Mackey–Arens theorem
The Mackey–Arens theorem is an important theorem in functional analysis that characterizes those locally convex vector topologies that have some given space of linear functionals as their continuous dual space.If 𝜏 is any other locally convex Hausdorff topological vector space topology on X, then we say that 𝜏 is compatible with duality between X and Y if when X is equipped with 𝜏, then it has Y as its continuous dual space.If we give X the weak topology 𝜎(X, Y) then X𝜎(X, Y) is a Hausdorff locally convex topological vector space (TVS) and 𝜎(X, Y) is compatible with duality between X and Y (i.e.We can now ask the question: what are all of the locally convex Hausdorff TVS topologies that we can place on X that are compatible with duality between X and Y?Mackey–Arens theorem[2] — Let X be a vector space and let 𝒯 be a locally convex Hausdorff topological vector space topology on X.