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.
  • 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 $X$ 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.

See also

• Mackey topology
• Topologies on spaces of linear maps

References

• Robertson, A.P.; W.J. Robertson (1964). Topological vector spaces. Cambridge Tracts in Mathematics. Vol. 53. Cambridge University Press. p. 81.
• Schaefer, Helmut H.; Wolff, Manfred P. (1999). Topological Vector Spaces. GTM. Vol. 8 (Second ed.). New York, NY: Springer New York Imprint Springer. pp. 132–133. ISBN 978-1-4612-7155-0. OCLC 840278135.
• Khaleelulla, S. M. (1982). Counterexamples in Topological Vector Spaces. Lecture Notes in Mathematics. Vol. 936. Berlin, Heidelberg, New York: Springer-Verlag. ISBN 978-3-540-11565-6. OCLC 8588370.