View source for Mackey–Arens theorem

The '''Mackey–Arens theorem''' is an important theorem in [[functional analysis]] that characterizes those [[locally convex]] [[Topological vector space|vector topologies]] that have some given space of [[linear functional]]s as their [[continuous dual space]]. 
According to Narici (2011), this profound result is central to [[Dual system|duality theory]]; a theory that is "the central part of the modern theory of topological vector spaces."{{sfn | Schaefer|Wolff| 1999 | p=122}} 

==Prerequisites==
{{Main|Polar topology|Mackey topology}}

Let {{mvar|X}} be a vector space and let {{mvar|Y}} be a vector subspace of the algebraic dual of {{mvar|X}} that [[Separating set|separates points]] on {{mvar|X}}. 
If {{math|𝜏}} is any other locally convex Hausdorff topological vector space topology on {{mvar|X}}, then we say that {{math|𝜏}} is '''compatible with duality between {{mvar|X}} and {{mvar|Y}}''' if when {{mvar|X}} is equipped with {{math|𝜏}}, then it has {{mvar|Y}} as its continuous dual space. 
If we give {{mvar|X}} the weak topology {{math|𝜎(''X'', ''Y'')}} then {{math|''X''<sub>𝜎(''X'', ''Y'')</sub>}} is a Hausdorff locally convex [[topological vector space]] (TVS) and {{math|𝜎(''X'', ''Y'')}} is compatible with duality between {{mvar|X}} and {{mvar|Y}} (i.e. <math>X_{\sigma(X, Y)}^{\prime} = \left( X_{\sigma(X, Y)} \right)^{\prime} = Y</math>). 
We can now ask the question: what are ''all'' of the locally convex Hausdorff TVS topologies that we can place on {{mvar|X}} that are compatible with duality between {{mvar|X}} and {{mvar|Y}}? 
The answer to this question is called the Mackey–Arens theorem.

==Mackey–Arens theorem==

{{Math theorem|name=Mackey–Arens theorem{{sfn|Trèves|2006|pp=196, 368–370}}|math_statement=
Let ''X'' be a vector space and let 𝒯 be a locally convex Hausdorff topological vector space topology on ''X''. Let {{math|''X''{{big|{{'}}}}}} denote the continuous dual space of ''X'' and let <math>X_{\mathcal{T}}</math> denote ''X'' with the topology 𝒯. Then the following are equivalent:
{{ordered list|
 | 𝒯 is identical to a <math>\mathcal{G}^{\prime}</math>-topology on ''X'', where <math>\mathcal{G}^{\prime}</math> is a covering of <{{math|''X''{{big|{{'}}}}}} consisting of convex, balanced, {{math|&sigma;(''X''{{big|{{'}}}}, ''X'')}}-compact sets with the properties that
{{ordered list|type=lower-roman|style=margin-left: 1em
   | If <math>G_1^{\prime}, G_2^{\prime} \in \mathcal{G}^{\prime}</math> then there exists a <math>G^{\prime} \in \mathcal{G}^{\prime}</math> such that <math>G_1^{\prime} \cup G_2^{\prime} \subseteq G^{\prime}</math>, and
   | If <math>G_1^{\prime} \in \mathcal{G}^{\prime}</math> and <math>\lambda</math> is a scalar then there exists a <math>G^{\prime} \in \mathcal{G}^{\prime}</math> such that <math>\lambda G_1^{\prime} \subseteq G^{\prime}</math>.
}}
 | The continuous dual of <math>X_{\mathcal{T}}</math> is identical to {{math|''X''{{big|{{'}}}}}}.
}}
And furthermore, 
{{ordered list
 | the topology 𝒯 is identical to the {{math|&epsilon;(''X'', ''X''{{big|{{'}}}})}} topology, that is, to the topology of uniform on convergence on the equicontinuous subsets of {{math|''X''{{big|{{'}}}}}}.
 | the Mackey topology {{math|&tau;(''X'', ''X''{{big|{{'}}}})}} is the finest locally convex Hausdorff TVS topology on ''X'' that is compatible with duality between ''X'' and <math>X_{\mathcal{T}}^{\prime}</math>, and
 | the weak topology {{math|&sigma;(''X'', ''X''{{big|{{'}}}})}} is the coarsest locally convex Hausdorff TVS topology on ''X'' that is compatible with duality between ''X'' and <math>X_{\mathcal{T}}^{\prime}</math>.
}}
}}

==See also==

* [[Dual system]]
* [[Mackey topology]]
* [[Polar topology]]

==References==

{{reflist}}

==Sources==

* {{Rudin Walter Functional Analysis|edition=2}}
* {{Narici Beckenstein Topological Vector Spaces|edition=2}}
* {{Schaefer Wolff Topological Vector Spaces|edition=2}}
* {{Trèves François Topological vector spaces, distributions and kernels}}

{{Duality and spaces of linear maps}}
{{Topological vector spaces}}
{{Functional analysis}}

{{DEFAULTSORT:Mackey-Arens theorem}}

[[Category:Theorems in functional analysis]]
[[Category:Lemmas]]
[[Category:Topological vector spaces]]
[[Category:Linear functionals]]