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]
    • In particular, all Banach spaces and Hilbert spaces are Mackey spaces.
  • 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 with continuous dual is a Mackey space if and only if each convex and -relatively compact subset of 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 -quasi-barrelled.

See also

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.
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