Mackey space

In

• All
  infrabarreled spaces ^[2]
  • Hence in particular all bornological spaces ^[1] and reflexive spaces
• All metrizable spaces.^[1]
  • In particular, all
    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.^[3]

Properties

• A locally convex space ${\displaystyle X}$ with continuous dual ${\displaystyle X'}$ is a Mackey space if and only if each convex and ${\displaystyle \sigma (X',X)}$-relatively compact subset of ${\displaystyle X'}$ 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
  quasi-barrelled
  , nor ${\displaystyle \sigma }$-quasi-barrelled.

See also

• Mackey topology
• Topologies on spaces of linear maps

References

• OCLC 17499190
  .
• OCLC 886098
  .
• Khaleelulla, S. M. (1982). Counterexamples in Topological Vector Spaces.
  OCLC 8588370
  .
• Robertson, A.P.; W.J. Robertson (1964). Topological vector spaces. Cambridge Tracts in Mathematics. Vol. 53. Cambridge University Press. p. 81.
• OCLC 840278135
  .

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