Supriya Ghosh (Editor)

Mackey space

Updated on
Edit
Like
Comment
Share on FacebookTweet on TwitterShare on LinkedInShare on Reddit

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.

Contents

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.
  • All Hausdorff locally convex barreled 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 ′ is a Mackey space if and only if each convex and σ ( X ′ , X ) -relatively compact subset of X ′ 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 σ -quasi-barrelled.

    • References

    Mackey space Wikipedia
    (Text) CC BY-SA


    Similar Topics