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Let X be a linear space over a field K = R or C, equipped with a metric ρ. It is proved that ρ is induced by a norm provided it is translation invariant, real scalar "separately" continuous, such that every 1-dimensional subspace of X is isometric to K in its natural metric, and (in the complex case) ρ(x, y) = ρ(ix, iy) for any x, y ∈ X.doi:10.1216/rmjm/1181068769 fatcat:htpzyndlrzcwnale5md4az5igm