# ${C}_{0}^{\mathrm{\infty}}(U)$ is not empty

Theorem. If $U$ is a non-empty open set in ${\mathbb{R}}^{n}$, then the set of
smooth functions with compact support ${C}_{0}^{\mathrm{\infty}}(U)$ is non-trivial (that is, it contains functions^{} other than the zero function).

*Remark.* This theorem may seem to be obvious at first sight.
A way to notice that it is not so obvious, is to formulate it for
analytic functions^{} with compact support: in that case, the result
does not hold; in fact, there are no nonconstant analytic
functions with compact support at all.
One important consequence of this theorem is the existence of partitions
of unity.

*Proof of the theorem:*
Let us first prove this for $n=1$:
If $$ be real numbers, then there exists a
smooth non-negative function $f:\mathbb{R}\to \mathbb{R}$, whose support (http://planetmath.org/SupportOfFunction) is the
compact set $[a,b]$.

To see this, let $\varphi :\mathbb{R}\to \mathbb{R}$ be the function defined on this page (http://planetmath.org/InfinitelyDifferentiableFunctionThatIsNotAnalytic), and let

$$f(x)=\varphi (x-a)\varphi (b-x).$$ |

Since $\varphi $ is smooth, it follows that $f$ is smooth. Also, from the definition of $\varphi $, we see that $\varphi (x-a)=0$ precisely when $x\le a$, and $\varphi (b-x)=0$ precisely when $x\ge b$. Thus the support of $f$ is indeed $[a,b]$.

Since $U$ is non-empty and
open there exists an $x\in U$ and $\epsilon >0$ such that
${B}_{\epsilon}(x)\subseteq U$. Let $f$ be smooth function^{}
such that $\mathrm{supp}f=[-\epsilon /2,\epsilon /2]$, and
let

$$h(z)=f({\parallel x-z\parallel}^{2}).$$ |

Since ${\parallel \cdot \parallel}^{2}$ (Euclidean norm) is smooth, the claim follows. $\mathrm{\square}$

Title | ${C}_{0}^{\mathrm{\infty}}(U)$ is not empty |
---|---|

Canonical name | Cinfty0UIsNotEmpty |

Date of creation | 2013-03-22 13:43:57 |

Last modified on | 2013-03-22 13:43:57 |

Owner | matte (1858) |

Last modified by | matte (1858) |

Numerical id | 17 |

Author | matte (1858) |

Entry type | Theorem |

Classification | msc 26B05 |