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In [[mathematics]], particularly in [[functional analysis]], a '''Mackey space''' is a [[locally convex topological vector space]] ''X'' such that the [[topology]] of ''X'' coincides with the [[Mackey topology]] τ(''X'',''X&prime;''), the [[finer topology|finest topology]] which still preserves the [[continuous dual]].
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==Examples==
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Examples of Mackey spaces include:
* All [[bornological space]]s.
* All Hausdorff locally convex [[barrelled space|quasi-barrelled]] (and hence all Hausdorff locally convex [[barrelled space]]s and all Hausdorff locally convex reflexive spaces).
* All Hausdorff locally convex [[metrizable space]]s.
 
==Properties==
* A locally convex space <math>X</math> with continuous dual <math>X'</math> is a Mackey space if and only if each convex and <math>\sigma(X', X)</math>-relatively compact subset of <math>X'</math> is equicontinuous.
* The [[Completion (metric space)|completion]] of a Mackey space is again a Mackey space.
* A separated quotient of a Mackey space is again a Mackey space.
* A Mackey space need not be separable, complete, quasi-barrelled, nor <math>\sigma</math>-quasi-barrelled.
 
== References ==
* {{cite book |last=Robertson |first=A.P. |coauthors= W.J. Robertson |title= Topological vector spaces |series=Cambridge Tracts in Mathematics |volume=53 |year=1964 |publisher= [[Cambridge University Press]] | page=81 }}
* {{cite book | author=H.H. Schaefer | title=Topological Vector Spaces | publisher=[[Springer-Verlag]] | series=[[Graduate Texts in Mathematics|GTM]] | volume=3 | date=1970 | isbn=0-387-05380-8 | pages=132–133 }}
* {{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=31, 41, 55-58 }}
 
{{Functional Analysis}}
 
[[Category:Topological vector spaces]]
 
{{mathanalysis-stub}}

Latest revision as of 20:14, 2 November 2014

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