(1) Let $(e_{1},\ldots,e_{r})$ be an orthogonal basis of $(V,\|\raisebox{1.72218pt}{.}\|)$ (cf. Proposition 1.3). Then

$$\|a_{1}e_{1}+\cdots+a_{r}e_{r}\|=\max\{|a_{1}|\|e_{1}\|,\ldots,|a_{r}|\|e_{r}\|\}$$

for all $a_{1},\ldots,a_{r}\in k$, so that

$$\|a_{1}e_{1}+\cdots+a_{r}e_{r}\|\in\{0,\|e_{1}\|,\ldots,\|e_{r}\|\}.$$

(2) First we assume that

$$\|e_{1}\|=\cdots=\|e_{r}\|=c.$$

Then, for any $v\in V$,

$$\|v\|=\begin{cases}c&\text{if $v\not=0$},\\ 0&\text{if $v=0$}.\end{cases}$$

Let us see that

$$\|a^{\prime}_{1}e_{1}+\cdots+a^{\prime}_{r}e_{r}\|^{\prime}=c\max\{|a^{\prime}_{1}|^{\prime},\ldots,|a^{\prime}_{r}|^{\prime}\}$$

for $a^{\prime}_{1},\ldots,a^{\prime}_{r}\in k^{\prime}$. Clearly we may assume that

$$(a^{\prime}_{1},\ldots,a^{\prime}_{r})\not=(0,\ldots,0).$$

We set $\gamma:=\max\{|a^{\prime}_{1}|^{\prime},\ldots,|a^{\prime}_{r}|^{\prime}\}$. We fix $\omega\in k^{\prime}$ with $|\omega|^{\prime}=\gamma$. By the assumption (i), for each $j=1,\ldots,r$, we can find $a_{j}\in k$ and $b^{\prime}_{j}\in k^{\prime}$ such that

$$a^{\prime}_{j}=a_{j}\omega+b^{\prime}_{j}\quad\text{and}\quad|b^{\prime}_{j}|^{\prime}<\gamma.$$

Note that

$$a^{\prime}_{1}e_{1}+\cdots+a^{\prime}_{r}e_{r}=\omega\left(\sum\nolimits_{j=1}^{r}a_{j}e_{j}\right)+b^{\prime}_{1}e_{1}+\cdots+b^{\prime}_{r}e_{r}.$$

Moreover, as $\sum_{j=1}^{r}a_{j}e_{j}\not=0$, we have

	$\displaystyle\left\|\omega\left(\sum\nolimits_{j=1}^{r}a_{j}e_{j}\right)\right\|^{\prime}$	$\displaystyle=\gamma\left\|\sum\nolimits_{j=1}^{r}a_{j}e_{j}\right\|=c\gamma$
and
	$\displaystyle\|b^{\prime}_{1}e_{1}+\cdots+b^{\prime}_{r}e_{r}\|^{\prime}$	$\displaystyle\leq c\max\{|b^{\prime}_{1}|^{\prime},\ldots,|b^{\prime}_{r}|^{\prime}\}<c\gamma.$

Therefore,

$$\|a^{\prime}_{1}e_{1}+\cdots+a^{\prime}_{r}e_{r}\|^{\prime}=c\gamma=c\max\{|a^{\prime}_{1}|^{\prime},\ldots,|a^{\prime}_{r}|^{\prime}\}.$$

In general, we take positive numbers $c_{1}<\cdots<c_{b}$ and non-empty subsets $I_{1},\ldots,I_{b}$ of $\{1,\ldots,r\}$ such that $\{\|e_{l}\|\mid l\in I_{s}\}=\{c_{s}\}$ for $s=1,\ldots,b$ and $I_{1}\cup\cdots\cup I_{b}=\{1,\ldots,r\}$. Note that $I_{s}\cap I_{s^{\prime}}=\emptyset$ for $s\not=s^{\prime}$. Let us consider

$$x=a^{\prime}_{1}e_{1}+\cdots+a^{\prime}_{r}e_{r}=\sum_{s=1}^{b}x_{s}\in V_{k^{\prime}}\quad(a^{\prime}_{1},\ldots,a^{\prime}_{r}\in k^{\prime}),$$

where $x_{s}=\sum_{l\in I_{s}}a^{\prime}_{l}e_{l}$. Note that $(e_{l})_{l\in I_{s}}$ forms an orthogonal basis of $\bigoplus_{l\in I_{s}}ke_{l}$ and $\|e_{l}\|=c_{s}$ for all $l\in I_{s}$. Therefore, by the above observation,

$$\left\|x_{s}\right\|^{\prime}=c_{s}\max_{l\in I_{s}}\{|a^{\prime}_{l}|^{\prime}\}=\max_{l\in I_{s}}\{\|a^{\prime}_{l}e_{l}\|^{\prime}\},$$

so that it is sufficient to see that

$$\|x\|^{\prime}=\max_{s=1,\ldots,b}\left\{\left\|x_{s}\right\|^{\prime}\right\}.$$

Clearly we may assume that $x\not=0$. We set

$$\Sigma:=\left\{s\in\{1,\ldots,b\}\mid x_{s}\not=0\right\}.$$

For $s,s^{\prime}\in\Sigma$ with $s\not=s^{\prime}$, we have $\|x_{s}\|^{\prime}\not=\|x_{s^{\prime}}\|^{\prime}$. Indeed, we choose $l_{s}\in I_{s}$ and $l_{s^{\prime}}\in I_{s^{\prime}}$ with $\left\|x_{s}\right\|^{\prime}=\|a^{\prime}_{l_{s}}e_{l_{s}}\|^{\prime}$ and $\left\|x_{s^{\prime}}\right\|^{\prime}=\|a^{\prime}_{l_{s^{\prime}}}e_{l_{s^{\prime}}}\|^{\prime}$. If $\|x_{s}\|^{\prime}=\|x_{s^{\prime}}\|^{\prime}$, then

$$\left|a^{\prime}_{l_{s}}/a^{\prime}_{l_{s^{\prime}}}\right|^{\prime}=\|e_{l_{s^{\prime}}}\|/\|e_{l_{s}}\|,$$

so that, by the assumption (ii), $\|e_{l_{s^{\prime}}}\|=\|e_{l_{s}}\|$, which is a contradiction. Therefore,

$$\|x\|^{\prime}=\left\|\sum\nolimits_{s\in\Sigma}x_{s}\right\|^{\prime}=\max_{s\in\Sigma}\{\|x_{s}\|^{\prime}\}=\max_{s=1,\ldots,b}\{\|x_{s}\|^{\prime}\},$$

as required. ∎