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 ${\displaystyle X}$be a normed space. The following are equivalent:

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

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

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