Remark 1

When $A$ is a commutative algebra, def. 1 is identical with that which was given in III, p. 37 (Remark 3); but when $A$ is not commutative, the order structure of the set of indices $\mathbf{N}$ intervenes in an essential way in def. 1; if $\sigma$ is any permutation of $\mathbf{N}$, nothing permits one to assert in general that the sequence $(\mathbf{x}_{\sigma(n)})$ is multipliable when the sequence $(\mathbf{x}_n)$ is; moreover, when these two sequences are multipliable, their products are in general different.