A metric characterization of normed linear spaces (Q2477983)

The authors prove that a metric \(\rho\) on a linear space \(X\) is induced by a norm provided that \(\rho\) is translation invariant, real continuous (i.e., for any \(x\in X\), the map \([0,1]\to X:t\mapsto tx\) is continuous), every one-dimensional affine subspace of \(X\) is isometric to the field of scalars, and (in the complex case) \(\rho(x,y)=\rho(ix,iy)\). An example is presented showing that the assumptions of this statement cannot be significantly weakened. The exposition is fairly self-contained; only the Mazur--Ulam theorem about isometries in linear normed spaces is used.