\begin{definition}[Definition:Schwartz Test Function]
Let $\phi : \R \to \C$ be a complex-valued function.
Let $\phi \in \map {\CC^\infty} \R$ be smooth.
Suppose $\phi$ is bounded in the following way:
:$\ds \forall m, l \in \N : \sup_{x \mathop \in \R} \size {x^l \map {\phi^{\paren m}} x} < \infty$
where $\phi^{\paren m}$ denotes the $m$-th derivative of $\phi$.
Then $\phi$ is known as a '''Schwartz test function'''.
{{NamedforDef|Laurent-Moïse Schwartz|cat = Schwartz}}
\end{definition}