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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.

Contents

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 , ⟨ , ⟩ ) , a dual topology on X is a locally convex topology τ so that

( X , τ ) ′ ≃ Y .

Here ( X , τ ) ′ denotes the continuous dual of ( X , τ ) and ( X , τ ) ′ ≃ Y means that there is a linear isomorphism

Ψ : Y → ( X , τ ) ′ , y ↦ ( x ↦ ⟨ x , y ⟩ ) .

(If a locally convex topology τ on X is not a dual topology, then either Ψ is not surjective or it is ill-defined since the linear functional x ↦ ⟨ x , y ⟩ is not continuous on X for some y .)

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 X ′ , and the finest topology is the Mackey topology, the topology of uniform convergence on all absolutely convex 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 ′

    References

    Dual topology Wikipedia
    (Text) CC BY-SA