Mackey space

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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:

Properties

  • A locally convex space [math]\displaystyle{ X }[/math] with continuous dual [math]\displaystyle{ X' }[/math] is a Mackey space if and only if each convex and [math]\displaystyle{ \sigma(X', X) }[/math]-relatively compact subset of [math]\displaystyle{ X' }[/math] 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 [math]\displaystyle{ \sigma }[/math]-quasi-barrelled.

See also

References

  1. 1.0 1.1 1.2 Bourbaki 1987, p. IV.4.
  2. Grothendieck 1973, p. 107.
  3. Schaefer (1999) p. 138
  4. Schaefer (1999) p. 133