Mackey space
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Wikipedia : http://en.wikipedia.org/wiki/Mackey_spaceIn 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]}
- 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
- Robertson, A.P.; W.J. Robertson (1964). Topological vector spaces. Cambridge Tracts in Mathematics. 53. Cambridge University Press. p. 81.
- H.H. Schaefer (1970). Topological Vector Spaces. GTM. 3. Springer-Verlag. pp. 132–133. ISBN 0-387-05380-8.
- S.M. Khaleelulla (1982). Counterexamples in Topological Vector Spaces. GTM. 936. Springer-Verlag. pp. 31, 41, 55–58. ISBN 978-3-540-11565-6.
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