Mackey–Arens theorem

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

{{Math theorem|name=Mackey–Arens theorem|math_statement= Let X be a vector space and let 𝒯 be a locally convex Hausdorff topological vector space topology on X. Let denote the continuous dual space of X and let X_{\mathcal{T}} denote X with the topology 𝒯. Then the following are equivalent: {{ordered list| | 𝒯 is identical to a \mathcal{G}^{\prime}-topology on X, where \mathcal{G}^{\prime} is a covering of G_1^{\prime}, G_2^{\prime} \in \mathcal{G}^{\prime} then there exists a G^{\prime} \in \mathcal{G}^{\prime} such that G_1^{\prime} \cup G_2^{\prime} \subseteq G^{\prime}, and | If G_1^{\prime} \in \mathcal{G}^{\prime} and \lambda is a scalar then there exists a G^{\prime} \in \mathcal{G}^{\prime} such that \lambda G_1^{\prime} \subseteq G^{\prime}. }} | The continuous dual of X_{\mathcal{T}} is identical to . }} And furthermore, {{ordered list | the topology 𝒯 is identical to the topology, that is, to the topology of uniform on convergence on the equicontinuous subsets of . | the Mackey topology is the finest locally convex Hausdorff TVS topology on X that is compatible with duality between X and X_{\mathcal{T}}^{\prime}, and | the weak topology is the coarsest locally convex Hausdorff TVS topology on X that is compatible with duality between X and X_{\mathcal{T}}^{\prime}. }} }} ==See also==