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.^[1]
• All Hausdorff locally convex barreled spaces.^[1]
• The product, locally convex direct sum, and the inductive limit of a family of Mackey spaces is a Mackey space.^[2]

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. 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.