Bornological space

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.

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.

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.

References

• 20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.
  
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• 20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.
  
  My blog: http://www.primaboinca.com/view_profile.php?userid=5889534
• 20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.
  
  My blog: http://www.primaboinca.com/view_profile.php?userid=5889534