First of all I am new to the field of embedding one manifold into another other.
I have recently come across with the paper "Embedding Riemannian manifolds by their heat kernel" by P. BERARD, G. BESSON, S. GALLOT (published in Geometric and Functional Analysis in 1984), who prove that one can embed a closed Riemannian manifold with certain assumptions on its Ricci curvature and its diameter into $\ell^2$.
My question is if there is other results on isometric embedding of a closed Riemannian manifold without any assumption on its Ricci curvature and its diameter into some infinite dimensional manifold (Banach, Hilbert, or Frechet manifolds)? Or into some $L^2$ space? The point is the ambient space must be an infinite dimensional manifold, not finite dimension. Or if not, what other "weaker" assumptions have to be imposed on the manifold?