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

${\displaystyle (X,\tau )'\simeq Y.}$

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

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

(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 ${\displaystyle x\mapsto \langle x,y\rangle }$ 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 ${\displaystyle (X,X')}$ 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