Smooth approximation theorem for differential manifolds: Difference between revisions

Revision as of 02:10, 2 February 2008

This fact is an application of the following pivotal fact/result/idea: Stone-Weierstrass theorem
View other applications of Stone-Weierstrass theorem OR Read a survey article on applying Stone-Weierstrass theorem

This article gives a result on smooth approximation; a result stating that a continuous map can be replaced by a smooth map satisfying similar properties

Statement

Let $M$ and $N$ be differential manifolds and define $C^{0} (M, N)$ to be the space of all continuous maps from $M$ to $N$ , endowed with the compact-open topology. Let $C^{\infty} (M, N)$ be the space of all smooth maps from $M$ to $N$ , viewed as a subset of $C^{0} (M, N)$ . Then $C^{\infty} (M, N)$ is dense in $C^{0} (M, N)$ .

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 {{applicationof|Stone-Weierstrass theorem}}
+{{smooth approximation result}}
 ==Statement==
 Let <math>M</math> and <math>N</math> be [[differential manifold]]s and define <math>C^0(M,N)</math> to be the space of all continuous maps from <math>M</math> to <math>N</math>, endowed with the [[compact-open topology]]. Let <math>C^\infty(M,N)</math> be the space of all [[smooth map]]s from <math>M</math> to <math>N</math>, viewed as a subset of <math>C^0(M,N)</math>. Then <math>C^\infty(M,N)</math> is dense in <math>C^0(M,N)</math>.