Mackey–Arens theorem

{{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 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_{𝜎(X, Y)}}} 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. $X\_\{\backslash sigma(X,\; Y)\}^\{\backslash prime\}\; =\; \backslash left(\; X\_\{\backslash sigma(X,\; Y)\}\; \backslash right)^\{\backslash prime\}\; =\; Y$).

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.

{{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 $X\_\{\backslash mathcal\{T\}\}$ denote X with the topology 𝒯. Then the following are equivalent:

{{ordered list|

| 𝒯 is identical to a $\backslash mathcal\{G\}^\{\backslash prime\}$-topology on X, where $\backslash mathcal\{G\}^\{\backslash prime\}$ is a covering of <{{math|X{{big|{{'}}}}}} consisting of convex, balanced, {{math|σ(X{{big|{{'}}}}, X)}}-compact sets with the properties that

{{ordered list|type=lower-roman|style=margin-left: 1em

| If $G\_1^\{\backslash prime\},\; G\_2^\{\backslash prime\}\; \backslash in\; \backslash mathcal\{G\}^\{\backslash prime\}$ then there exists a $G^\{\backslash prime\}\; \backslash in\; \backslash mathcal\{G\}^\{\backslash prime\}$ such that $G\_1^\{\backslash prime\}\; \backslash cup\; G\_2^\{\backslash prime\}\; \backslash subseteq\; G^\{\backslash prime\}$, and

| If $G\_1^\{\backslash prime\}\; \backslash in\; \backslash mathcal\{G\}^\{\backslash prime\}$ and $\backslash lambda$ is a scalar then there exists a $G^\{\backslash prime\}\; \backslash in\; \backslash mathcal\{G\}^\{\backslash prime\}$ such that $\backslash lambda\; G\_1^\{\backslash prime\}\; \backslash subseteq\; G^\{\backslash prime\}$.

}}

| The continuous dual of $X\_\{\backslash mathcal\{T\}\}$ is identical to {{math|X{{big|{{'}}}}}}.

}}

And furthermore,

{{ordered list

| the topology 𝒯 is identical to the {{math|ε(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|τ(X, X{{big|{{'}}}})}} is the finest locally convex Hausdorff TVS topology on X that is compatible with duality between X and $X\_\{\backslash mathcal\{T\}\}^\{\backslash prime\}$, and

| the weak topology {{math|σ(X, X{{big|{{'}}}})}} is the coarsest locally convex Hausdorff TVS topology on X that is compatible with duality between X and $X\_\{\backslash mathcal\{T\}\}^\{\backslash prime\}$.

}}

}}

• Dual system
• Mackey topology
• Polar topology

{{reflist}}

• {{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