Английская Википедия:Countably quasi-barrelled space
In functional analysis, a topological vector space (TVS) is said to be countably quasi-barrelled if every strongly bounded countable union of equicontinuous subsets of its continuous dual space is again equicontinuous. This property is a generalization of quasibarrelled spaces.
Definition
A TVS X with continuous dual space <math>X^{\prime}</math> is said to be countably quasi-barrelled if <math>B^{\prime} \subseteq X^{\prime}</math> is a strongly 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 quasi-barrelled if and only if each bornivorous 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-quasi-barrelled space
A TVS with continuous dual space <math>X^{\prime}</math> is said to be σ-quasi-barrelled if every strongly bounded (countable) sequence in <math>X^{\prime}</math> is equicontinuous.Шаблон:Sfn
Sequentially quasi-barrelled space
A TVS with continuous dual space <math>X^{\prime}</math> is said to be sequentially quasi-barrelled if every strongly convergent sequence in <math>X^{\prime}</math> is equicontinuous.
Properties
Every countably quasi-barrelled space is a σ-quasi-barrelled space.
Examples and sufficient conditions
Every barrelled space, every countably barrelled space, and every quasi-barrelled space is countably quasi-barrelled and thus also σ-quasi-barrelled space.Шаблон:Sfn The strong dual of a distinguished space and of a metrizable locally convex space is countably quasi-barrelled.Шаблон:Sfn
Every σ-barrelled space is a σ-quasi-barrelled space.Шаблон:Sfn Every DF-space is countably quasi-barrelled.Шаблон:Sfn A σ-quasi-barrelled space that is sequentially complete is a σ-barrelled space.Шаблон:Sfn
There exist σ-barrelled spaces that are not Mackey spaces.Шаблон:Sfn There exist σ-barrelled spaces (which are consequently σ-quasi-barrelled spaces) that are not countably quasi-barrelled spaces.Шаблон:Sfn There exist sequentially complete Mackey spaces that are not σ-quasi-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
