Definition:Uniform Absolute Convergence of Product/General Definition/Definition 2

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