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.

Definition[edit]

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[edit]

  • 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[edit]

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[edit]

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

References[edit]