Mackey space

Template:Short description 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 Template:Sfn and more generally all infrabarreled spaces Template:Sfn
  • Hence in particular all bornological spaces Template:Sfn and reflexive spaces
• All metrizable spaces.Template:Sfn
  • 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.

Properties

• A locally convex space $X$ with continuous dual $X^{\prime }$ is a Mackey space if and only if each convex and $\sigma (X^{\prime },X)$-relatively compact subset of $X^{\prime }$ is equicontinuous.
• The completion of a Mackey space is again a Mackey space.
• A separated quotient of a Mackey space is again a Mackey space.
• A Mackey space need not be separable, complete, quasi-barrelled, nor $\sigma$-quasi-barrelled.

See also

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References

• Template:Bourbaki Topological Vector Spaces Part 1 Chapters 1–5
• Template:Grothendieck Topological Vector Spaces
• Template:Khaleelulla Counterexamples in Topological Vector Spaces
• Template:Cite book
• Template:Schaefer Wolff Topological Vector Spaces