Corollary 2.14 (00HJ) · ScalingStacks

00HJ

Corollary 2.14. Let $(V,\lVert\mathord{\cdot}\rVert_{V})$ be a finite-dimensional ultrametrically normed vector space over $k$ . Suppose that $(k,|\cdot|)$ is discretely valued. If $\{e_{i}\}_{i\in\{1,\dots,n\}}$ is a basis of $V$ such that $\{\log\lVert e_{i}\rVert\}_{i\in\{1,\dots,n\}}$ are $\mathbb{Q}$ -independent in $\mathbb{R}/H(k,\lvert\mathord{\cdot}\rvert)$ , then $\{e_{i}\}_{i\in\{1,\dots,n\}}$ is an orthogonal basis.

00HK

Proof. For any $f=(f_{1},\dots,f_{n})\in(k^{\times})^{n}$ , the numbers $\{\lvert f_{i}\rvert\lVert e_{i}\rVert\}_{i\in\{1,\dots,r\}}$ are distinct, otherwise there exist $i,j\in\{1,\dots,n\},i\neq j$ such that

\log\lVert e_{i}\rVert-\log\lVert e_{j}\rVert=\log\Big|\frac{f_{i}}{f_{j}}\Big|\in\log\lvert k^{\times}\rvert

which contradicts the assumption of $\mathbb{Q}$ -independence. Hence

\Big\|\sum_{i\in\{1,\dots,n\}}f_{i}e_{i}\Big\|=\max_{0\leq i\leq n}\lVert f_{i}e_{i}\rVert

by Lemma 2.13. ∎