### Abstract

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 H_{1} ⊂ H_{2} ⊂ ⋯ ⊂ 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 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.

Original language | English (US) |
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Pages (from-to) | 299-320 |

Number of pages | 22 |

Journal | Journal of Mathematical Analysis and Applications |

Volume | 285 |

Issue number | 1 |

DOIs | |

State | Published - Sep 1 2003 |

### All Science Journal Classification (ASJC) codes

- Analysis
- Applied Mathematics

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## Cite this

^{1}-Weierstrass for compact sets in Hilbert space.

*Journal of Mathematical Analysis and Applications*,

*285*(1), 299-320. https://doi.org/10.1016/S0022-247X(03)00427-X