Mackey topology
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) |
Comments
References
[a1] | H. Jarchow, "Locally convex spaces" , Teubner (1981) (Translated from German) |
Mackey topology. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Mackey_topology&oldid=13416