Tame Fréchet space

Definition

A graded Fréchet space is a Fréchet space $F$ with a sequence of seminorms $\norm{\cdot}_0 \le \norm{\cdot}_1 \le \norm{\cdot}_2 \le \cdots$ that defines its topology. Let $L \colon F \to G$ be a continuous linear map between graded Fréchet spaces. It is tame if there are integers $r, b \ge 0$ and constants $C_n$ with $\norm{L f}_n \le C_n \norm{f}_{n + r}$ for all $f \in F$ and all $n \ge b$.

For a Banach space $B$, let $\Sigma(B)$ be the space of sequences $f = (f_k)_{k \ge 0}$ in $B$ with $$\norm{f}_n = \sum_{k \ge 0} e^{n k} \norm{f_k}_B < \infty \quad \text{for all } n \ge 0,$$ graded by these norms. A graded Fréchet space $F$ is tame if there are a Banach space $B$ and tame linear maps $L \colon F \to \Sigma(B)$ and $M \colon \Sigma(B) \to F$ with $M \circ L = \id_F$.