Can every separable Banach space be isometrically embedded in $l^2$ ? Or at least in $l^p$ for some $1\le p<\infty$?

Question

Can every separable Banach space be isometrically embedded in $l^2$ ? Or at least in $l^p$ for some $1\le p<\infty$ ?

I only know that any separable Banach space is isometrically isomorphic to a linear subspace of $l^{\infty}$.

Please help . Thanks in advance

Answer 1

Of course not. Every closed subspace of $\ell_2$ is isometric to some Hilbert space.

Answer 2

$c_0$ does not embed in $\ell_p$ for $1\leqslant p<\infty$ as it is not weakly sequentially complete.