Banach spaces that have normal structure and are isomorphic to a Hilbert space

We prove that given a Hilbert space $ \left( {E,\vert\vert \cdot \vert\vert} \right)$ a norm on $ E$, $ 1/\beta \left\vert x \right\vert \leqslant \left\Vert x \right\Vert \leqslant \left\vert x \right\vert$, if $ 1 \leqslant \beta &lt; \sqrt 2 $ <IMG WIDTH="72" HEIGHT="41" ALIGN="MIDDLE" BORDER="0" SRC="images/img9.gif" ALT="$ \left( {E,\vert \cdot \vert} \right)$"> satisfies a convexity property from which normal structure follows.