Schwarz space

$\{f\in C^{\infty }({\mathbb{R}}^n,\mathbb{C})|\forall \alpha ,\beta \in {\mathbb{N}}^n,\Vert f{\Vert }_{\alpha ,\beta }<\infty \}$, where

• $\alpha ,\beta \in {\mathbb{R}}^n$
• $C^{\infty }({\mathbb{R}}^n,\mathbb{C})$ is the set of infinitely differentiable functions $f:{\mathbb{R}}^n\to \mathbb{C}$
• $\Vert f{\Vert }_{\alpha ,\beta }:={\sup }_{x\in {\mathbb{R}}^n}|(x^{\alpha }{D}^{\beta }f)(x)|$

Intuitively, all derivatives of $f$ decay to zero faster than any polynomial, as $|x|\to \infty$.