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. They are named after George Mackey.



Examples of locally convex spaces that are Mackey spaces include:


  • 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.[4]
  • 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



  1. ^ a b c Bourbaki 1987, p. IV.4.
  2. ^ Grothendieck 1973, p. 107.
  3. ^ Schaefer (1999) p. 138
  4. ^ Schaefer (1999) p. 133
  • Bourbaki, Nicolas (1987) [1981]. Topological Vector Spaces: Chapters 1–5. Éléments de mathématique. Translated by Eggleston, H.G.; Madan, S. Berlin New York: Springer-Verlag. ISBN 3-540-13627-4. OCLC 17499190.
  • Grothendieck, Alexander (1973). Topological Vector Spaces. Translated by Chaljub, Orlando. New York: Gordon and Breach Science Publishers. ISBN 978-0-677-30020-7. OCLC 886098.
  • 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.
  • 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.