Definition:Uniform Absolute Convergence of Product

Definition

General 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.

Complex Functions

Let $X$ be a set.

Let $\sequence {f_n}$ be a sequence of bounded mappings $f_n: X \to \C$.

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 $\ds \prod_{n \mathop = 1}^\infty \paren {1 + \size {f_n} }$ 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.

Definition 3

The infinite product $\ds \prod_{n \mathop = 1}^\infty \paren {1 + f_n}$ converges uniformly absolutely if and only if there exists $n_0 \in \N$ such that:

$(1): \quad \map {f_n} x \ne -1$ for $n \ge n_0$ and $x \in X$

and:

$(2): \quad$ The series $\ds \sum_{n \mathop = n_0}^\infty \map \log {1 + f_n}$ is uniformly absolutely convergent.

Also presented as

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