View source for Mackey topology

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''$\tau(F,G)$ 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 [[#References|[1]]]. The Mackey topology is the strongest of the separated locally convex topologies (cf. [[Locally convex topology|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|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====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  G.W. Mackey,   "On convex topological linear spaces"  ''Trans. Amer. Math. Soc.'' , '''60'''  (1946)  pp. 519–537</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  N. Bourbaki,   "Elements of mathematics. Topological vector spaces" , Addison-Wesley  (1977)  (Translated from French)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  H.H. Schaefer,   "Topological vector spaces" , Springer  (1971)</TD></TR></table>



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====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  H. Jarchow,   "Locally convex spaces" , Teubner  (1981)  (Translated from German)</TD></TR></table>