Remark

If $z = \sum_{i=1}^n z_i e_i$ then $\psi(z_1, \ldots, z_n) = |z|$ is a positively homogeneous continuous function on $\mathbf{R}^n$. Setting $\mu_i = e'_i \circ m = h_i \cdot |m|$, by definition (Ch. V, §5, No. 9) $\psi(\mu_1, \ldots, \mu_n) = \psi(h_1, \ldots, h_n) \cdot |m| = |h| \cdot |m| = |m|$.