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.^[1][2]

Dixmier-Ng theorem.^[1] 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.

Applications

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.^[3]

References

1. Ng, Kung-fu (December 1971), "On a theorem of Dixmier", Mathematica Scandinavica, 29: 279–280, doi:10.7146/math.scand.a-11054
2. Dixmier, J. (December 1948), "Sur un théorème de Banach", Duke Mathematical Journal, 15 (4): 1057–1071, doi:10.1215/s0012-7094-48-01595-6
3. Godefroy, G.; Kalton, N. J. (2003), "Lipschitz-free Banach spaces", Studia Mathematica, 159 (1): 121–141, doi:10.4064/sm159-1-6