= Compact norming dual-pair criterion
Let $X$ be a <Banach space> and let $Z$ be a <linear subspace> of $X^*$ that separates points, with the inherited <operator norm>. If the <closed unit ball> $B_X$ is compact for $\sigma(X,Z)$, then the evaluation map $J:X\to Z^*$ is an <isometric isomorphism of normed spaces>.
For a unit-ball element $\Phi\in Z^*$, the <Hahn-Banach theorem> extends $\Phi$ to an element of $X^{**}$ of norm at most one. <Goldstine theorem> approximates that extension weak-star by elements of $B_X$. Restricting the pairings to $Z$ shows that $J(B_X)$ is weak-star dense in $B_{Z^*}$. The assumed compactness also makes $J(B_X)$ weak-star compact and hence closed, so $J(B_X)=B_{Z^*}$.
Separation makes $J$ injective. Equality of the two unit balls makes it surjective and norm preserving: a smaller image norm could otherwise place a vector outside $B_X$ in the image of $B_X$. This proves the claimed dual representation without assuming in advance that $Z$ is norming.
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