# Smooth approximation theorem for differential manifolds

From Diffgeom

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 and be differential manifolds and define to be the space of all continuous maps from to , endowed with the compact-open topology. Let be the space of all smooth maps from to , viewed as a subset of . Then is dense in .

## Applications

- Smooth homotopy theorem: This theorem has two parts: any continuous map of differential manifolds is homotopy-equivalent to a smooth map, and that if two continuous maps are homotopic, then they are smoothly homotopic.