Let $B$ be a Banach space. It is not necessarily true that there exists a Hilbert space $H$ linearly isometric to $B$.
However, is it true that there exists a Hilbert space $H$ homeomorphic to $B$?
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Sign up to join this communityLet $B$ be a Banach space. It is not necessarily true that there exists a Hilbert space $H$ linearly isometric to $B$.
However, is it true that there exists a Hilbert space $H$ homeomorphic to $B$?
Yes, but this is quite a deep result. Two infinite-dimensional Banach spaces $X$ and $Y$ are homeomorphic iff $d(X)=d(Y)$, where the density $d(X)$ is the minimal size of a dense subset of $X$.
So any separable infinite-dimensional Banach space is homeomorphic to the Hilbert space $\ell^2$ (and even to $\mathbb{R}^\omega$, because the result extends to locally convex completely metrisable TVS's as well). And for higher densities we have Hilbert spaces $\ell_2(\kappa)$ as models. Finite dimensional we only have the $\mathbb{R}^n$ up to homeomorphism, which are already Hilbert spaces.