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.

The notation and terminology used in this paper are introduced in the following papers: [20], [23], [4], [24], [6], [5], [19], [3], [10], [1], [18], [7], [21], [22], [11], [8], [9], [25], [13], [15], [16], [17], [12], [14], and [2]. For simplicity, we adopt the following rules: n, k denote natural numbers, x, X, Z denote sets, g, r denote real numbers, S denotes a real normed space, r1 denotes ...

The articles [16], [19], [20], [2], [21], [4], [9], [3], [1], [11], [15], [5], [17], [18], [10], [7], [8], [6], [13], [22], [12], and [14] provide the notation and terminology for this paper. We use the following convention: n is a natural number, x, X, X1 are sets, and s, r, p are real numbers. Let S, T be 1-sorted structures. A partial function from S to T is a partial function from the carri...