A Banach space isomorphic to his dual is an Hilbert space?

Question

By the Riesz representation theorem, we know that the Hilbert space $\mathcal{H}$ is isomorphic to his dual. Is the converse true ? Does the fact that a Banach space $E$ is isomorphic to his dual implies that $E$ is an Hilbert space ? If it is not the case, does anyone have an example ?

Answer 1

No. For any reflexive Banach space $X$ consider $X \oplus X^*$. Then $(X \oplus X^*)^* = X^* \oplus X$, which is of course isomorphic to $X \oplus X^*$ but not necessarily a Hilbert space.