C-Weierstrass for compact sets in Hilbert space |
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Authors: | H Movahedi-Lankarani R Wells |
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Institution: | a Department of Mathematics and Statistics, Penn State Altoona, Altoona, PA 16601-3760, USA b Department of Mathematics, Penn State University, University Park, PA 16802, USA |
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Abstract: | The C1-Weierstrass approximation theorem is proved for any compact subset X of a Hilbert space . 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 such that ?n?1Hn is dense in and πn(X)=X∩Hn for each n?1. Here, is the orthogonal projection. It is also shown that when X is compact convex with 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 , an extension of the C1-Weierstrass approximation theorem is proved for C1 maps with compact derivatives. |
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Keywords: | C1 embedding Weierstrass Spherically compact C1-topology Tangent space Paratingent Quasibundle |
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