Mackey space
{{short description|Mathematics concept}}
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 {{sfn|Bourbaki|1987|p=IV.4}} and more generally all infrabarreled spaces {{sfn|Grothendieck|1973|p=107}}
- Hence in particular all bornological spaces {{sfn|Bourbaki|1987|p=IV.4}} and reflexive spaces
- All metrizable spaces.{{sfn|Bourbaki|1987|p=IV.4}}
- 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.Schaefer (1999) p. 138
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.Schaefer (1999) p. 133
- 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
- {{annotated link|Mackey topology}}
- {{annotated link|Topologies on spaces of linear maps}}
References
{{sfn whitelist|CITEREFBourbaki1987}}
{{reflist}}
- {{Bourbaki Topological Vector Spaces Part 1 Chapters 1–5}}
- {{Grothendieck Topological Vector Spaces}}
- {{Khaleelulla Counterexamples in Topological Vector Spaces}}
- {{cite book |last=Robertson |first=A.P. |author2=W.J. Robertson |title= Topological vector spaces |series=Cambridge Tracts in Mathematics |volume=53 |year=1964 |publisher= Cambridge University Press | page=81 }}
- {{Schaefer Wolff Topological Vector Spaces|edition=2|pp=132–133}}
{{Duality and spaces of linear maps}}
{{Topological vector spaces}}
{{Boundedness and bornology}}
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