Compact norming dual-pair criterion
= Compact norming dual-pair criterion
Let $Z\subseteq X^*$ separate points of a Banach space $X$. If the closed unit ball of $X$ is compact for $\sigma(X,Z)$ and $Z$ has the norm inherited from $X^*$, then the evaluation map $X\to Z^*$ is an isometric isomorphism. <Goldstine theorem> makes its unit-ball image weak-star dense, while the assumed compactness makes that image closed.