Convex-block separation criterion for reflexivity

A Banach space $X$ is reflexive exactly when, for every $\theta >0$ and every sequence $(x_i)$ in $B_X$, some convex combination of a finite initial segment lies within $\theta$ of a convex combination of the remaining tail. In a reflexive space both convex hulls approximate a common weak cluster point. The converse is the convex-block form of the 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 $X$.