PlanetMath: $C^\infty_0(U)$ is not empty

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$C^\infty_0(U)$ is not empty

(Theorem)

Theorem. If is a non-empty open set in $\mathbb{R}^n$ , then the set of smooth functions with compact support $C^\infty_0(U)$ 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 : If be real numbers, then there exists a smooth non-negative function $f:\mathbb{R}\to \mathbb{R}$ , whose support is the compact set .

To see this, let $\phi\colon \mathbb{R}\to \mathbb{R}$ be the function defined on this page, and let

$\displaystyle f(x) = \phi(x-a) \phi(b-x).$

Since $\phi$ is smooth, it follows that

is smooth. Also, from the definition of $\phi$ , we see that $\phi(x-a)=0$ precisely when $x\le a$ , and $\phi(b-x)=0$ precisely when $x\ge b$ . Thus the support of

is indeed

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

$\displaystyle h(z) = f(\Vert x-z \Vert^2).$

Since $\lVert\cdot \rVert^2$ (Euclidean norm) is smooth, the claim follows. $\Box$

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Cross-references: Euclidean norm, smooth function, open, compact set, smooth, real numbers, proof, existence of partitions of unity, consequence, compact, analytic functions, obvious, functions, smooth functions with compact support, open set
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This is version 14 of $C^\infty_0(U)$ is not empty, born on 2003-07-05, modified 2006-05-27.
Object id is 4422, canonical name is Cinfty_0UIsNotEmpty.
Accessed 3199 times total.

Classification:

AMS MSC:

26B05 (Real functions :: Functions of several variables :: Continuity and differentiation questions)

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