Definition:Uniform Convergence of Product

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Definition for Mappings to a Field

Let $X$ be a set.

Let $\struct {\mathbb K, \norm {\,\cdot\,} }$ be a valued field.

Let $\mathbb K$ be complete.

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


Definition 1

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

the sequence of partial products of $\ds \prod_{n \mathop = n_0}^\infty f_n$ converges uniformly
$\ds \inf_{x \mathop \in X} \norm {\prod_{n \mathop = n_0}^\infty \map {f_n} x} \ne 0$.


Definition 2

The infinite product $\ds \prod_{n \mathop = 1}^\infty f_n$ converges uniformly if and only if it converges in the normed algebra of bounded mappings $X \to \mathbb K$.


Definition for Continuous Mappings on a Compact Space

Let $X$ be a compact topological space.

Let $\struct {\mathbb K, \norm{\,\cdot\,}}$ be a complete valued field.

Let $\sequence {f_n}$ be a sequence of continuous mappings $f_n: X \to \mathbb K$.


The infinite product $\ds \prod_{n \mathop = 1}^\infty f_n$ converges uniformly if and only if there exists $n_0 \in \N$ such that the sequence of partial products of $\ds \prod_{n \mathop = n_0}^\infty f_n$ converges uniformly and is nonzero.


Definition for Mappings to a Normed Algebra

Let $X$ be a set.

Let $\mathbb K$ be a complete division ring with norm $\norm {\, \cdot \,}_{\mathbb K}$.

Let $A$ be a normed unital algebra over $\mathbb K$ with norm $\norm {\, \cdot \,}$.

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


The infinite product $\ds \prod_{n \mathop = 1}^\infty f_n$ converges uniformly if and only if it converges in the normed algebra of bounded mappings $X \to A$.


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