Definition:Uniform Absolute Convergence of Product/General Definition/Definition 1

Definition
Let $X$ be a set.

Let $\mathbb K$ be a field with absolute value $\left\vert{\, \cdot \,}\right\vert$.

Let $\left \langle {f_n} \right \rangle$ be a sequence of bounded mappings $f_n: X \to \mathbb K$.

The infinite product $\displaystyle \prod_{n \mathop = 1}^\infty \left({1 + f_n}\right)$ converges uniformly absolutely the sequence of partial products of $\displaystyle \prod_{n \mathop = 1}^\infty(1+|f_n|)$ converges uniformly.