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.
  • Schaefer, Helmut H.; Wolff, Manfred P. (1999). Topological Vector Spaces. GTM. 8 (Second ed.). New York, NY: Springer New York Imprint Springer. pp. 132–133. ISBN 978-1-4612-7155-0. OCLC 840278135.CS1 maint: ref=harv (link)
  • Khaleelulla, S. M. (July 1, 1982). Written at Berlin Heidelberg. Counterexamples in Topological Vector Spaces. Lecture Notes in Mathematics. 936. Berlin New York: Springer-Verlag. ISBN 978-3-540-11565-6. OCLC 8588370.CS1 maint: ref=harv (link) CS1 maint: date and year (link)
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