# Isometric characterization of infinite-dimensional Hilbert spaces within Banach spaces

Let $$H=\ell^2(\mathbb{N})$$ be the separable infinite-dimensional Hilbert space. Is it the unique separable infinite-dimensional Banach space with the property that all its closed infinite-dimensional linear subspaces are isometric to it?

There is a result of Gowers saying that if one replaces in the question above isometry by isomorphism, then $$H$$ is indeed unique. Also, the Gowers' theorem implies that if there is another Banach space with this property, it must be isomorphic to $$H$$.

Gowers' result that you invoke is the solution to the longstanding "Homogeneous spaces problem". A Banach space $$B$$ is homogeneous if each of its infinite-dimensional closed subspaces is isomorphic to $$B$$.
Gowers showed, building on existing results, that $$\ell^2(\mathbb N)$$ is the only homogeneous Banach space. (*)