Definition:Uniform Convergence of Product/Compact Space

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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.