Definition 1

One says that the sequence $(x_n)_{n \geq N}$ is multipliable in the normed algebra $A$ if the mapping $J \mapsto p_J$ has a limit following the filter of sections of the set $\mathscr{F}(\mathbf{N})$ of finite subsets of $\mathbf{N}$, ordered by the relation $\subset$; this limit is called the product of the sequence $(x_n)_{n \in \mathbf{N}}$, and is denoted $\prod_{n \in \mathbf{N}} x_n$ (or simply $\prod_n x_n$); the $x_n$ are called the factors of this product.

Def. 1 is equivalent to the following one: the sequence $(x_n)$ is multipliable and has product $p$ if, for every $\varepsilon > 0$, there exists a finite subset $J_0$ of $\mathbf{N}$ such that, for every finite subset $J \supset J_0$ of $\mathbf{N}$, one has $\|p_J - p\| \leq \varepsilon$.