Non-Hilbert Banach space isomorphic to $\ell_2$ I know that $\ell_2$ is the only separable Hilbert space of infinite dimension up to isometric isomorphism, so in particular, any separable Hilbert space of infinite dimension is isomorphic to $\ell_2$.
So my question is, can someone give an example of a Banach space isomorphic to $\ell_2$ but not isometrically isomorphic to it?
I know, for what is said above, that this space cannot be a Hilbert space but I can't think of any example.

For example, let $X=\ell_2(\{0,1,2,\dots\})$ and define a norm on $X$ by $$||x||=|x_0|+\left(\sum_{n=1}^\infty|x_n|^2\right)^{1/2}.$$
The identity map is an isomorphism onto $\ell_2$. A simple way to make certain that $X$ is not isometric to $\ell_2$ is to verify that the parallelogram law fails: $$||e_0+e_1||^2+||e_0-e_1||^2=8,$$while $$2||e_0||^2+2||e_1||^2=4.$$
