Definition:Uniform Convergence of Product

Definition
Let $X$ be a set.

Let $\mathbb K$ be a field with absolute value $\left\vert{\, \cdot \,}\right\vert$.

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

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
 * $\displaystyle \inf_{x \mathop \in X} \left\vert{ \prod_{n \mathop = n_0}^\infty f_n \left({x}\right)}\right\vert \ne 0$.

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 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
 * Logarithm of Infinite Product of Complex Functions
 * Uniform Product of Continuous Functions is Continuous
 * Uniform Product of Analytic Functions is Analytic