if is equicontinuous. Importantly, a set of continuous linear functionals on a TVS is equicontinuous if and only if it is contained in the polar of some neighborhood of the origin in (i.e. ). Since a TVS's topology is completely determined by the open neighborhoods of the origin, this means that via operation of taking the polar of a set, the collection of equicontinuous subsets of "encode" all information about 's topology (i.e. distinct TVS topologies on produce distinct collections of equicontinuous subsets, and given any such collection one may recover the TVS original topology by taking the polars of sets in the collection). Thus uniform convergence on the collection of equicontinuous subsets is essentially "convergence on the topology of ". Mackey topology Suppose that is a locally convex Hausdorff space. If is metrizable or barrelled then 's original topology is identical to the Mackey topology Topologies compatible with pairings 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 is compatible with duality between and if when is equipped with then it has as its continuous dual space. If is given the weak topology then is a Hausdorff locally convex topological vector space (TVS) and is compatible with duality between and (i.e. ). The question arises: what are all of the locally convex Hausdorff TVS topologies that can be placed on that are compatible with duality between and ? The answer to this question is called the Mackey–Arens theorem. See also References Topology of function spaces