Serre's Renorming Problem

Abstract

Let $\mathbb{K}$ be a non-archimedean complete valued field and let $(E,\|.\| )$ be a non-archimedean normed space over $\mathbb{K}$. Let $\left\vert \mathbb{K}^{\times }\right\vert :=\left\{ \left\vert \lambda \right\vert :\lambda \in \mathbb{K}\backslash \left\{ 0\right\} \right\} $ and $\left\Vert E^{\times}\right\Vert :=\left\{ \left\Vert x\right\Vert :x\in E\backslash \left\{0\right\} \right\}$. By Serre's Renorming Problem we mean the following question: can one for every non-archimedean normed space $E$ introduce a norm $\|.\| _{\bullet }$ on $E$ that is equivalent to the given norm in the sense that it determines the same topology and has the property $\|E^{\times }\|_{\bullet }=\left\vert \mathbb{K}^{\times }\right\vert $? An affirmative solution of this problem is presented.