Let \(k\in \mathbb{N}\). The Sobolev space \(H^k(\mathbb{Z}^d)\) is defined by
\[ H^k(\mathbb{Z}^d)=\{f\in \ell^2(\mathbb{Z}^d)\mid \partial^\alpha f\in \ell^2(\mathbb{Z}^d), \lvert \alpha\rvert\le k\}. \]The map
\begin{equation*} \lVert f\rVert_{H^k(\mathbb{Z}^d)}^2=\sum_{\lvert \alpha\rvert\le k} \lVert \partial^\alpha f\rVert_{\ell^2(\mathbb{Z}^d)}^2, \end{equation*}is a norm on \(H^k(\mathbb{Z}^d)\).