Английская Википедия: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.Шаблон:Sfn
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 <math>B</math> of a topological vector space (TVS) <math>X</math> is called bornivorous if it absorbs all bounded subsets of <math>X</math>; that is, if for each bounded subset <math>S</math> of <math>X,</math> there exists some scalar <math>r</math> such that <math>S \subseteq r B.</math> 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.Шаблон:SfnШаблон:Sfn
Characterizations
If <math>X</math> is a Hausdorff locally convex space then the canonical injection from <math>X</math> into its bidual is a topological embedding if and only if <math>X</math> is infrabarrelled.Шаблон:Sfn
A Hausdorff topological vector space <math>X</math> is quasibarrelled if and only if every bounded closed linear operator from <math>X</math> into a complete metrizable TVS is continuous.Шаблон:Sfn By definition, a linear <math>F : X \to Y</math> operator is called closed if its graph is a closed subset of <math>X \times Y.</math>
For a locally convex space <math>X</math> with continuous dual <math>X^{\prime}</math> the following are equivalent:
- <math>X</math> is quasibarrelled.
- Every bounded lower semi-continuous semi-norm on <math>X</math> is continuous.
- Every <math>\beta(X', X)</math>-bounded subset of the continuous dual space <math>X^{\prime}</math> is equicontinuous.
If <math>X</math> is a metrizable locally convex TVS then the following are equivalent:
- The strong dual of <math>X</math> is quasibarrelled.
- The strong dual of <math>X</math> is barrelled.
- The strong dual of <math>X</math> is bornological.
Properties
Every quasi-complete infrabarrelled space is barrelled.Шаблон:Sfn
A locally convex Hausdorff quasibarrelled space that is sequentially complete is barrelled.Шаблон:Sfn
A locally convex Hausdorff quasibarrelled space is a Mackey space, quasi-M-barrelled, and countably quasibarrelled.Шаблон:Sfn
A locally convex quasibarrelled space that is also a σ-barrelled space is necessarily a barrelled space.Шаблон:Sfn
A locally convex space is reflexive if and only if it is semireflexive and quasibarrelled.Шаблон:Sfn
Examples
Every barrelled space is infrabarrelled.Шаблон:Sfn A closed vector subspace of an infrabarrelled space is, however, not necessarily infrabarrelled.Шаблон:Sfn
Every product and locally convex direct sum of any family of infrabarrelled spaces is infrabarrelled.Шаблон:Sfn Every separated quotient of an infrabarrelled space is infrabarrelled.Шаблон:Sfn
Every Hausdorff barrelled space and every Hausdorff bornological space is quasibarrelled.Шаблон:Sfn Thus, every metrizable TVS is quasibarrelled.
Note that there exist quasibarrelled spaces that are neither barrelled nor bornological.Шаблон:Sfn There exist Mackey spaces that are not quasibarrelled.Шаблон:Sfn There exist distinguished spaces, DF-spaces, and <math>\sigma</math>-barrelled spaces that are not quasibarrelled.Шаблон:Sfn
The strong dual space <math>X_b^{\prime}</math> of a Fréchet space <math>X</math> is distinguished if and only if <math>X</math> is quasibarrelled.[1]
Counter-examples
There exists a DF-space that is not quasibarrelled.Шаблон:Sfn
There exists a quasibarrelled DF-space that is not bornological.Шаблон:Sfn
There exists a quasibarrelled space that is not a σ-barrelled space.Шаблон:Sfn
See also
References
Bibliography
- Шаблон:Adasch Topological Vector Spaces
- Шаблон:Berberian Lectures in Functional Analysis and Operator Theory
- Шаблон:Bourbaki Topological Vector Spaces Part 1 Chapters 1–5
- Шаблон:Conway A Course in Functional Analysis
- Шаблон:Edwards Functional Analysis Theory and Applications
- Шаблон:Grothendieck Topological Vector Spaces
- Шаблон:Hogbe-Nlend Bornologies and Functional Analysis
- Шаблон:Husain Khaleelulla Barrelledness in Topological and Ordered Vector Spaces
- Шаблон:Jarchow Locally Convex Spaces
- Шаблон:Khaleelulla Counterexamples in Topological Vector Spaces
- Шаблон:Köthe Topological Vector Spaces I
- Шаблон:Köthe Topological Vector Spaces II
- Шаблон:Narici Beckenstein Topological Vector Spaces
- Шаблон:Schaefer Wolff Topological Vector Spaces
- Шаблон:Swartz An Introduction to Functional Analysis
- Шаблон:Trèves François Topological vector spaces, distributions and kernels
- Шаблон:Wilansky Modern Methods in Topological Vector Spaces
- Шаблон:Wong Schwartz Spaces, Nuclear Spaces, and Tensor Products
Шаблон:Topological vector spaces Шаблон:Boundedness and bornology Шаблон:Functional analysis
- ↑ Gabriyelyan, S.S. "On topological spaces and topological groups with certain local countable networks (2014)
