Mackey space
# 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. 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.
