For $v=a_{1}e_{1}+\cdots+a_{r}e_{r}\in V$ and $a\in k^{\times}$,

so that $\|v\|_{\mathscr{V}}=\max\{|a_{1}|,\ldots,|a_{r}|\}$. ∎

Clearly we may assume that $G\not=\{0\}$, so that $G\cap\mathbb{R}_{>0}\not=\emptyset$. We set $\delta=\inf(G\cap\mathbb{R}_{>0})$. If $\delta\in G\cap\mathbb{R}_{>0}$, then $G=\mathbb{Z}\delta$. Indeed, for $g\in G$, let $n$ be an integer such that $n\leq g/\delta<n+1$. Thus $0\leq g-n\delta<\delta$, and hence $g=n\delta$. Therefore, $G$ is discrete.

Next we assume that $\delta\not\in G\cap\mathbb{R}_{>0}$. Then there is a sequence $\{\delta_{n}\}_{n=1}^{\infty}$ in $G\cap\mathbb{R}_{>0}$ such that $\delta_{n}>\delta_{n+1}$ for all $n$ and $\lim_{n\to\infty}\delta_{n}=\delta$. If we set $a_{n}=\delta_{n}-\delta_{n+1}$, then $a_{n}\in G\cap\mathbb{R}_{>0}$ and $\lim_{n\to\infty}a_{n}=0$. For an open interval $(\alpha,\beta)$ of $\mathbb{R}$ ($\alpha<\beta$), we choose $a_{n}$ and an integer $m$ such that $a_{n}<\beta-\alpha$ and $m<\beta/a_{n}\leq m+1$. Then we have $ma_{n}<\beta$ and

so that $ma_{n}\in(\alpha,\beta)\cap G$. Thus $G$ is dense. ∎

The first assertion is obvious because, for $a\in k^{\times}$, $av\in\mathscr{V}$ if and only if $\|v\|\leq|a|^{-1}$.

For $v\in V$, let $a\in k^{\times}$ with $av\in\mathscr{V}$. Then $\|av\|\leq 1$, that is, $\|v\|\leq|a|^{-1}$, and hence $\|v\|\leq\|v\|_{\mathscr{V}}$.

Finally we consider the second inequality, that is, $\|v\|_{\mathscr{V}}\leq|\alpha|\|v\|$ for $v\in V$. Clearly we may assume that $v\not=0$. As $|\alpha|^{-1}<1$, there is $\epsilon>0$ with $|\alpha|^{-1}e^{\epsilon}<1$. By the first assertion, we can choose $b\in k^{\times}$ such that $\|v\|\leq|b|\leq e^{\epsilon}\|v\|_{\mathscr{V}}$. If $\|v\|<|b\alpha^{-1}|$, then

Thus $1\leq e^{\epsilon}|\alpha|^{-1}$. This is a contradiction, so that $\|v\|\geq|b\alpha^{-1}|$. Therefore,

as required. ∎

(1) Let $(e^{\prime}_{1},\ldots,e^{\prime}_{r})$ be an orthogonal basis of $V$ with respect to $\|\raisebox{1.72218pt}{.}\|_{\mathscr{V}}$ (cf. Proposition 1.3). As $|\raisebox{1.72218pt}{.}|$ is discrete, there is $\lambda_{i}\in k^{\times}$ with $|\lambda_{i}|=\|e^{\prime}_{i}\|_{\mathscr{V}}$. If we set $e_{i}=\lambda_{i}^{-1}e^{\prime}_{i}$ for $i=1,\ldots,r$, then $(e_{1},\ldots,e_{r})$ forms an orthonormal basis of $V$ with respect to $\|\raisebox{1.72218pt}{.}\|_{\mathscr{V}}$. Therefore,

Thus we have (1) because $\mathfrak{o}_{k}$ is noetherian.

(2) follows from Lemma 1.16. ∎

By Lemma 1.15, we can find a sequence $\{\beta_{n}\}_{n=1}^{\infty}$ such that $|\beta_{n}|>1$ and $\lim_{n\to\infty}|\beta_{n}|=1$. On the other hand, by Lemma 1.16,

Therefore the assertion follows. ∎

Let $(e_{1},\ldots,e_{r})$ be an $e^{-\epsilon/2}$-orthogonal basis of $V$ with respect to $\|\raisebox{1.72218pt}{.}\|$ (cf. Proposition 1.3). As $\|\raisebox{1.72218pt}{.}\|=\|\raisebox{1.72218pt}{.}\|_{\mathscr{V}}$ by Proposition 1.18, we can find $\lambda_{i}\in k^{\times}$ such that $\|e_{i}\|\leq|\lambda_{i}|\leq e^{\epsilon/2}\|e_{i}\|$ for each $i$. We set $\omega_{i}:=\lambda_{i}^{-1}e_{i}$ ($i=1,\ldots,r$) and $\mathscr{V}^{\prime}:=\mathfrak{o}_{k}\omega_{1}+\cdots+\mathfrak{o}_{k}\omega_{r}$. Note that $\omega_{i}\in\mathscr{V}$ for all $i$, that is, $\mathscr{V}^{\prime}$ is a sub-lattice of $\mathscr{V}$ and $\mathscr{V}^{\prime}$ is finitely generated over $\mathfrak{o}_{k}$. For $c_{1},\ldots,c_{r}\in k$, by Proposition 1.14,

so that we have $\|\raisebox{1.72218pt}{.}\|_{\mathscr{V}^{\prime}}\leq e^{\epsilon}\|\raisebox{1.72218pt}{.}\|$. ∎