There are plenty of isometric embeddings of metric spaces in Banach spaces. Nevertheless, I have been unable to find any significant result on isometric embeddings into Hilbert spaces. My question is: how can one recognize those metric spaces that are isometrically embeddable into Hilbert spaces?

Nash's theorem shows that many Riemannian manifolds are. My question is much simpler, though, since I do not require finite-dimensional Hilbert spaces. On the other hand, I'm curious about embedding infinite-dimensional Riemannian manifolds.

If I do not require a finite dimension for the receiving space, can one produce a significantly simpler (and clearer) proof of Nash's theorem? (One that could be extended to infinite-dimensional Riemannian manifolds, maybe?)

There are plenty of isometric embeddings of metric spaces in Banach spaces. Nevertheless, I have been unable to find any significant result on isometric embeddings into Hilbert spaces. My question is: how can one recognize those metric spaces that are isometrically embeddable into Hilbert spaces?

Later edit: I have removed two paragraphs from my original question, which created a lot of confusion among those who answered it. I take responsibility for mixing "metric spaces" isometries and "differential geometry" isometries. I apologize.