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′), the finest topology which still preserves the continuous dual.

Examples

Examples of Mackey spaces include:

• All bornological spaces.
• All Hausdorff locally convex quasi-barrelled (and hence all Hausdorff locally convex barrelled spaces and all Hausdorff locally convex reflexive spaces).
• All Hausdorff locally convex metrizable spaces.
• All Hausdorff locally convex barreled 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.