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. ^[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 ${\displaystyle B}$ of a topological vector space (TVS) ${\displaystyle X}$ is called bornivorous if it absorbs all bounded subsets of ${\displaystyle X}$; that is, if for each bounded subset ${\displaystyle S}$ of ${\displaystyle X,}$ there exists some scalar ${\displaystyle r}$ such that ${\displaystyle S\subseteq rB.}$ 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. ^{[2] [3]}

Characterizations

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

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

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

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

If ${\displaystyle X}$ is a metrizable locally convex TVS then the following are equivalent:

1. The strong dual of ${\displaystyle X}$ is quasibarrelled.
2. The strong dual of ${\displaystyle X}$ is barrelled.
3. The strong dual of ${\displaystyle X}$ 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 ${\displaystyle \sigma }$-barrelled spaces that are not quasibarrelled. ^[3]

The strong dual space ${\displaystyle X_{b}^{\prime }}$ of a Fréchet space ${\displaystyle X}$ is distinguished if and only if ${\displaystyle X}$ 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

• Barrelled space  – Type of topological vector space
• Reflexive space  – Locally convex topological vector space
• Semi-reflexive space

References

1. Schaefer & Wolff 1999, p. 142.
2. Jarchow 1981, p. 222.
3. Khaleelulla 1982, pp. 28–63.
4. Narici & Beckenstein 2011, pp. 488–491.
5. Adasch, Ernst & Keim 1978, p. 43.
6. Khaleelulla 1982, p. 28.
7. Khaleelulla 1982, pp. 35.
8. Schaefer & Wolff 1999, p. 194.
9. Adasch, Ernst & Keim 1978, pp. 70–73.
10. Gabriyelyan, S.S. "On topological spaces and topological groups with certain local countable networks (2014)

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