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.
References
- Schaefer (1999) p. 132
- Schaefer (1999) p. 138
- 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.
- 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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