# Definition:Locally Uniform Absolute Convergence of Product

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