is not empty
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.
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|
|Date of creation||2013-03-22 13:43:57|
|Last modified on||2013-03-22 13:43:57|
|Last modified by||matte (1858)|