Definition:Zero-Limit Sequence in Schwartz Space

Definition
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:
 * $\phi_n \stackrel \SS {\longrightarrow} \mathbf 0$