= Solution
Use $\mathbb T^n=\mathbb R^n/(2\pi\mathbb Z)^n$ with normalized <Haar measure> and Fourier basis $e_k(x)=e^{ik\cdot x}$, $k\in\mathbb Z^n$. For any real $s$, the <periodic Sobolev space> is
$$
\boxed{H_s(\mathbb T^n)=\left\{u\in\mathcal D'(\mathbb T^n):
\|u\|_s^2=\sum_{k\in\mathbb Z^n}(1+|k|^2)^s|\widehat u(k)|^2<\infty\right\}.}
$$
Weighted <Fourier coefficients> identify it isometrically with $\ell^2$, so it is a <Hilbert space> and complete. This definition includes negative orders and distributions. For nonnegative integer $s$ the norm is equivalent to the sum of squared L2 norms of weak derivatives through order $s$.
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