Let $n \in \mathbb{N}$, let $U \subset \mathbb{R}^n$ be open, let $m \in \mathbb{R}$, and let $0 \leq \delta \leq \rho \leq 1$. For $\xi \in \mathbb{R}^n$, write $\langle \xi \rangle := (1 + |\xi|^2)^{1/2}$. For $x \in \mathbb{R}^n$ and $A \subset \mathbb{R}^n$, write $\operatorname{dist}(x,A) := \inf\{|x-y| : y \in A\}$, with the convention $\operatorname{dist}(x,\varnothing)=\infty$. For $y \in \mathbb{R}^n$ and $R>0$, write $\overline{B}(y,R) := \{z \in \mathbb{R}^n : |z-y| \leq R\}$. For each compact set $K \subset U$ and each $N \in \mathbb{N}_0$, define the extended seminorm

\begin{align*} p_{K,N}^{(m,\rho,\delta)}: C^\infty(U \times \mathbb{R}^n;\mathbb{C}) \to [0,\infty] \end{align*}

by

\begin{align*} p_{K,N}^{(m,\rho,\delta)}(a) := \max_{|\alpha| + |\beta| \leq N} \sup_{(x,\xi) \in K \times \mathbb{R}^n} \langle \xi \rangle^{-m+\rho|\beta|-\delta|\alpha|} |\partial_x^\alpha \partial_\xi^\beta a(x,\xi)|, \end{align*}

where $\alpha,\beta \in \mathbb{N}_0^n$ are multi-indices and $a: U \times \mathbb{R}^n \to \mathbb{C}$ is smooth. Define $S_{\rho,\delta}^m(U \times \mathbb{R}^n)$ to be the complex [vector space](/page/Vector%20Space) of all $a \in C^\infty(U \times \mathbb{R}^n;\mathbb{C})$ such that $p_{K,N}^{(m,\rho,\delta)}(a)<\infty$ for every compact set $K \subset U$ and every $N \in \mathbb{N}_0$. Equipped with the locally convex topology generated by the finite-valued restrictions of the seminorms $p_{K,N}^{(m,\rho,\delta)}$ to $S_{\rho,\delta}^m(U \times \mathbb{R}^n)$, the space $S_{\rho,\delta}^m(U \times \mathbb{R}^n)$ is a [Fréchet space](/page/Fr%C3%A9chet%20Space) over $\mathbb{C}$.