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

Examples of locally convex spaces that are Mackey spaces include:

• All barrelled spaces ^[1] and more generally all infrabarreled spaces ^[2]
  • Hence in particular all bornological spaces ^[1] and reflexive spaces
• All metrizable spaces.^[1]
  • In particular, all Fréchet spaces, including all Banach spaces and specifically Hilbert spaces, are Mackey spaces.
• The product, locally convex direct sum, and the inductive limit of a family of Mackey spaces is a Mackey space.^[3]

Properties

• A locally convex space ${\displaystyle X}$ with continuous dual ${\displaystyle X'}$ is a Mackey space if and only if each convex and ${\displaystyle \sigma (X',X)}$-relatively compact subset of ${\displaystyle X'}$ 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 ${\displaystyle \sigma }$-quasi-barrelled.

See also

• Mackey topology
• Topologies on spaces of linear maps

References

• 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.

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