Smooth finite dimensional embeddings

We give necessary and sufficient conditions for a norm-compact subset of a Hilbert space to admit a C-1 embedding into a finite dimensional Euclidean space. Using quasibundles, we prove a structure theorem saying that the str atum of n-dimensional points is contained in an n-dimensional C-1 submanifo ld of the ambient Hilbert space. This work sharpens and extends earlier res ults of G. Glaeser on paratingents. As byproducts we obtain smoothing theor ems for compact subsets of Hilbert space and disjunction theorems for local ly compact subsets of Eudidean space.