Let $T^n$ be the $n$-torus, and define $$H^s(T^n):=L^2_s(\mathbb{Z}^n),$$ where given $\xi \in \mathbb{Z}^n$, $\mu_s(\{\xi\}):=(1+|\xi|^2)^s$.
This is essentially the definition given in Warner. However, it is not clear to me how one would go on to define $H^s(\Omega)$, $\Omega \subset \mathbb{R}^n$ open, from this. My quesion is exactly this: how do we do this? Are there references?