Definition:Locally Uniform Convergence of Product/Definition 2

Definition
Let $X$ be a weakly locally compact topological space.

Let $\mathbb K$ be a field with absolute value $\left\vert{\, \cdot \,}\right\vert$.

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

The infinite product $\displaystyle \prod_{n \mathop = 1}^\infty f_n$ converges locally uniformly it converges uniformly on every compact subspace of $X$.