Proposition 2

A family $(z_i)$ of complex numbers $z_i = r_i (\cos \theta_i + i \sin \theta_i)$ is multipliable if and only if the family $(r_i)$ of absolute values of the $z_i$ is multipliable in $\mathbf{R}_+^*$ and the family $(\theta_i)$ of amplitudes of the $z_i$ is summable in the group of angles $\mathfrak{A}$.

In view of the structure of the group $\mathbf{C}^*$ ($\S$ 1, no. 3, Proposition 1), the proposition is an immediate consequence of Proposition 4 of Chapter III, $\S$ 5, no. 4.

If we map each angle $\theta$ to that one of its measures (to any given base $a$) which belongs to the interval $] - \frac{1}{2} a, \frac{1}{2} a ]$, we have a local isomorphism of $\mathfrak{A}$ with $\mathbf{R}$ ($\S$ 2, no. 2); since $\lim \theta_i = 0$ with respect to the filter of complements of finite subsets of $I$, we may replace the condition (in the statement of Proposition 2) that the family $\theta_i$ should be summable in $\mathfrak{A}$ by the condition that the family $(t_i)$ of measures of the angles $\theta_i$ which belong to $] - \frac{1}{2} a, \frac{1}{2} a ]$ should be summable in $\mathbf{R}$.

The following theorem gives another criterion for a family of complex numbers, put in the form $(1 + u_i)$, to be multipliable in $\mathbf{C}^*$. (It generalizes Theorem 4 of Chapter IV, $\S$ 7, no. 4; see also Chapter IX, Appendix, no. 2, Proposition 1):