Mackey space

Examples

• All bornological spaces.
• All Hausdorff locally convex quasi-barrelled.
• All Hausdorff locally convex metrizable spaces.
• * In particular, all Banach spaces and Hilbert spaces are Mackey 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 with continuous dual is a Mackey space if and only if each convex and -relatively compact subset of 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 -quasi-barrelled.