Remark 1

When $A$ is a commutative algebra, Definition 1 is identical with that given in Chapter III, § 5, no. 1, Remark 3; but when $A$ is not commutative, the order structure of the index set $\mathbf{N}$ is essentially involved in Definition 1. If $\sigma$ is an arbitrary permutation of $\mathbf{N}$, we cannot in general assert that the sequence $(x_{\sigma(n)})$ is multipliable if the sequence (x_n) is multipliable; and if both sequences are multipliable, their products will in general be different.