Remark

When the rank of the system (21) is infinite, the condition of Theorem 8 is no longer sufficient. For example, suppose that the $x_i^*$ are the coordinate forms on the space $E = K_s^{(I)}$, $I$ being infinite (§ 2, no. 6); as the $x_i^*$ are linearly independent, the condition of Theorem 8 holds for every family $(\eta_i)$ but the system (21) only has solutions if the family $(\eta_i)$ has finite support.

A system (21) is always of finite rank if it has only a finite number of equations and its rank is then at most equal to the number of equations (no. 2, Proposition 1). Similarly, if $E$ is of finite dimension $n$ (which for a system (22) corresponds to the case where there is only a finite number n of unknowns), its dual $E^*$ is of dimension $n$ and hence the rank of system (21) is at most equal to $n$ (no. 3, Corollary 4 to Proposition 4). From this we deduce: