A characterization of normed spaces (Q924243)

Previously, \textit{T. Oikhberg} and \textit{H. Rosenthal} [Rocky Mt. J. Math. 37, No. 2, 597--608 (2007; Zbl 1138.46006)] proved that if \(d\) is a translation invariant metric on \(X\) such that the multiplication by real scalars is continuous and every one-dimensional subspace of \(X\) is isometric to \(\mathbb R\), then \(d\) is induced by some norm on \(X\). This paper shows that this result can be obtained without a continuity assumption provided that the space is at least two-dimensional.