Let $k^{\prime}$ be an extension field of $k$, and let $|\raisebox{1.72218pt}{.}|^{\prime}$ be a complete absolute value of $k^{\prime}$ which is an extension of $|\raisebox{1.72218pt}{.}|$. We set $V_{k^{\prime}}:=V\otimes_{k}k^{\prime}$. Identifying $V_{k^{\prime}}$ with

$$\mathrm{Hom}_{k}(\mathrm{Hom}_{k}(V,k),k^{\prime}),$$

we can give a norm $\|\raisebox{1.72218pt}{.}\|_{k^{\prime}}$ of $V_{k^{\prime}}$, that is,

$$\|v^{\prime}\|_{k^{\prime}}=\sup\left\{\frac{|(\phi\otimes 1)(v^{\prime})|^{\prime}}{\|\phi\|^{\vee}}\,\Big|\,\phi\in V^{\vee}\right\}.$$

The norm $\|\raisebox{1.72218pt}{.}\|_{k^{\prime}}$ is called the scalar extension of $\|\raisebox{1.72218pt}{.}\|$. Note that $\|v\otimes 1\|_{k^{\prime}}=\|v\|$ for $v\in V$. Indeed, by Corollary 1.7,

$$\|v\otimes 1\|_{k^{\prime}}=\sup\left\{\frac{|\phi(v)|}{\|\phi\|^{\vee}}\,\Big|\,\phi\in V^{\vee}\right\}=\|v\|.$$