# 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 $$H$$.

## 1 Answer

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. (*)

Your question puts a stronger assumption, namely isometry instead of isomorphy, but Gowers' theorem only needs "isomorphic" to yield the conclusion. Afterwards one may exploit that isomorphic Hilbert spaces are isometric. So your question is answered by yes.

* Cf page 113 in Bollobás' report

• I am aware of the Gowers' theorem. I mentioned it to point out that any Banach space satisfying the condition from my question is necessarily isomorphic to a Hilbert space. However if I read you correctly and you are suggesting that isomorphic Hilbert spaces are isometric, then that's certainly not right. So I don't see how Gowers' theorem answers my question. – user446046 Jun 2 at 9:06
• I mean if a Banach space is isomorphic to a Hilbert space, then it certainly doesn't have to be isometric to it. – user446046 Jun 2 at 10:27