In functional analysis and related areas of mathematics, the Mackey topology, named after George Mackey, is the finest topology for a topological vector space which still preserves the continuous dual.
In other words the Mackey topology does not make linear functions continuous which were discontinuous in the default topology.
A topological vector space (TVS) is called a Mackey space if its topology is the same as the Mackey topology.
The Mackey topology is the opposite of the weak topology, which is the coarsest topology on a topological vector space which preserves the continuity of all linear functions in the continuous dual.
The Mackey–Arens theorem states that all possible dual topologies are finer than the weak topology and coarser than the Mackey topology.
is the polar topology defined on
is endowed with the Mackey topology then it will be denoted by
if no ambiguity can arise.
A linear map
is said to be Mackey continuous (with respect to pairings
The definition of the Mackey topology for a topological vector space (TVS) is a specialization of the above definition of the Mackey topology of a pairing.
is a TVS with continuous dual space
then the evaluation map
is called the canonical pairing.
The Mackey topology on a TVS
induced by the canonical pairing
obtained by using the set of all weak*-compact disks in
is endowed with the Mackey topology then it will be denoted by
if no ambiguity can arise.
A linear map
between TVSs is Mackey continuous if
Every metrizable locally convex
with continuous dual
carries the Mackey topology, that is
ν = τ
or to put it more succinctly every metrizable locally convex space is a Mackey space.
Every Hausdorff barreled locally convex space is Mackey.
Every Fréchet space
ν = τ
The Mackey topology has an application in economies with infinitely many commodities.