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

Definition
Let $X$ be a set.

Let $\struct {\mathbb K, \norm{\,\cdot\,}}$ be a valued field.

Let $\sequence {f_n} $ 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 $\displaystyle \prod_{n \mathop = 1}^\infty(1+ \norm{f_n})$ converges uniformly.