We give necessary and sufficient conditions for a norm-compact subset of a Hubert space to admit a C1 embedding into a finite dimensional Euclidean space. Using quasibundles, we prove a structure theorem saying that the stratum of n-dimensional points is contained in an n-dimensional C1 submanifold of the ambient Hubert space. This work sharpens and extends earlier results of G. Glaeser on paratingents. As byproducts we obtain smoothing theorems for compact subsets of Hubert space and disjunction theorems for locally compact subsets of Euclidean space.
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