replaced with without changing the resulting topology. The following theorem gives ways in which can be modified without changing the resulting -topology on It is because of this theorem that many authors often require that also satisfy the following additional conditions: The union of any two sets is contained in some set ; All scalar multiples of every belongs to Some authors further assume that every belongs to some set because this assumption suffices to ensure that the -topology is Hausdorff. Convergence of nets and filters If is a net in then in the -topology on if and only if for every or in words, if and only if for every the net of linear functionals on converges uniformly to on ; here, for each the linear functional is defined by If then in the -topology on if and only if for all A filter on converges to an element in the -topology on if converges uniformly to on each Properties The results in the article Topologies on spaces of linear maps can be applied to polar topologies. Throughout, is a pairing of vector spaces over the field and is a non-empty collection of -bounded subsets of Hausdorffness We say that covers if every point in belong to some set in We say that is total in if the linear span of is dense in Proof of (2): If then we're done, so assume otherwise. Since the -topology on is a TVS topology, it suffices to show that the set is closed in Let be non-zero, let be defined by for all and let Since distinguishes points of there exists some (non-zero) such that where (since is surjective) it can be assumed without loss of generality that The set is a -open subset of that is not empty (since it contains ). Since is a -dense subset of there exists some and some such that Since so that where is a subbasic closed neighborhood of the origin in the -topology on ■ Examples of polar topologies induced by a pairing Throughout, will be a pairing of vector spaces over the field and will be a non-empty collection of -bounded subsets of The following table will omit mention of The topologies are listed in an order that roughly corresponds with coarser topologies first and the finer topologies last; note that some of these topologies may be out of order e.g. and