We assume that the absolute value $|\raisebox{1.72218pt}{.}|$ of $k$ is trivial. Let $(V,\|\raisebox{1.72218pt}{.}\|)$ be a finite-dimensional normed vector space over $(k,|\raisebox{1.72218pt}{.}|)$. Then we have the following:

1. (1)

   The set $\{\|v\|\mid v\in V\}$ is a finite set.

2. (2)

   Let $k^{\prime}$ be a field and $|\raisebox{1.72218pt}{.}|^{\prime}$ a complete and non-trivial absolute value of $k^{\prime}$ such that $k\subseteq k^{\prime}$ and $|\raisebox{1.72218pt}{.}|^{\prime}$ is an extension of $|\raisebox{1.72218pt}{.}|$. Let $\mathfrak{o}_{k^{\prime}}$ be the valuation ring of $(k^{\prime},|\raisebox{1.72218pt}{.}|^{\prime})$ and $\mathfrak{m}_{k^{\prime}}$ the maximal ideal of $\mathfrak{o}_{k^{\prime}}$. We assume the following:

   1. (i)

      The natural map $k\to\mathfrak{o}_{k^{\prime}}$ induces an isomorphism $k\overset{\sim}{\longrightarrow}\mathfrak{o}_{k^{\prime}}/\mathfrak{m}_{k^{\prime}}$.

   2. (ii)

      If an equation $|a^{\prime}|^{\prime}=\|v\|/\|v^{\prime}\|$ holds for some $a^{\prime}\in{k^{\prime}}^{\times}$ and $v,v^{\prime}\in V\setminus\{0\}$, then $\|v\|=\|v^{\prime}\|$.

   Let $\|\raisebox{1.72218pt}{.}\|^{\prime}$ be a norm of $V_{k^{\prime}}:=V\otimes_{k}k^{\prime}$ over $(k^{\prime},|\raisebox{1.72218pt}{.}|^{\prime})$ such that $\|v\|=\|v\otimes 1\|^{\prime}$ for all $v\in V$. If $(e_{1},\ldots,e_{r})$ is an orthogonal basis of $(V,\|\raisebox{1.72218pt}{.}\|)$, then $(e_{1},\ldots,e_{r})$ forms an orthogonal basis of $(V_{k^{\prime}},\|\raisebox{1.72218pt}{.}\|^{\prime})$. In particular, $\|\raisebox{1.72218pt}{.}\|^{\prime}=\|\raisebox{1.72218pt}{.}\|_{k^{\prime}}$.