In functional analysis, the Dixmier-Ng Theorem is a characterization of when a normed space is in fact a dual Banach space.
Dixmier-Ng Theorem. Let be a normed space. The following are equivalent: There exists a Hausdorff locally convex topology on so that the closed unit ball, , of is -compact. 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 (endowed with the Lipschitz constant as norm) is a dual Banach space.