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.