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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.
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| == History ==
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| Barrelled spaces were introduced by {{harvs|last=Bourbaki|authorlink=Nicolas Bourbaki|year=1950|txt}}.
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| == Examples == | |
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| * In a [[semi normed vector space]] the closed [[unit ball]] is a barrel.
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| * Every [[locally convex topological vector space]] has a [[neighbourhood basis]] consisting of barrelled sets.
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| * [[Fréchet space]]s, and in particular [[Banach space]]s, are barrelled, but generally a [[normed vector space]] is ''not'' barrelled.
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| * [[Montel space]]s are barrelled. Consequently, strong duals of Montel spaces are barrelled (since they are Montel spaces).
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| * [[locally convex space]]s which are [[Baire space]]s are barrelled.
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| == Properties ==
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| For a [[locally convex space]] <math>X</math> with continuous dual <math>X'</math> the following are equivalent:
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| * <math>X</math> is barrelled,
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| * <math>X</math> carries the [[strong topology (polar topology)|strong topology]] <math>\beta(X, X')</math>,
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| * every lower semi-continuous semi-norm on <math>X</math> is continuous,
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| * every <math>\sigma(X', X)</math>-bounded subset of the continuous dual space <math>X'</math> is equicontinuous.
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| In addition,
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| * Every sequentially complete quasibarrelled space is barrelled.
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| * A barrelled space need not be [[Montel space|Montel]], complete, metrizable, unordered Baire-like, nor the inductive limit of Banach spaces.
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| ==Quasi-barrelled spaces==
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| 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.
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| For a [[locally convex space]] <math>X</math> with continuous dual <math>X'</math> the following are equivalent:
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| * <math>X</math> is quasi-barrelled,
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| * every bounded lower semi-continuous semi-norm on <math>X</math> is continuous,
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| * every <math>\beta(X', X)</math>-bounded subset of the continuous dual space <math>X'</math> is equicontinuous.
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| ==References==
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| <references/> | |
| *{{cite journal
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| | last = Bourbaki | first = Nicolas | authorlink = Nicolas Bourbaki
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| | journal = [[Annales de l'Institut Fourier]]
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| | language = French
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| | mr = 0042609
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| | pages = 5–16 (1951)
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| | title = Sur certains espaces vectoriels topologiques
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| | url = http://www.numdam.org/item?id=AIF_1950__2__5_0
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| | volume = 2
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| | year = 1950}}
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| * {{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}}
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| * {{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 }}
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| * {{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 }}
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| {{Functional Analysis}}
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| [[Category:Topological vector spaces]]
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| [[fr:Ensemble tonnelé]]
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