# 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:

## 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.[3]
• 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.

## References

1. ^ a b Schaefer (1999) p. 132
2. ^ Schaefer (1999) p. 138
3. ^ Schaefer (1999) p. 133