 Compact norming dual-pair criterion

Let $X$ be a Banach space and let $Z$ be a linear subspace of $X^{\ast }$ 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 $J:X\to Z^{\ast }$ is an isometric isomorphism of normed spaces.

For a unit-ball element $\Phi \in Z^{\ast }$, the Hahn-Banach theorem extends $\Phi$ to an element of $X^{\ast \ast }$ 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 $B_{Z^{\ast }}$. The assumed compactness also makes $J(B_X)$ weak-star compact and hence closed, so $J(B_X)=B_{Z^{\ast }}$.

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.