 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 isWeighted 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 $s$.