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In [[functional analysis]] and related areas of [[mathematics]], the '''Mackey topology''', named after [[George Mackey]], is the [[finer topology|finest topology]] for a [[topological vector space]] which still preserves the [[continuous dual]]. In other words the Mackey topology does not make linear functions continuous which were discontinuous in the default topology. A [[topological vector space]] (TVS) is called a '''[[Mackey space]]''' if its topology is the same as the Mackey topology.

The Mackey topology is the opposite of the [[weak topology]], which is the [[Comparison of topologies|coarsest topology]] on a [[topological vector space]] which preserves the continuity of all linear functions in the continuous dual.

The [[Mackey–Arens theorem]] states that all possible [[Dual topology|dual topologies]] are finer than the weak topology and coarser than the Mackey topology.

==Definition==
{{Main|Polar topology|Dual system}}

===Definition for a pairing===

Given a [[Dual pair|pairing]] <math>(X, Y, b),</math> the '''Mackey topology''' on <math>X</math> induced by <math>(X, Y, b),</math> denoted by <math>\tau(X, Y, b),</math> is the [[polar topology]] defined on <math>X</math> by using the set of all [[Dual system#Weak topology|<math>\sigma(Y, X, b)</math>-compact]] [[absolutely convex|disks]] in <math>Y.</math> 

When <math>X</math> is endowed with the Mackey topology then it will be denoted by <math>X_{\tau(X, Y, b)}</math> or simply <math>X_{\tau(X, Y)}</math> or <math>X_{\tau}</math> if no ambiguity can arise. 

A linear map <math>F : X \to W</math> is said to be '''Mackey continuous''' (with respect to pairings <math>(X, Y, b)</math> and <math>(W, Z, c)</math>) if <math>F : (X, \tau(X, Y, b)) \to (W, \tau(W, Z, c))</math> is continuous.

===Definition for a topological vector space===

The definition of the Mackey topology for a [[topological vector space]] (TVS) is a specialization of the above definition of the Mackey topology of a pairing. 
If <math>X</math> is a TVS with [[continuous dual space]] <math>X^{\prime},</math> then the evaluation map <math>\left(x, x^{\prime}\right) \mapsto x^{\prime}(x)</math> on <math>X \times X^{\prime}</math> is called the '''canonical pairing'''. 

The '''Mackey topology''' on a TVS <math>X,</math> denoted by <math>\tau\left(X, X^{\prime}\right),</math> is the Mackey topology on <math>X</math> induced by the canonical pairing <math>\left\langle X, X^{\prime} \right\rangle.</math> 

That is, the Mackey topology is the [[polar topology]] on <math>X</math> obtained by using the set of all [[Weak* topology|weak*-compact]] [[Absolutely convex|disks]] in <math>X^{\prime}.</math> 
When <math>X</math> is endowed with the Mackey topology then it will be denoted by <math>X_{\tau\left(X, X^{\prime}\right)}</math> or simply <math>X_{\tau}</math> if no ambiguity can arise. 

A linear map <math>F : X \to Y</math> between TVSs is '''Mackey continuous''' if <math>F : \left(X, \tau\left(X, X^{\prime}\right)\right) \to \left(Y, \tau\left(Y, Y^{\prime}\right)\right)</math> is continuous. 

==Examples==

Every [[Metrizable topological vector space|metrizable]] [[Locally convex topological vector space|locally convex]] <math>(X, \nu)</math> with continuous dual <math>X^{\prime}</math> carries the Mackey topology, that is <math>\nu = \tau\left(X, X^{\prime}\right)</math> or to put it more succinctly every metrizable locally convex space is a [[Mackey space]].

Every [[Hausdorff space|Hausdorff]] [[Barrelled space|barreled]] locally convex space is Mackey.

Every [[Fréchet space]] <math>(X, \nu)</math> carries the Mackey topology and the topology coincides with the [[strong topology]], that is <math>\nu = \tau\left(X, X^{\prime}\right) = \beta\left(X, X^{\prime}\right).</math> 

==Applications==

The Mackey topology has an application in economies with infinitely many commodities.<ref>{{Cite journal|last1=Bewley|first1=T. F.|title=Existence of equilibria in economies with infinitely many commodities|journal=Journal of Economic Theory|year=1972|volume=4|issue=3|pages=514–540|doi=10.1016/0022-0531(72)90136-6}}</ref>

==See also==

* {{annotated link|Dual system}}
* {{annotated link|Mackey space}}
* {{annotated link|Polar topology}}
* {{annotated link|Strong topology}}
* {{annotated link|Topologies on spaces of linear maps}}
* {{annotated link|Weak topology}}

==Citations==

{{reflist}}

==Bibliography==

* {{cite book|last=Bourbaki|first=Nicolas|authorlink=Nicolas Bourbaki|title=Topological vector spaces|series=Elements of mathematics|publisher=Addison–Wesley|year=1977}}
* {{Bourbaki Topological Vector Spaces Part 1 Chapters 1–5}} <!-- {{sfn|Bourbaki|1987|p=}} -->
* {{cite journal|last=Mackey|first=G.W.|authorlink=George Mackey|title=On convex topological linear spaces|journal=Trans. Amer. Math. Soc.|volume=60|year=1946|pages=519–537|doi=10.2307/1990352|issue=3|publisher=Transactions of the American Mathematical Society, Vol. 60, No. 3|jstor=1990352|pmc=1078623 }}
* {{Narici Beckenstein Topological Vector Spaces|edition=2}} <!-- {{sfn|Narici|Beckenstein|2011|p=}} -->
* {{cite book|last=Robertson|first=A.P.|author2=W.J. Robertson|title=Topological vector spaces|series=Cambridge Tracts in Mathematics|volume=53|year=1964|publisher=[[Cambridge University Press]]|page=62}}
* {{cite book|last=Schaefer|first=Helmut H. | authorlink = Helmut H. Schaefer |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=131}}
* {{Schaefer Wolff Topological Vector Spaces|edition=2}} <!-- {{sfn|Schaefer|Wolff|1999|p=}} -->
* {{springer|id=M/m062080|title=Mackey topology|author=A.I. Shtern}}

{{Duality and spaces of linear maps}}
{{Topological vector spaces}}
{{Boundedness and bornology}}
{{Functional analysis}}
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[[Category:Topological vector spaces]]