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 $X$ 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.