Mackey space

In mathematics, particularly in functional analysis, a Mackey space is a locally convex topological vector space X such that the topology of X coincides with the Mackey topology τ(X,X&prime;), the finest topology which still preserves the continuous dual. They are named after George Mackey.

Examples
Examples of locally convex spaces that are Mackey spaces include:
 * All barrelled spaces and more generally all infrabarreled spaces
 * Hence in particular all bornological spaces and reflexive spaces
 * All metrizable spaces.
 * In particular, all Fréchet spaces, including all Banach spaces and specifically Hilbert spaces, are Mackey spaces.
 * The product, locally convex direct sum, and the inductive limit of a family of Mackey spaces is a Mackey space.

Properties

 * A locally convex space $$X$$ with continuous dual $$X'$$ is a Mackey space if and only if each convex and $$\sigma(X', X)$$-relatively compact subset of $$X'$$ is equicontinuous.
 * The completion of a Mackey space is again a Mackey space.
 * A separated quotient of a Mackey space is again a Mackey space.
 * A Mackey space need not be separable, complete, quasi-barrelled, nor $$\sigma$$-quasi-barrelled.