Differentiation and Integration in Several Real Variables/Test functions

We will consider two classes of test functions, namely bump functions and Schwartz functions.

Definition (Schwartz function):

Let $n\in \mathbb {N}$. Then the Schwartz functions on $\mathbb {R} ^{n}$ are the set ${\mathcal {S}}(\mathbb {R} ^{n})$ defined in the following way:

${\mathcal {S}}(\mathbb {R} ^{n}):=\left\{u\in C^{\infty }(\mathbb {R} ^{n}){\big |}\forall \alpha ,\beta \in \mathbb {N} _{0}^{n}:\left\|x^{\alpha }\partial ^{\beta }u\right\|_{\infty }<\infty \right\}$.

Definition (bump function):

Let $n\in \mathbb {N}$. A bump function on $\mathbb {R} ^{n}$ is a function $u\in C^{\infty }(\mathbb {R} ^{n})$ such that $\operatorname {supp} u$, the support of $u$, is compact.

Proposition (test functions are integrable):

Let $u\in {\mathcal {S}}(\mathbb {R} ^{n})$ or $u\in {\mathcal {D}}(\mathbb {R} ^{n})$. Then $u\in L^{1}(\mathbb {R} ^{n})$.

{{proof|Suppose first $u\in {\mathcal {S}}(\mathbb {R} ^{n})$. We know that on ${\overline {B}}_{1}(0)$, $|u(x)|$ attains a maximum by Weierstraß' theorem.