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.
Definition
Definition for a pairing Given a pairing ( X , Y , b ) , {\displaystyle (X,Y,b),} the Mackey topology on X {\displaystyle X} induced by ( X , Y , b ) , {\displaystyle (X,Y,b),} denoted by τ ( X , Y , b ) , {\displaystyle \tau (X,Y,b),} is the polar topology defined on X {\displaystyle X} by using the set of all σ ( Y , X , b ) {\displaystyle \sigma (Y,X,b)} -compact disks in Y . {\displaystyle Y.} When X {\displaystyle X} is endowed with the Mackey topology then it will be denoted by X τ ( X , Y , b ) {\displaystyle X_{\tau (X,Y,b)}} or simply X τ ( X , Y ) {\displaystyle X_{\tau (X,Y)}} or X τ {\displaystyle X_{\tau }} if no ambiguity can arise. A linear map F : X → W {\displaystyle F:X\to W} is said to be Mackey continuous (with respect to pairings ( X , Y , b ) {\displaystyle (X,Y,b)} and ( W , Z , c ) {\displaystyle (W,Z,c)} ) if F : ( X , τ ( X , Y , b ) ) → ( W , τ ( W , Z , c ) ) {\displaystyle F:(X,\tau (X,Y,b))\to (W,\tau (W,Z,c))} is continuous.
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