Definition:Uniform Absolute Convergence of Product/Complex Functions/Definition 3

Definition
Let $X$ be a set.

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

The infinite product $\displaystyle \prod_{n \mathop = 1}^\infty \left({1 + f_n}\right)$ converges uniformly absolutely there exists $n_0 \in \N$ such that:
 * $(1): \quad f_n \left({x}\right) \ne -1$ for $n \ge n_0$ and $x \in X$

and:
 * $(2): \quad$ The series $\displaystyle \sum_{n \mathop = n_0}^\infty \log \left({1 + f_n}\right)$ is uniformly absolutely convergent.