(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. ∎