Definition:Uniform Absolute Convergence of Product/General Definition
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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.