional analysis and related areas of mathematics, 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 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 ⊆ 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.

Properties

A locally convex Hausdorff quasibarrelled space that is sequentially complete is barrelled. A locally convex Hausdorff quasibarrelled space is a Mackey space, quasi-M-barrelled, and countably quasibarrelled.

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

A locally convex quasi-barreled space that is also a 𝜎-barrelled space is a barrelled space.

Characterizations

A Hausdorff TVS X is quasibarrelled if and only if every bounded closed linear operator from X into a complete metrizable TVS is continuous.

• Recall that a linear F : X → Y operator is called closed if its graph is a closed subset of X × Y.

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

• X is quasi-barrelled,
• every bounded lower semi-continuous semi-norm on X is continuous,
• every $\beta (X',X)$-bounded subset of the continuous dual space $X'$ 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.

Examples and sufficient conditions

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

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

Counter-examples

There exists a DF-space that is not quasibarrelled. There exists a quasibarrelled DF-space that is not bornological. There exists a quasi-barreled space that is not a 𝜎-barrelled space.

See also

• Barrelled space
• Countably barrelled space
• Countably quasi-barrelled space
• Infrabarreled space
• Uniform boundedness principle#Generalisations

References

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