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 ${\displaystyle X}$ 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.^[3]
• 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[]

• Robertson, A.P.; W.J. Robertson (1964). Topological vector spaces. Cambridge Tracts in Mathematics. 53. Cambridge University Press. p. 81.
• Schaefer, Helmut H.; (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.
• Khaleelulla, S. M. (1982). Counterexamples in Topological Vector Spaces. Lecture Notes in Mathematics. 936. Berlin, Heidelberg, New York: Springer-Verlag. ISBN 978-3-540-11565-6. OCLC 8588370.

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