# Mackey topology

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on a space , being in duality with a space (over the same field)

The topology of uniform convergence on the convex balanced subsets of that are compact in the weak topology (defined by the duality between and ). It was introduced by G.W. Mackey [1]. The Mackey topology is the strongest of the separated locally convex topologies (cf. Locally convex topology) which are compatible with the duality between and (that is, separated locally convex topologies on such that the set of all continuous linear functionals on endowed with the topology coincides with ). The families of sets in which are bounded relative to the Mackey topology and bounded relative to the weak topology coincide. A convex subsets of is equicontinuous when is endowed with the Mackey topology if and only if it is relatively compact in the weak topology. If a separated locally convex space is barrelled or bornological (in particular, metrizable) and is its dual, then the Mackey topology on (being dual with ) coincides with the initial topology on . For pairs of spaces () in duality the Mackey topology is not necessarily barrelled or metrizable. A weakly-continuous linear mapping of a separated locally convex space into a separated locally convex space is continuous relative to the Mackey topologies and . A locally convex space is called a Mackey space if the topology on is . Completions, quotient spaces and metrizable subspaces, products, locally convex direct sums, and inductive limits of families of Mackey spaces are Mackey spaces. If is a Mackey space and is a weakly-continuous mapping of into a locally convex space , then is a continuous linear mapping of into . If is a quasi-complete Mackey space and the space dual to equipped with the strong -topology is semi-reflexive, then is reflexive.

#### References

 [1] G.W. Mackey, "On convex topological linear spaces" Trans. Amer. Math. Soc. , 60 (1946) pp. 519–537 [2] N. Bourbaki, "Elements of mathematics. Topological vector spaces" , Addison-Wesley (1977) (Translated from French) [3] H.H. Schaefer, "Topological vector spaces" , Springer (1971)