Definition 1

The sequence $(x_n)_{n \in \mathbf{N}}$ is said to be multipliable in the normed algebra $A$ if the mapping $J \to p_J$ has a limit with respect to the filter of sections of the set $\mathfrak{F}(\mathbf{N})$ 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 by $\prod_{n \in \mathbf{N}} x_n$ (or simply $\prod_n x_n$); the $x_n$ are called the factors of this product.

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