- 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 X,X') with X a topological vector space and X' itscontinuous dual the Mackey topology au(X,X') is apolar topology defined on X by using the set of allabsolutely convex and weakly-compact sets in X'.Examples
* Every
metrisable locally convex space X, au) with continuous dual X' carries the Mackey topology, that is au = au(X, X'), or to put it more succinctly everyMackey space carries the Mackey topology
* EveryFréchet space X, au) carries the Mackey topology and the topology coincides with thestrong topology , that is au = au(X, X') = eta(X, X')See also
*
polar topology
*weak topology
*strong topology References
*springer|id=M/m062080|title=Mackey topology|author=A.I. Shtern
*
*
*
*
Wikimedia Foundation. 2010.