James reflexivity criterion

James's theorem says that a Banach space is reflexive exactly when every continuous linear functional attains its supremum on the closed unit ball. Its separation proof also yields the equivalent convex-block criterion: nonreflexivity produces a bounded sequence whose initial and tail convex hulls remain uniformly separated.

For finite-dimensional subspaces $E\subseteq X^{\ast \ast }$ and $F\subseteq X^{\ast }$, the principle of local reflexivity gives an almost-isometric map $T:E\to X$ that fixes $E\cap X$ and preserves the pairings with $F$. It lets finite-dimensional bidual separation data be realized inside $X$.