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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

Approximation theorems for Nash mappings and Nash manifolds


Author: Masahiro Shiota
Journal: Trans. Amer. Math. Soc. 293 (1986), 319-337
MSC: Primary 58A07
MathSciNet review: 814925
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Abstract: Let $ 0 \leq r < \infty $. A Nash function on $ {\mathbf{R}^n}$ is a $ {C^r}$ function whose graph is semialgebraic. It is shown that a $ {C^r}$ Nash function is approximated by a $ {C^\omega}$ Nash one in a strong topology defined in the same way as the usual topology on the space $ \mathcal{S}$ of rapidly decreasing $ {C^\infty}$ functions. A $ {C^r}$ Nash manifold in $ {\mathbf{R}^n}$ is a semialgebraic $ {C^r}$ manifold. We also prove that a $ {C^r}$ Nash manifold for $ r \ge 1$ is approximated by a $ {C^\omega}$ Nash manifold, from which we can classify all $ {C^r}$ Nash manifolds by $ {C^r}$ Nash diffeomorphisms.


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Additional Information

DOI: http://dx.doi.org/10.1090/S0002-9947-1986-0814925-3
PII: S 0002-9947(1986)0814925-3
Keywords: Nash manifold, real algebraic geometry, semialgebraic set
Article copyright: © Copyright 1986 American Mathematical Society