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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

On the nonexistence of a projection from functions of $ x$ to functions of $ x\sp{n}$


Author: P. Milman
Journal: Proc. Amer. Math. Soc. 63 (1977), 87-90
MSC: Primary 58D15; Secondary 26A93
MathSciNet review: 0440600
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Abstract: The subspace $ {\phi ^ \ast }{C^\infty }({{\mathbf{R}}^1}) \subset {C^\infty }({{\mathbf{R}}^1})$ of all $ {C^\infty }$ functions of $ \phi (x) = {x^n},n = 1,2,3, \ldots $, is a closed subspace of $ {C^\infty }({{\mathbf{R}}^1})$ by Glaeser's Composition Theorem. We prove that for $ n > 2$ there does not exist a linear continuous projection $ \pi $ from $ {C^\infty }({{\mathbf{R}}^1})$ onto $ {\phi ^ \ast }{C^\infty }({{\mathbf{R}}^1})$.


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Additional Information

DOI: http://dx.doi.org/10.1090/S0002-9939-1977-0440600-3
PII: S 0002-9939(1977)0440600-3
Keywords: Frechét space, Glaeser's Composition Theorem, open mapping theorem
Article copyright: © Copyright 1977 American Mathematical Society