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.[1][2]

Properties

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 quasi-barreled 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]

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.[5] By definition, 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 the following are equivalent:

  1. X is quasi-barrelled.
  2. Every bounded lower semi-continuous semi-norm on X is continuous.
  3. Every -bounded subset of the continuous dual space 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.[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 -barrelled spaces that are not quasibarrelled.[2]

The strong dual space of a Fréchet space is a Distinguished space if and only if it is quasibarrelled.[7]

Counter-examples

There exists a DF-space that is not quasibarrelled.[2] There exists a quasibarrelled DF-space that is not bornological.[2] There exists a quasi-barreled space that is not a 𝜎-barrelled space.[2]

See also

References
    1. Jarchow 1981, p. 222.
    2. Khaleelulla 1982, pp. 28-63.
    3. Khaleelulla 1982, p. 28.
    4. Khaleelulla 1982, pp. 35.
    5. Adasch, Ernst & Keim 1978, p. 43.
    6. Adasch, Ernst & Keim 1978, pp. 70-73.
    7. Gabriyelyan, S.S. "On topological spaces and topological groups with certain local countable networks (2014)

    Bibliography
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