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 ${\displaystyle X}$ is a locally convex topology ${\displaystyle \tau }$ so that

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

Here ${\displaystyle (X,\tau )'}$ denotes the continuous dual of ${\displaystyle (X,\tau )}$ 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 ${\displaystyle \tau }$ on ${\displaystyle X}$ is not a dual topology, then either ${\displaystyle \Psi }$ is not surjective or it is ill-defined since the linear functional ${\displaystyle x\mapsto \langle x,y\rangle }$ is not continuous on ${\displaystyle X}$ for some ${\displaystyle y}$.)

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 ${\displaystyle X'}$, and the finest topology is the Mackey topology, the topology of uniform convergence on all absolutely convex weakly compact subsets of ${\displaystyle X'}$.

Mackey–Arens theorem

Given a dual pair ${\displaystyle (X,X')}$ with ${\displaystyle X}$ a locally convex space and ${\displaystyle X'}$ its continuous dual, then ${\displaystyle \tau }$ is a dual topology on ${\displaystyle X}$ if and only if it is a topology of uniform convergence on a family of absolutely convex and weakly compact subsets of ${\displaystyle X'}$

See also

• Polar topology