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.{{citation

| last = Dixmier | first = J. | author-link = Jacques Dixmier

| date = December 1948

| doi = 10.1215/s0012-7094-48-01595-6

| issue = 4

| journal = Duke Mathematical Journal

| pages = 1057–1071

| title = Sur un théorème de Banach

| volume = 15}}

: Dixmier-Ng theorem.{{citation

| last = Ng | first = Kung-fu

| date = December 1971

| doi = 10.7146/math.scand.a-11054

| journal = Mathematica Scandinavica

| pages = 279–280

| title = On a theorem of Dixmier

| volume = 29}} Let $X$ be a normed space. The following are equivalent:

1. There exists a Hausdorff locally convex topology $\backslash tau$ on $X$ so that the closed unit ball, $\backslash mathbf\{B\}\_X$, of $X$ is $\backslash tau$-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 $\backslash tau$ to the Weak-* topology. That 1. implies 2. is an application of the Bipolar theorem.