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 the following statements are equivalent
| (1) | $f$ is a D3953: Schwartz function |
| (2) | \begin{equation} \forall \, N \in \mathbb{N} : \forall \, \alpha \in \mathbb{N}^D : \lim_{|x| \to \infty} |x|^N \partial^{\alpha} f(x) = 0 \end{equation} |
| (3) | \begin{equation} \forall \, N \in \mathbb{N} : \sup_{|\beta| \leq N} \Vert (1 + |x|^2)^N \partial^{\beta} f \Vert_{\infty} < \infty \end{equation} |
| (4) | \begin{equation} \forall \, N \in \mathbb{N} : \sum_{|\beta| \leq n} \Vert (1 + |x|)^N \partial^{\beta} f \Vert_{\infty} < \infty \end{equation} |
| (5) | \begin{equation} \forall \, \beta \in \mathbb{N}^D : \forall \, N \in \mathbb{N} : \partial^{\beta} f(x) = O(|x|^{-N}) \end{equation} |
| (6) | \begin{equation} \forall \, \beta \in \mathbb{N}^D : \forall \, N \in \mathbb{N} : |x|^N \partial^{\beta} f(x) = O(1) \end{equation} |