Superreflexive Banach space

A Banach space is superreflexive when every Banach space finitely representable in it is a reflexive Banach space. Equivalently, it admits an equivalent uniformly convex norm, or every one of its ultrapowers is reflexive.

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 $X$.

A Banach space $X$ is superreflexive exactly when, for every $\theta >0$, some $N$ forces every sequence $(x_i{)}_{i=1}^N\subseteq B_X$ to have two convex combinations, one before and one after a cut, at distance below $\theta$. Failure produces a nonreflexive ultrapower; finite representability transfers a violating finite sequence back from any nonreflexive space finitely representable in $X$.

A Banach space $X$ is superreflexive exactly when the distortions $c_X(D_n)$ of the finite diamond graphs tend to infinity. A uniformly separated convex-block sequence recursively realizes all branches of $D_n$ with bounded distortion, while uniform convexity forces quantitative collapse across repeated diamonds.