In the Wikipedia paper on Hadamard spaces, it is written that every flat Hadamard space is isometric to a closed convex subset of a Hilbert space. Looking through references provided by this Wikipedia paper, I found that this fact is proved as Theorem 7.2 in the lecture notes of Jacob Lurie. Since those notes contain no references and no historical information, it is not clear:
To whom this important result should be attributed? And when it was proved for the first time?
Remark. I am asking this question because, the mentioned Theorem 7.2 in Lurie notes implies the following nice characterization of metric spaces, which are isometric to Hilbert spaces:
Theorem. A metric space $(X,d)$ is isometric to a Hilbert space if and only if it is nonempty, complete, and satisfies the following two (first-order) conditions:
- $\forall x,y\in X\;\exists z\in X\;\; \big[d(x,z)=d(x,y)+d(y,z)=2d(x,y)\big]$;
- $\forall x,y\in X\;\exists m\in X\;\forall z\in X\;\big[d(z,m)^2=\tfrac12d(x,z)^2+\tfrac12d(y,z)^2-\tfrac 14d(x,y)^2\big]$.
I hope that this characterization is known (to the specialists). So the question is to whom should it be attributed?
Is this theorem the simplest (in a reasonable sense) metric characterization of Hilbert spaces?
