 Principle of local reflexivity

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 $X$.