is not empty
Theorem. If is a non-empty open set in , then the set of
smooth functions with compact support 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 , whose support (http://planetmath.org/SupportOfFunction) is the compact set .
To see this, let be the function defined on this page (http://planetmath.org/InfinitelyDifferentiableFunctionThatIsNotAnalytic), and let
Since is smooth, it follows that is smooth. Also, from the definition of , we see that precisely when , and precisely when . Thus the support of is indeed .
Since is non-empty and
open there exists an and such that
. Let be smooth function
such that , and
let
Since (Euclidean norm) is smooth, the claim follows.
Title | 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 |