View source for 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 <math>n \in \mathbb N</math>. Then the '''Schwartz functions''' on <math>\mathbb R^n</math> are the set <math>\mathcal S(\mathbb R^n)</math> defined in the following way:
:<math>\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\}</math>.}}

{{definition|bump function|Let <math>n \in \mathbb N</math>. A '''bump function''' on <math>\mathbb R^n</math> is a function <math>u \in C^\infty(\mathbb R^n)</math> such that <math>\operatorname{supp} u</math>, the [[General Topology/Pointed spaces and support#support|support]] of <math>u</math>, is compact.}}

{{proposition|test functions are integrable|Let <math>u \in \mathcal S(\mathbb R^n)</math> or <math>u \in \mathcal D(\mathbb R^n)</math>. Then <math>u \in L^1(\mathbb R^n)</math>.}}

{{proof|Suppose first <math>u \in \mathcal S(\mathbb R^n)</math>. We know that on <math>\overline B_1(0)</math>, <math>|u(x)|</math> attains a maximum by [[Weierstraß' theorem]].

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