Dual topology
In functional analysis and related areas of mathematics a dual topology is a locally convex topology on a dual pair, two vector spaces with a bilinear form defined on them, so that one vector space becomes the continuous dual of the other space. The different dual topologies for a given dual pair are characterized by the Mackey–Arens theorem. All locally convex topologies with their continuous dual are trivially a dual pair and the locally convex topology is a dual topology. Several topological properties depend only on the dual pair and not on the chosen dual topology and  thus it is often possible to substitute a complicated dual topology by a simpler one. Definition. Given a dual pair formula_1, a dual topology on formula_2 is a locally convex topology formula_3 so that Here formula_5 denotes the continuous dual of formula_6 and formula_7 means that there is a linear isomorphism Characterization of dual topologies. The Mackey–Arens theorem, named after George Mackey and Richard Arens, characterizes all possible dual topologies on a locally convex space. The theorem shows that the coarsest dual topology is the weak topology, the topology of uniform convergence on all finite subsets of formula_15, and  the finest topology is the Mackey topology, the topology of uniform convergence on all absolutely convex weakly compact subsets of formula_15. Mackey–Arens theorem. Given a dual pair formula_17 with formula_2 a locally convex space and formula_15 its continuous dual, then formula_3 is a dual topology on formula_2 if and only if it is a topology of uniform convergence on a family of absolutely convex and weakly compact subsets of formula_15