Smooth Finite Dimensional Embeddings

http://dx.doi.org/10.4153/CJM-1999-027-1
Canad. J. Math. 51(1999), 585-615

  Published:1999-06-01
 Printed: Jun 1999

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 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.