- Mackey topology
In
functional analysis and related areas ofmathematics , the Mackey topology, named afterGeorge Mackey , is the finest topology for atopological vector space which still preserves thecontinuous dual . In other words the Mackey topology does not make linear functions continuous which were discontinuous in the default topology.The Mackey topology is the opposite of the
weak topology , which is the coarsest topology on atopological 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.Definition
Given a
dual pair with a topological vector space and itscontinuous dual the Mackey topology is apolar topology defined on by using the set of allabsolutely convex and weakly-compact sets in .Examples
* Every
metrisable locally convex space with continuous dual carries the Mackey topology, that is , or to put it more succinctly everyMackey space carries the Mackey topology
* EveryFréchet space carries the Mackey topology and the topology coincides with thestrong topology , that isSee also
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polar topology
*weak topology
*strong topology References
*springer|id=M/m062080|title=Mackey topology|author=A.I. Shtern
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