When are maps preserving semi-inner products linear?
We observe that every map between finite-dimensional normed spaces of the same dimension that respects fixed semi-inner products must be automatically a linear isometry. Moreover, we construct a uniformly smooth Banach space and a mapping that respects the semi-inner products, yet it is non-linear. This demonstrates that there is no immediateextension of the former result to infinite dimensions, even under an extra hypothesis of uniform smoothness. Regrettably, this result refutes a claim concerning smooth spaces from a recent paper [Aequationes Math. (2020)].