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.

Dixmier-Ng Theorem. Let [math]\displaystyle{ X }[/math] be a normed space. The following are equivalent:

1. There exists a Hausdorff locally convex topology [math]\displaystyle{ \tau }[/math] on [math]\displaystyle{ X }[/math] so that the closed unit ball, [math]\displaystyle{ \mathbf{B}_X }[/math], of [math]\displaystyle{ X }[/math] is [math]\displaystyle{ \tau }[/math]-compact.
2. There exists a Banach space [math]\displaystyle{ Y }[/math] so that [math]\displaystyle{ X }[/math] is isometrically isomorphic to the dual of [math]\displaystyle{ Y }[/math].

That 2. implies 1. is an application of the Banach–Alaoglu theorem, setting [math]\displaystyle{ \tau }[/math] to the Weak-* topology. That 1. implies 2. is an application of the Bipolar theorem.

Applications

Let [math]\displaystyle{ M }[/math] be a pointed metric space with distinguished point denoted [math]\displaystyle{ 0_M }[/math]. The Dixmier-Ng Theorem is applied to show that the Lipschitz space [math]\displaystyle{ \text{Lip}_0(M) }[/math] of all real-valued Lipschitz functions from [math]\displaystyle{ M }[/math] to [math]\displaystyle{ \mathbb{R} }[/math] that vanish at [math]\displaystyle{ 0_M }[/math] (endowed with the Lipschitz constant as norm) is a dual Banach space.