What is Schwartz class?

The Schwartz class $\mathcal S(\R)$ consists of the $\mathcal C^\infty$ functions $f$ with $\sup_x\abs{x^m f^{(n)}(x)} < \infty$ for all $m, n \geq 0$ (all derivatives decay faster than any power). Examples: $\eu^{-ax^2}$, $\mathcal C_c^\infty$. Clearly $\mathcal S \subseteq L^p$ for every $p$ (bound by $C(1 + x^2)^{-1}$), and $\mathcal S$ is stable under derivatives, multiplication by polynomials, and products.