A precision on the concept of strict convexity in non-Archimedean analysis (Q2019316)

A non-Archimedean normed space \(X\) over a field \(K\) is called strictly convex if \(|2|= 1\) and for any pair of vectors \(x, y \in X\), \(\|x\|= \|y\|= \|x + y\|\) ensures that \(x = y\). \textit{A. Kubzdela} [Nonlinear Anal., Theory Methods Appl., Ser. A, Theory Methods 75, No. 4, 2060--2068 (2012; Zbl 1248.46050)] showed that the only non-Archimedean strictly convex space over a field with a non-trivial valuation is the zero space. The authors prove that the only non-Archimedean strictly convex spaces are the zero space and the one-dimensional linear space over \(\mathbb{Z}/3\mathbb{Z}\) with any of its trivial norms.