Definition:Uniform Convergence of Product/Compact Space

Definition

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.

This article, or a section of it, needs explaining. In particular: What does "nonzero" mean? To be consistent with Definition:Uniform Convergence of Product, it should be "nowhere zero" You can help $\mathsf{Pr} \infty \mathsf{fWiki}$ by explaining it. To discuss this page in more detail, feel free to use the talk page. When this work has been completed, you may remove this instance of {{Explain}} from the code.