Definition:Uniform Convergence of Product/Compact Space

Definition
Let $X$ be a compact topological space.

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 continuous mappings $f_n: X \to \mathbb K$.

Remark
In contrast to locally uniform convergence and absolutely uniform convergence, the notion of uniform convergence of a product is more delicate, which is why one usually restricts to continuous functions on a compact space.

Also see

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