Dixmier–Ng theorem


In functional analysis, the Dixmier–Ng theorem is a characterization of when a normed space is in fact a dual Banach space. It was proven by Kung-fu Ng, who called it a variant of a theorem proven earlier by Jacques Dixmier.
  1. There exists a Hausdorff locally convex topology on so that the closed unit ball,, of is -compact.
  2. There exists a Banach space so that is isometrically isomorphic to the dual of.
That 2. implies 1. is an application of the Banach–Alaoglu theorem, setting to the Weak-* topology. That 1. implies 2. is an application of the Bipolar theorem.

Applications

Let be a pointed metric space with distinguished point denoted. The Dixmier-Ng Theorem is applied to show that the Lipschitz space of all real-valued Lipschitz functions from to that vanish at is a dual Banach space.