Let \(U\subset \mathbb{R}^n\) be open. We denote the space of \(k\)-times continuously differentiable function with \(C^k(U)\).
Equipped with the norm
\begin{equation*} \lVert u\rVert_{C^k(U)} = \sum_{\lvert \alpha\rvert\le k} \lVert D^{\alpha}u\rVert_{\infty}j \end{equation*}\(C^k(U)\) is a normed vector space .