Schwartz space
= Schwartz space
{c}
{title2=$\mathcal S(\mathbb R^n)$}
{wiki}
The Schwartz space $\mathcal S(\mathbb R^n)$ consists of smooth functions whose derivatives decay faster than every inverse polynomial. Its Fréchet topology is generated by the seminorms
$$
p_{\alpha,\beta}(\varphi)=\sup_{x\in\mathbb R^n}|x^\alpha D^\beta\varphi(x)|.
$$
It is dense in every $H^s(\mathbb R^n)$.