C1-Weierstrass for compact sets in Hilbert space

✍ Scribed by H. Movahedi-Lankarani; R. Wells

• Publisher: Elsevier Science
• Year: 2003
• Tongue: English
• Weight: 219 KB
• Volume: 285
• Category: Article
• ISSN: 0022-247X
• DOI: 10.1016/s0022-247x(03)00427-x

✦ Synopsis

The C 1 -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

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 C 1 (X) is complete if and only if the C 1 -Whitney extension theorem holds for X. Finally, for compact subsets of H, an extension of the C 1 -Weierstrass approximation theorem is proved for C 1 maps H → H with compact derivatives.