\begin{definition}[Definition:Convergent Sequence/Schwartz Space]
Let $\map \SS \R$ be the Schwartz space.
Let $\sequence {\phi_n}_{n \mathop \in \N}$ be a sequence in $\map \SS \R$.
Let $\phi \in \map \SS \R$ be a Schwartz test function.
Suppose:
:$\ds \forall l, m \in \N : \lim_{n \mathop \to \infty} \sup_{x \mathop \in \R} \size {x^l \map {\phi_n^{\paren m} } x} = 0$
where:
:$\phi^{\paren m}$ denotes the $m$th derivative of $\phi$
:$\sup$ denotes the supremum.
Then the sequence $\sequence {\phi_n}_{n \mathop \in \N}$ '''converges to $\mathbf 0$ in $\map \SS \R$'''.
This can be denoted:
:$\phi_n \stackrel \SS {\longrightarrow} \mathbf 0$
\end{definition}