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. According to Narici (2011), this profound result is central to duality theory; a theory that is "the central part of the modern theory of topological vector spaces." Prerequisites 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. ). 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 See also Dual system Mackey topology Polar topology References Sources Functional analysis Lemmas Topological vector spaces