Let $(V,\lVert\mathord{\cdot}\rVert)$ be a normed vector space. Let $(k^{\prime},\lvert\mathord{\cdot}\rvert^{\prime})$ be a complete valued field extension of $(k,\lvert\mathord{\cdot}\rvert)$. Set $V_{k^{\prime}}$ to be $V\otimes_{k}k^{\prime}$, which can be identified with $\mathrm{Hom}_{k}(\mathrm{Hom}_{k}(V,k),k^{\prime})$. The norm

$$\forall v^{\prime}\in V_{k^{\prime}},\ \lVert v^{\prime}\rVert_{k^{\prime}}:=\sup\Big\{\frac{\lvert(\ell\otimes 1)(v^{\prime})\rvert^{\prime}}{\lVert\ell\rVert^{\vee}},\ \ell\in V^{\vee}\Big\}$$

defined via this identification is called the scalar extension of $\lVert\mathord{\cdot}\rVert$.