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.


 * Dixmier-Ng theorem. Let $$X$$ be a normed space. The following are equivalent:
 * 1)  There exists a Hausdorff locally convex topology $$\tau$$ on $$X$$ so that the closed unit ball, $$\mathbf{B}_X$$, of $$X$$ is $$\tau$$-compact.
 * 2) There exists a Banach space $$Y$$ so that $$X$$ is isometrically isomorphic to the dual of $$Y$$.

That 2. implies 1. is an application of the Banach–Alaoglu theorem, setting $$\tau$$ to the Weak-* topology. That 1. implies 2. is an application of the Bipolar theorem.

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