# 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 ${\displaystyle X}$ with continuous dual ${\displaystyle X'}$ is a Mackey space if and only if each convex and ${\displaystyle \sigma (X',X)}$-relatively compact subset of ${\displaystyle X'}$ 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 ${\displaystyle \sigma }$-quasi-barrelled.

## References

1. ^ a b Schaefer (1999) p. 132
2. ^ Schaefer (1999) p. 138
3. ^ Schaefer (1999) p. 133
• Robertson, A.P.; W.J. Robertson (1964). Topological vector spaces. Cambridge Tracts in Mathematics. 53. Cambridge University Press. p. 81.
• H.H. Schaefer (1970). Topological Vector Spaces. GTM. 3. Springer-Verlag. pp. 132–133. ISBN 0-387-05380-8.
• S.M. Khaleelulla (1982). Counterexamples in Topological Vector Spaces. GTM. 936. Springer-Verlag. pp. 31, 41, 55–58. ISBN 978-3-540-11565-6.