Definition:Uniform Absolute Convergence of Product/General Definition

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 $f_n: X \to \mathbb K$.

Definition 1

The infinite product $\ds \prod_{n \mathop = 1}^\infty \paren {1 + f_n}$ converges uniformly absolutely if and only if the sequence of partial products of $\ds \prod_{n \mathop = 1}^\infty \paren {1 + \norm {f_n} }$ converges uniformly.

Definition 2

The infinite product $\ds \prod_{n \mathop = 1}^\infty \paren {1 + f_n}$ converges uniformly absolutely if and only if the series $\ds \sum_{n \mathop = 1}^\infty f_n$ converges uniformly absolutely.