DF-space

1. X is a countably quasi-barrelled space, and
2. X possesses a fundamental sequence of bounded.

Properties

• Let X be a DF-space and let V be a convex balanced subset of X. Then V is a neighborhood of the origin if and only if for every convex, balanced, bounded subset, is a 0-neighborhood in B.
• * Thus, a linear map from a DF-space into a locally convex space is continuous if its restriction to each bounded subset of the domain is continuous.
• The strong dual of a DF-space is a Fréchet space.

  Sufficient conditions

• The strong dual of a metrizable locally convex space is a DF-space. Hence:
• * Every normed space is a DF-space.
• * Every Banach space is a DF-space.
• * Every infrabarreled space possessing a fundamental sequence of bounded sets is a DF-space.
• Every separated quotient of a DF-space is a DF-space.

  Examples