Dixmier–Ng theorem explained

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:\tau on X so that the closed unit ball, B _{X} , of X is \tau -compact.

1. 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 Lip _{0(M)} of all real-valued Lipschitz functions from M to R that vanish at 0 _{M} (endowed with the Lipschitz constant as norm) is a dual Banach space