Английская Википедия:Countably barrelled space
In functional analysis, a topological vector space (TVS) is said to be countably barrelled if every weakly bounded countable union of equicontinuous subsets of its continuous dual space is again equicontinuous. This property is a generalization of barrelled spaces.
Definition
A TVS X with continuous dual space <math>X^{\prime}</math> is said to be countably barrelled if <math>B^{\prime} \subseteq X^{\prime}</math> is a weak-* bounded subset of <math>X^{\prime}</math> that is equal to a countable union of equicontinuous subsets of <math>X^{\prime}</math>, then <math>B^{\prime}</math> is itself equicontinuous.Шаблон:Sfn A Hausdorff locally convex TVS is countably barrelled if and only if each barrel in X that is equal to the countable intersection of closed convex balanced neighborhoods of 0 is itself a neighborhood of 0.Шаблон:Sfn
σ-barrelled space
A TVS with continuous dual space <math>X^{\prime}</math> is said to be σ-barrelled if every weak-* bounded (countable) sequence in <math>X^{\prime}</math> is equicontinuous.Шаблон:Sfn
Sequentially barrelled space
A TVS with continuous dual space <math>X^{\prime}</math> is said to be sequentially barrelled if every weak-* convergent sequence in <math>X^{\prime}</math> is equicontinuous.Шаблон:Sfn
Properties
Every countably barrelled space is a countably quasibarrelled space, a σ-barrelled space, a σ-quasi-barrelled space, and a sequentially barrelled space.Шаблон:Sfn An H-space is a TVS whose strong dual space is countably barrelled.Шаблон:Sfn
Every countably barrelled space is a σ-barrelled space and every σ-barrelled space is sequentially barrelled.Шаблон:Sfn Every σ-barrelled space is a σ-quasi-barrelled space.Шаблон:Sfn
A locally convex quasi-barrelled space that is also a 𝜎-barrelled space is a barrelled space.Шаблон:Sfn
Examples and sufficient conditions
Every barrelled space is countably barrelled.Шаблон:Sfn However, there exist semi-reflexive countably barrelled spaces that are not barrelled.Шаблон:Sfn The strong dual of a distinguished space and of a metrizable locally convex space is countably barrelled.Шаблон:Sfn
Counter-examples
There exist σ-barrelled spaces that are not countably barrelled.Шаблон:Sfn There exist normed DF-spaces that are not countably barrelled.Шаблон:Sfn There exists a quasi-barrelled space that is not a 𝜎-barrelled space.Шаблон:Sfn There exist σ-barrelled spaces that are not Mackey spaces.Шаблон:Sfn There exist σ-barrelled spaces that are not countably quasi-barrelled spaces and thus not countably barrelled.Шаблон:Sfn There exist sequentially barrelled spaces that are not σ-quasi-barrelled.Шаблон:Sfn There exist quasi-complete locally convex TVSs that are not sequentially barrelled.Шаблон:Sfn
See also
References
- Шаблон:Khaleelulla Counterexamples in Topological Vector Spaces
- Шаблон:Narici Beckenstein Topological Vector Spaces
- Шаблон:Schaefer Wolff Topological Vector Spaces
- Шаблон:Trèves François Topological vector spaces, distributions and kernels
- Шаблон:Cite book
Шаблон:Topological vector spaces
