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Mackey space
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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
[edit]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
[edit]- 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
[edit]References
[edit]- ^ a b c Bourbaki 1987, p. IV.4.
- ^ Grothendieck 1973, p. 107.
- ^ Schaefer (1999) p. 138
- ^ 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.
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