I'm currently trying to prove, that we can use any function $F \in C^\infty (\mathbb{C}^k , \mathbb C)$ with $F=0$ to map Schwartz functions $f_j \in \mathcal S(\mathbb R ^d)$ to themselves. So
$$F \circ (f_1, ... ,f_k) \in \mathcal S (\mathbb R ^d)$$
and that this Map $\mathcal S (\mathbb R ^d)^k \to \mathcal S (\mathbb R ^d)$ is continous.
I need to show, that $\sup |x^\alpha D^\beta F \circ(f_1, ..., f_k)| < \infty$ and I guess it's enough to show this for $|\beta| = 1$. Hence, show $$\sup |x^\alpha \sum_{j=1}^k D_j F(f_1(x), ..., f_k(x)) D_{x_i}f_j(x)|< \infty $$ where $ |x^\alpha D_{x_i}f_j(x)| < \infty$ because $f_j \in \mathcal S (\mathbb R^d)$. Now I need to apply $F(0) = 0$ but I only have the derivative of $F$.
Is this correct? And how do I finish the proof?