We call a linear map \(u\colon \mathcal{S}(\mathbb{R}^d)\to \mathbb{R}\) is called tempered distribution if there is an number \(k\in \mathbb{N}\) and a constant \(C_k>0\) such that for every Schwartz function \(\varphi\in \mathcal{S}(\mathbb{R}^d)\)
\[ \lvert u(\varphi)\rvert \le C_k \sup_{\lvert \alpha\rvert+\lvert \beta\rvert\le k} \lVert \varphi\rVert_{\alpha,\beta}. \]The space of tempered distributions is denoted by \(\mathcal{S}'(\mathbb{R}^d)\).