*This is the answer to the original question (Not the one which is posted now).*

Look in two papers, mine and the paper of Enrico Le Donne.

You are looking for spaces which admit length-preserving embedding into Hilbert space.
In my paper I prove that a compact length spaces which (roughly) admit a length-preserving map into Euclidean $m$-space has to be inverse limits of $m$-dimensional polyhedral spaces.

The infinite dimensional case is easier; it can be done along the same lines; in this case the dimension of polyhedral spaces will go to infinity. It seems that if a compact space admits a length-preserving map into infinite dimensional Hilbert space then it can be perturbed into length-preserving embedding. Enrico considers length-preserving embedding in finite dimensional case (which is much harder).

**About the last question.** Nash's theorem can be indeed simplified considerably if the target space is infinite dimensional. You can apply the same procedure as in the proof of Nash--Kuiper theorem, but since the dimension is infinite, you can use a parallel frame each time. This way you get a sequence of maps $f_1,f_2,\dots$ which converges to isometric embedding and such that the $k$-th coordinate of $f_n$ is the same for all large $n$. This way you can get better regularity of the limit --- instead of $C^1$, you should get $C^\infty$ (am I right?).

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