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 , a dual topology on is a locally convex topology so that
Here denotes the continuous dual of and means that there is a linear isomorphism
(If a locally convex topology on is not a dual topology, then either is not surjective or it is ill-defined since the linear functional is not continuous on for some .)
Properties
- Theorem (by Mackey): Given a dual pair, the bounded sets under any dual topology are identical.
- Under any dual topology the same sets are barrelled.
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 , and the finest topology is the Mackey topology, the topology of uniform convergence on all absolutely convex weakly compact subsets of .
Mackey–Arens theorem
Given a dual pair with a locally convex space and its continuous dual, then is a dual topology on if and only if it is a topology of uniform convergence on a family of absolutely convex and weakly compact subsets of