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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.
- Definition for a pairing
- Definition: Given a pairing (X, Y, b), the Mackey topology on X induced by (X, Y, b), denoted by τ(X, Y, b), is the polar topology defined on X by using the set of all 𝜎(X, Y, b)-compact disks in Y.
When X is endowed with the Mackey topology then it will be denoted by Xτ(X, Y, b) or simply Xτ(X, Y) or Xτ if no ambiguity can arise.
- Definition: We say that the linear map F : X → W is Mackey continuous (with respect to pairings (X, Y, b) and (W, Z, c)) if F : (X, τ(X, Y, b)) → (W, τ(W, Z, c)) is continuous.
- Definition for a topological vector space
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 X is a TVS with continuous dual space X', then the evaluation map (x, x') ↦ x'(x) on X × X' is called the canonical pairing.
- Definition: The Mackey topology on a TVS X, denoted by τ(X, X'), is the Mackey topology on X induced by the canonical pairing ⟨X, X'⟩.
That is, the Mackey topology is the polar topology on X obtained by using the set of all weak*-compact disks in X'. When X is endowed with the Mackey topology then it will be denoted by Xτ(X, X') or simply Xτ if no ambiguity can arise.
- Definition: We say that the linear map F : X → Y between TVSs is Mackey continuous if F : (X, τ(X, X')) → (Y, τ(Y, Y')) is continuous.
- Every metrisable locally convex space (X, 𝒯) with continuous dual X' carries the Mackey topology, that is 𝒯 = τ(X, X'), or to put it more succinctly every metrisable locally convex space is a Mackey space.
- Every Hausdorff barreled locally convex space is Mackey.
- Every Fréchet space (X, 𝒯) carries the Mackey topology and the topology coincides with the strong topology, that is 𝒯 = τ(X, X') = β(X, X').
The Mackey topology has an application in economies with infinitely many commodities.
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- A.I. Shtern (2001) , "Mackey topology", Encyclopedia of Mathematics, EMS Press
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