Infunctional analysis and related areas ofmathematics adual topology is alocally convex topology on avector space that is induced by thecontinuous dual of the vector space, by means of thebilinear form (also calledpairing) associated with thedual pair.

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 thedual pair and not on the chosen dual topology and thus it is often possible to substitute a complicated dual topology by a simpler one.

Given adual pair${\displaystyle (X,Y,⟨,⟩)}$, adual topology on${\displaystyle X}$ is alocally convex topology${\displaystyle \tau }$ so that

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

Here${\displaystyle (X,\tau {)}^{\prime }}$ denotes thecontinuous dual of${\displaystyle (X,\tau )}$ and${\displaystyle (X,\tau {)}^{\prime }\simeq Y}$ means that there is alinear isomorphism

${\displaystyle \mathrm{\Psi }:Y\to (X,\tau {)}^{\prime },y\mapsto (x\mapsto ⟨x,y⟩).}$

(If a locally convex topology${\displaystyle \tau }$ on${\displaystyle X}$ is not a dual topology, then either${\displaystyle \mathrm{\Psi }}$ is not surjective or it is ill-defined since the linear functional${\displaystyle x\mapsto ⟨x,y⟩}$ is not continuous on${\displaystyle X}$ for some${\displaystyle y}$.)

• Theorem (byMackey): Given a dual pair, thebounded sets under any dual topology are identical.
• Under any dual topology the same sets arebarrelled.

TheMackey–Arens theorem, named afterGeorge Mackey andRichard Arens, characterizes all possible dual topologies on alocally convex space.

The theorem shows that thecoarsest dual topology is theweak topology, the topology of uniform convergence on all finite subsets of${\displaystyle X^{\prime }}$, and thefinest topology is theMackey topology, the topology of uniform convergence on allabsolutely convex weakly compact subsets of${\displaystyle X^{\prime }}$.

Given adual pair${\displaystyle (X,X^{\prime })}$ with${\displaystyle X}$ a locally convex space and${\displaystyle X^{\prime }}$ itscontinuous dual, then${\displaystyle \tau }$ is a dual topology on${\displaystyle X}$if and only if it is atopology of uniform convergence on a family ofabsolutely convex andweakly compact subsets of${\displaystyle X^{\prime }}$

• Polar topology

• Bogachev, Vladimir I; Smolyanov, Oleg G. (2017).Topological Vector Spaces and Their Applications.Springer Monographs in Mathematics. Cham, Switzerland: Springer International Publishing.ISBN 978-3-319-57117-1.OCLC 987790956.

• Topology of function spaces
• Linear functionals