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In [[functional analysis]] and related areas of [[mathematics]], '''barrelled spaces''' are Hausdorff [[topological vector spaces]] for which every barrelled set in the space is a [[neighbourhood (topology)|neighbourhood]] for the [[zero vector]]. A '''barrelled set''' or a '''barrel''' in a topological vector space is a [[Set (mathematics)|set]] which is [[Convex set|convex]], [[balanced set|balanced]], [[absorbing set|absorbing]] and [[closed set|closed]]. Barrelled spaces are studied because a form of the [[Banach–Steinhaus theorem]] still holds for them. | |||
== History == | |||
== | Barrelled spaces were introduced by {{harvs|last=Bourbaki|authorlink=Nicolas Bourbaki|year=1950|txt}}. | ||
== Examples == | |||
< | * In a [[semi normed vector space]] the closed [[unit ball]] is a barrel. | ||
* Every [[locally convex topological vector space]] has a [[neighbourhood basis]] consisting of barrelled sets. | |||
* [[Fréchet space]]s, and in particular [[Banach space]]s, are barrelled, but generally a [[normed vector space]] is ''not'' barrelled. | |||
* [[Montel space]]s are barrelled. Consequently, strong duals of Montel spaces are barrelled (since they are Montel spaces). | |||
* [[locally convex space]]s which are [[Baire space]]s are barrelled. | |||
== Properties == | |||
For a [[locally convex space]] <math>X</math> with continuous dual <math>X'</math> the following are equivalent: | |||
* <math>X</math> is barrelled, | |||
* <math>X</math> carries the [[strong topology (polar topology)|strong topology]] <math>\beta(X, X')</math>, | |||
* every lower semi-continuous semi-norm on <math>X</math> is continuous, | |||
* every <math>\sigma(X', X)</math>-bounded subset of the continuous dual space <math>X'</math> is equicontinuous. | |||
In addition, | |||
* Every sequentially complete quasibarrelled space is barrelled. | |||
* A barrelled space need not be [[Montel space|Montel]], complete, metrizable, unordered Baire-like, nor the inductive limit of Banach spaces. | |||
==Quasi-barrelled spaces== | |||
A [[topological vector space]] <math>X</math> for which every barrelled bornivorous set in the space is a [[neighbourhood (topology)|neighbourhood]] of <math>0</math> is called a quasi-barrelled space, where a set is bornivorous if it absorbs all bounded subsets of <math>X</math>. Every barrelled space is quasi-barrelled. | |||
For a [[locally convex space]] <math>X</math> with continuous dual <math>X'</math> the following are equivalent: | |||
* <math>X</math> is quasi-barrelled, | |||
* 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'</math> is equicontinuous. | |||
==References== | |||
<references/> | |||
*{{cite journal | |||
| last = Bourbaki | first = Nicolas | authorlink = Nicolas Bourbaki | |||
| journal = [[Annales de l'Institut Fourier]] | |||
| language = French | |||
| mr = 0042609 | |||
| pages = 5–16 (1951) | |||
| title = Sur certains espaces vectoriels topologiques | |||
| url = http://www.numdam.org/item?id=AIF_1950__2__5_0 | |||
| volume = 2 | |||
| year = 1950}} | |||
* {{cite book |last1=Robertson |first1=Alex P. |first2= Wendy J.|last2=Robertson |title= Topological vector spaces |series=Cambridge Tracts in Mathematics |volume=53 |year=1964 |publisher= [[Cambridge University Press]] | pages=65–75}} | |||
* {{cite book | last = Schaefer | first = Helmut H. | year = 1971 | title = Topological vector spaces | series=[[Graduate Texts in Mathematics|GTM]] | volume=3 | publisher = Springer-Verlag | location = New York | isbn = 0-387-98726-6 | page=60 }} | |||
* {{cite book | author=S.M. Khaleelulla | title=Counterexamples in Topological Vector Spaces | publisher=[[Springer-Verlag]] | series=[[Graduate Texts in Mathematics|GTM]] | volume=936 | date=1982 | isbn=978-3-540-11565-6 | pages=28-46 }} | |||
{{Functional Analysis}} | |||
[[Category:Topological vector spaces]] | |||
[[fr:Ensemble tonnelé]] | |||
Revision as of 12:13, 29 January 2014
In functional analysis and related areas of mathematics, barrelled spaces are Hausdorff topological vector spaces for which every barrelled set in the space is a neighbourhood for the zero vector. A barrelled set or a barrel in a topological vector space is a set which is convex, balanced, absorbing and closed. Barrelled spaces are studied because a form of the Banach–Steinhaus theorem still holds for them.
History
Barrelled spaces were introduced by Template:Harvs.
Examples
- In a semi normed vector space the closed unit ball is a barrel.
- Every locally convex topological vector space has a neighbourhood basis consisting of barrelled sets.
- Fréchet spaces, and in particular Banach spaces, are barrelled, but generally a normed vector space is not barrelled.
- Montel spaces are barrelled. Consequently, strong duals of Montel spaces are barrelled (since they are Montel spaces).
- locally convex spaces which are Baire spaces are barrelled.
Properties
For a locally convex space with continuous dual the following are equivalent:
- is barrelled,
- carries the strong topology ,
- every lower semi-continuous semi-norm on is continuous,
- every -bounded subset of the continuous dual space is equicontinuous.
In addition,
- Every sequentially complete quasibarrelled space is barrelled.
- A barrelled space need not be Montel, complete, metrizable, unordered Baire-like, nor the inductive limit of Banach spaces.
Quasi-barrelled spaces
A topological vector space for which every barrelled bornivorous set in the space is a neighbourhood of is called a quasi-barrelled space, where a set is bornivorous if it absorbs all bounded subsets of . Every barrelled space is quasi-barrelled.
For a locally convex space with continuous dual the following are equivalent:
- is quasi-barrelled,
- every bounded lower semi-continuous semi-norm on is continuous,
- every -bounded subset of the continuous dual space is equicontinuous.
References
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