Let $J$ be a set (usually countable). Let $t_j$, $j\in J$, be variables in ${\mathbb R}/2\pi i{\mathbb Z}.$ Put $u_j=\exp(it_j),$ $j\in J.$ Introduce the following semi-norms on the space of Fourier polynomials ${\mathbb C}[u_j^{\pm 1}|j\in J]:$ for $N\geq 0,$ put $$||f||_N=\max_{t\in (S^1)^J} \sum _{|\alpha|=N} |\partial ^{[\alpha ]} f(t)|$$ where the sum is taken over multi-indices $\alpha=(\alpha_j|j\in J)$; $\alpha_j\geq 0;$ $|\alpha|=\sum_{j\in J}\alpha_j;$ and $\partial ^{[\alpha ]}=\prod_{j\in J} \frac{1}{\alpha_j !}(\frac{\partial}{\partial t_j})^{\alpha_j}$.

Let $C^\infty ((S^1)^J)$ be the completion of ${\mathbb C}[u_j^{\pm 1}|j\in J]$ in the topology defined by these semi-norms.

**Question**. How to describe $C^\infty((S^1)^J)$ in terms of the Fourier coefficients $a_n$?

sureyouwantto consider these metrics? The usefulness of Cᵏ lies in a bound for k⸣th derivatives leading to abound on variationof k-1⸣st derivatives. However (if I did not miss something) you essentially work in (a class of) metrics on a torus for which the diameter of the torus is infinite. Would not it be better to “allow more metrics on the torus” (leading to more metrics on C∞) so that the estimate mentioned above works? $\endgroup$ – Ilya Zakharevich Nov 11 '19 at 5:08