$\begingroup$This is a rather simple question, but I don't have many resources I've been able to consult in this regard (meaning that a Google search hasn't yielded any meaningful results). I just want to clarify, in the setting of Schwartz spaces, is the following definition correct:
A function $f\in C^{\infty}(\mathbb{R}^N)$ is said to be of tempered growth if, for all multi-indices $\alpha\in\mathbb{N}_0^N$, $\exists C_\alpha>0$, and $\exists M_\alpha\in\mathbb{N}$, such that $|\partial^\alpha f(x)|\leq C_a(1 + |x|)^{M_\alpha}$, $\forall x\in\mathbb{R}^N$.
Any corrections or validations are greatly appreciated.
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