> [!definition] > > Let $f \in C^\infty(\real^d)$ be a [[Space of Smooth Functions|smooth function]]. If for every [[Multi-Index|multi-index]] $\alpha$, there exists $C_{f, \alpha} \ge 0$ and $N_{f, \alpha} \ge 0$ such that > $ > \abs{\partial^\alpha f(x)} \le C_{f, \alpha} (1 + \abs{x})^{N_{f, \alpha}} > $ > then $f$ is **slowly increasing**. If $f$ is slowly increasing, then the map $\cs \to \cs$ on the [[Schwartz Space|Schwartz space]] defined by $\phi \mapsto f\phi$ is a [[Bounded Linear Map|continuous linear map]]. > > *Proof*. Let $\alpha$ be a multi-index. Let $C_\alpha = \max_{\beta + \gamma = \alpha}\frac{\alpha !}{\beta! \gamma!}$, $N_f = \max_{\gamma \le \alpha}N_{f, \gamma}$, and $C_f = \max_{\gamma \le \alpha}C_{f, \gamma}$, then > $ > \begin{align*} > \abs{\partial^\alpha(f\phi)(x)} &\le \sum_{\beta + \gamma = \alpha}\frac{\alpha!}{\beta!\gamma!}\abs{(\partial^\beta f)(x)(\partial^\gamma \phi)(x)} \\ > &\le C_{\alpha}\sum_{\gamma \le \alpha}C_f(1 + \abs{x})^{N_f}\abs{(\partial^\gamma \phi)(x)} \\ > \norm{f \phi}_{(0, \alpha)} &\le C_{\alpha}C_f\sum_{\gamma \le \alpha}\norm{\phi}_{(N_f, \gamma)} \\ > \norm{f \phi}_{(N, \alpha)} &\le C_{\alpha}C_f\sum_{\gamma \le \alpha}\norm{\phi}_{(N+N_f, \gamma)} > \end{align*} > $