Serre's Renorming Problem
Copyright policy )Serre's Renorming Problem
The article is devoted to Serre's renorming problem. The problem is about a non-archimedean normed field $K$ complete relative to the metric induced by its norm $|.|$. Does each non-archimedean normed space $E$ have a norm $\|.\|^*$ which is equivalent to the given norm $\|.\|$ on $E$, such that $\|E\|^*=|K|$? \textit{J.-P. Serre} [Publ. Math., Inst. Hautes Étud. Sci. 12, 69--85 (1962; Zbl 0104.33601)] solved it for discretely valued $K$ in 1962. \textit{A.~C.~M. von Rooij} [``Notes on $p$-adic Banach spaces'', Katholieke Universiteit Nijmegen, Report 7633 (1976)] solved it for densely valued $K$ in 1976. This article answers affirmatively the remaining cases.
Serre's renorming problem, non-Archimedean normed spaces, Functional analysis over fields other than \(\mathbb{R}\) or \(\mathbb{C}\) or the quaternions; non-Archimedean functional analysis
Serre's renorming problem, non-Archimedean normed spaces, Functional analysis over fields other than \(\mathbb{R}\) or \(\mathbb{C}\) or the quaternions; non-Archimedean functional analysis
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