Proof. [0260]

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Proof.

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

	$\displaystyle av\in\mathscr{V}$	$\displaystyle\Longleftrightarrow\text{$aa_{i}\in\mathfrak{o}_{k}$ for all $i=1,\ldots,r$}$
		$\displaystyle\Longleftrightarrow\text{$\|a_{i}\|\leq\|a\|^{-1}$ for all $i=1,\ldots,r$}$
		$\displaystyle\Longleftrightarrow\text{$\max\{\|a_{1}\|,\ldots,\|a_{r}\|\}\leq\|a\|^{-1}$},$

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