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

(X, \tau)' \simeq Y.

Here (X, \tau)' denotes the continuous dual of (X,\tau) and (X, \tau)' \simeq Y means that there is a linear isomorphism

\Psi : Y \to (X, \tau)',\quad y \mapsto (x \mapsto \langle x, y\rangle).

(If a locally convex topology \tau on X is not a dual topology, then either \Psi is not surjective or it is ill-defined since the linear functional x \mapsto \langle x, y\rangle is not continuous on X for some y.)

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 X', and the finest topology is the Mackey topology, the topology of uniform convergence on all weakly compact subsets of X'.

Mackey–Arens theorem[edit]

Given a dual pair (X, X') with X a locally convex space and X' its continuous dual, then \tau 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'

References[edit]