C1-Weierstrass for compact sets in Hilbert space

The C1-Weierstrass approximation theorem is proved for any compact subset X of a Hilbert space H. The same theorem is also proved for Whitney 1-jets on X when X satisfies the following further condition: There exist finite dimensional linear subspaces H1 ⊂ H2 ⊂··· ⊂ H such that n 1 Hn is dense in span{X} and πn(X) = X ∩ Hn for each n 1. Here, πn :H→Hn is the orthogonal projection. It is also shown that when X is compact convex with span{X} = H and satisfies the above condition, then C1(X) is complete if and only if the C1-Whitney extension theorem holds for X. Finally, for compact subsets of H, an extension of the C1-Weierstrass approximation theorem is proved for C1 maps H→H with compact derivatives.  2003 Elsevier Inc. All rights reserved