Absolute Value of Uniformly Convergent Product

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Theorem

Let $X$ be a compact topological space.

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

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

Let the infinite product $\displaystyle \prod_{n \mathop = 1}^\infty f_n$ converge uniformly to $f$.


Then $\displaystyle \prod_{n \mathop = 1}^\infty \norm{f_n}$ converges uniformly to $ \norm{f}$.


Proof


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