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Concept
Mackey topology
Summary
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.
Polar topology and Dual system
Given a pairing the Mackey topology on induced by denoted by is the polar topology defined on by using the set of all -compact disks in
When is endowed with the Mackey topology then it will be denoted by or simply or if no ambiguity can arise.
A linear map is said to be Mackey continuous (with respect to pairings and ) if is continuous.
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.
If is a TVS with continuous dual space then the evaluation map on is called the canonical pairing.
The Mackey topology on a TVS denoted by is the Mackey topology on induced by the canonical pairing
That is, the Mackey topology is the polar topology on obtained by using the set of all weak*-compact disks in
When is endowed with the Mackey topology then it will be denoted by or simply if no ambiguity can arise.
A linear map between TVSs is Mackey continuous if is continuous.
Every metrizable locally convex with continuous dual carries the Mackey topology, that is or to put it more succinctly every metrizable locally convex space is a Mackey space.
Every Hausdorff barreled locally convex space is Mackey.
Official source
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