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

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

Definition (bump function):

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

Proposition (test functions are integrable):

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

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