Definition:Uniform Convergence of Product

Definition for Mappings to a Field
Let $X$ be a set.

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

Let $\mathbb K$ be complete.

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

Definition 1
The infinite product $\displaystyle \prod_{n \mathop = 1}^\infty f_n$ converges uniformly there exists $n_0 \in \N$ such that:
 * The sequence of partial products of $\displaystyle \prod_{n \mathop = n_0}^\infty f_n$ converges uniformly
 * $\displaystyle \inf_{x \mathop \in X} \left\vert{ \prod_{n \mathop = n_0}^\infty f_n \left({x}\right)}\right\vert \ne 0$.

Definition 2
The infinite product $\displaystyle \prod_{n \mathop = 1}^\infty f_n$ converges uniformly it converges in the normed algebra of bounded mappings $X \to \mathbb K$.

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 $\|\cdot\|$.

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

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

Remark
As with locally uniform convergence, the notion of uniform convergence of a product is more delicate, which is why one usually restricts to bounded mappings or continuous mappings on a compact space.

Note that Continuous Function on Compact Subspace of Euclidean Space is Bounded.

Desirable consequences hereof are:
 * Partial Products of Uniformly Convergent Product Converge Uniformly
 * Uniform Convergence of Product Does Not Depend on Finite Number of Factors

Also see

 * Equivalence of Definitions of Uniformly Convergent Product
 * Definition:Uniform Absolute Convergence of Product
 * Definition:Locally Uniform Convergence of Product, a concept that is more often used
 * Uniform Product of Continuous Functions is Continuous
 * Uniform Product of Analytic Functions is Analytic