Hamilton's paper "The Inverse Function theorem of Nash and Moser" (1982, Bull. Amer. Math. Soc, vol. 7, n. 1, page $137$) proves that $C^{\infty}(M)$ is a tame Fréchet space when $M$ is a compact manifold. It was asked [here on MO](https://mathoverflow.net/questions/138535/are-smooth-functions-tame) if this space is tame in the non-compact case, and in the second answer, there was an argument that for a space to be tame, every sufficiently large taming semi-norm would be a norm (which is false in $C^{\infty}(M)$ for the non-compact case), but this is true for $\mathcal{S}(\mathbb{R}^n)$ (the space of functions that are rapidly decreasing, along with its derivatives). So to disprove tameness, a different argument would be necessary.


So the question is, is $\mathcal{S}(\mathbb{R}^n)$ a tame Fréchet space (with the usual Fréchet topology)? Is there a reference? 



Is $\mathcal{S}(\mathbb{R}^n)$ a tame Fréchet space?
Using Fourier series, $C^\infty(S^1)$ is ismorphic to the sequence space
$$s=\{(x\_k)\_{k\in \mathbb N}\in \mathbb C^{\mathbb N}: \sum\_{k=1}^\infty k^{2n} |x\_k|^2 <\infty \text{ for all $n\in\mathbb N$}\}$$
and this space is isomorphic to $\mathscr S(\mathbb R^d)$.
This is very classical, a modern treatment (with much information about structural properties of Fréchet spaces) is in the book *Introduction to Functional Analysis* of Meise and Vogt.


