Let $\mathbb{R}^D$ be a D5630: Set of euclidean real numbers such that

(i)	$f : \mathbb{R}^D \to \mathbb{C}$ is an D1493: Infinitely differentiable function
(ii)	$\mathbb{N}^D \subseteq \mathbb{R}^D$ is a D5179: Set of euclidean natural numbers

Then $f$ is a Schwartz function if and only if \begin{equation} \forall \, \alpha, \beta \in \mathbb{N}^D : \sup_{x \in \mathbb{R}^D} |x^{\alpha} \partial^{\beta} f(x)| < \infty \end{equation}

Let $\mathbb{R}^D$ be a D5630: Set of euclidean real numbers such that

(i)	$f : \mathbb{R}^D \to \mathbb{C}$ is an D1493: Infinitely differentiable function
(ii)	$\mathbb{N}^D \subseteq \mathbb{R}^D$ is a D5179: Set of euclidean natural numbers

Then $f$ is a Schwartz function if and only if \begin{equation} \forall \, \alpha, \beta \in \mathbb{N}^D : \Vert x^{\alpha} \partial^{\beta} f(x) \Vert_{\infty} < \infty \end{equation}