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 $(X, Y, \langle , \rangle)$, a dual topology on X is a locally convex topology τ so that

$(X, \tau)' \simeq Y.$

That is the continuous dual of (X,τ) is equal to Y up to linear isomorphism.

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 spaces.

The theorem shows that the coarsest dual topology is the weak topology, the topology of uniform convergence on all finite subsets of X', and the finest topology is the Mackey topology, the topology of uniform convergence on all weakly compact subsets of X'.

Mackey–Arens theorem

Given a dual pair (X,X') with X a locally convex space and X' its continuous dual then τ is a dual topology on X if and only if it is a topology of uniform convergence on a family of absolutely convex and weakly compact subsets of X'