 Compact norming dual-pair criterion

Let $Z\subseteq X^{\ast }$ 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 $X^{\ast }$, then the evaluation map $X\to Z^{\ast }$ is an isometric isomorphism. Goldstine theorem makes its unit-ball image weak-star dense, while the assumed compactness makes that image closed.