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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 stratum of 
$n$-dimensional points is contained in an $n$-dimensional
${{C}^{1}}$
submanifold of the ambient Hilbert space. This work sharpens and extends earlier results of 
$\text{G}$. Glaeser on paratingents. As byproducts we obtain smoothing theorems for compact subsets of Hilbert space and disjunction theorems for locally compact subsets of Euclidean space.
