Definition:Local Uniform Absolute Convergence of Product

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Definition

Let $X$ be a topological space.

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

Let $\sequence {f_n}$ be a sequence of mappings $f_n: X \to \mathbb K$.


Then the infinite product $\displaystyle \prod_{n \mathop = 1}^\infty f_n$ converges locally uniformly absolutely if and only if every point of $X$ has a neighborhood on which it converges uniformly absolutely.


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