# Mackey topology

$\tau(F,G)$ on a space $F$, being in duality with a space $G$ (over the same field)

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