Definition:Uniform Absolute Convergence of Product

Definition
Let $S$ 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 mappings $f_n: S \to \mathbb K$.

Then the infinite product $\displaystyle \prod_{n \mathop = 1}^\infty \left({1 + f_n}\right)$ converges uniformly absolutely the product $\displaystyle \prod_{n \mathop = 1}^\infty \left({1 + \left\lvert{f_n}\right\rvert}\right)$ converges uniformly.

Also presented as
The infinite product $\displaystyle \prod_{n \mathop = 1}^\infty f_n$ is uniformly absolutely convergent  $\displaystyle \prod_{n \mathop = 1}^\infty \left({1 + \left\lvert{f_n - 1}\right\rvert}\right)$ is uniformly convergent.

Also see

 * Definition:Absolute Convergence of Product
 * Definition:Uniform Convergence of Product
 * Definition:Local Uniform Absolute Convergence of Product
 * Uniformly Absolutely Convergent Product is Uniformly Convergent