If $H^*$ is isomorphic with $H$ is H always a Hilbert space?

Question

If $H$ is a Hilbert space then $H^*$ is isomorphic with $H$. I am asking if we have a vector space H equipped with inner product (,) and $H^*$ is isomorphic with $H$ is it true to say that $H$ is Hilbert? Edit:I am also interested for cases that the norm of H is not the ordinary norm given from inner product

Answer 1

The answer is No.
A non Hilbert space can be isometric (not just isomorphic) with his Dual. For example $X:=(\ell^2,\lVert $.$\rVert_{\ell^2} +\lVert $.$\rVert_{\infty})$, $X\cong X^*$ ,and obviously is not a hilbert space.