= Finite-dimensional non-Archimedean norm equivalence over a complete field
{title2=$c\max_i|a_i|\leq\|\sum_i a_i e_i\|\leq C\max_i|a_i|$}
Let $K$ be complete for a <Non-Archimedean absolute value> and let $V$ be a finite-dimensional <vector space> with a norm satisfying $\|av\|=|a|\|v\|$ and the <ultrametric inequality>. All such norms are equivalent to the maximum of coordinates in any <basis>. The upper bound is immediate. For the lower bound, induct on dimension. The span $W$ of all but the last <basis> vector is complete by induction, hence closed for the given norm. Thus $\delta=\inf_{w\in W}\|e_n-w\|>0$. For $v=w+ae_n$, $\|v\|\geq |a|\delta$; also $\|w\|\leq\max(\|v\|,|a|\|e_n\|)$. The induction bound on coordinates of $w$ now bounds all coordinates by a constant times $\|v\|$. This proof needs no <local compactness>.
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