00KP Lemma 3.13. Let (K,|⋅|K)(K,\lvert\mathord{\cdot}\rvert_{K}) be a complete ultrametric valued field extension of (k,|⋅|)(k,\lvert\mathord{\cdot}\rvert). Then for any {rj}j∈{0,…,d}\{r_{j}\}_{j\in\{0,\dots,d\}} elements in ℝ+\mathbb{R}_{+}, one has inf∑j∈{0,…,d}κj=1maxj{|κj|K⋅rj}=minj∈{0,…,d}{rj},\inf_{\sum_{j\in\{0,\dots,d\}}\kappa_{j}=1}\max_{j}\Big\{\lvert\kappa_{j}\rvert_{K}\cdot r_{j}\Big\}=\min_{j\in\{0,\dots,d\}}\{r_{j}\}, where {κj}j∈{0,…,d}\{\kappa_{j}\}_{j\in\{0,\dots,d\}} are elements in KK.
00KQ Proof. On the one hand, let j0∈{0,…,d}j_{0}\in\{0,\dots,d\} be an index such that rj0r_{j_{0}} is minimal. By taking κj=0\kappa_{j}=0 for j≠j0j\neq j_{0} and κj0=1\kappa_{j_{0}}=1, one sees that inf∑j∈{0,…,d}κj=1maxj{|κj|K⋅rj}≤rj0=minj∈{0,…,d}{rj}.\inf_{\sum_{j\in\{0,\dots,d\}}\kappa_{j}=1}\max_{j}\Big\{\lvert\kappa_{j}\rvert_{K}\cdot r_{j}\Big\}\leq r_{j_{0}}=\min_{j\in\{0,\dots,d\}}\{r_{j}\}. On the other hand, by the ultrametricity of |⋅|K\lvert\mathord{\cdot}\rvert_{K}, if ∑jκj=1\sum_{j}\kappa_{j}=1, then there exist at least one j1∈{0,…,d}j_{1}\in\{0,\dots,d\} such that |κj1|≥1\lvert\kappa_{j_{1}}\rvert\geq 1, so inf∑j∈{0,…,d}κj=1maxj{|κj|K⋅rj}≥inf∑j∈{0,…,d}κj=1|κj1|K⋅rj1≥inf∑j∈{0,…,d}κj=1rj1=rj1≥minj∈{0,…,d}{rj}.\begin{split}\inf_{\sum_{j\in\{0,\dots,d\}}\kappa_{j}=1}\max_{j}\Big\{\lvert\kappa_{j}\rvert_{K}\cdot r_{j}\Big\}&\geq\inf_{\sum_{j\in\{0,\dots,d\}}\kappa_{j}=1}\lvert\kappa_{j_{1}}\rvert_{K}\cdot r_{j_{1}}\\ &\geq\inf_{\sum_{j\in\{0,\dots,d\}}\kappa_{j}=1}r_{j_{1}}=r_{j_{1}}\geq\min_{j\in\{0,\dots,d\}}\{r_{j}\}.\end{split} Hence the two sides are equal. ∎