Superreflexive tensor product spaces

Abstract

The aim of this note is to prove that, given two superreflexive Banach spaces $X$ and $Y$, then $X{\hat{\otimes }}_{\ddot{I}}€Y$ is superreflexive if and only if either $X$ or $Y$ is finite-dimensional. In a similar way, we prove that $X{\hat{\otimes }}_{\epsilon }Y$ is superreflexive if and only if either $X$ or $Y$ is finite-dimensional.