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 H_n is dense in span{X} and π_n (X) = X ∩ H_n for each n ≥ 1. Here, π_n : ℍ → H_n 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.