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

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

  • 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 the following are equivalent:

  • X is quasi-barrelled,
  • every bounded lower semi-continuous semi-norm on X is continuous,
  • 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]

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]

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See also

References
  1. Jarhow 1981, p. 222.
  2. Khaleelulla 1982, pp. 28-63.
  3. Khaleelulla 1982, p. 28.
  4. Khaleelulla 1982, pp. 35.
  5. Adasch 1978, p. 43.
  6. Adasch 1978, pp. 70-73.


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