Dual topology

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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.

1 Definition
2 Properties
3 Characterization of dual topologies
- 3.1 Mackey-Arens theorem

[edit] 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.

[edit] 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.

[edit] 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 $X'$ .