Definition:Locally Uniform Convergence of Product/Definition 1

Definition
Let $T = \left({S, \tau}\right)$ be a weakly locally compact topological space.

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

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

The infinite product $\displaystyle \prod_{n \mathop = 1}^\infty f_n$ converges locally uniformly every point of $T$ has a compact neighborhood on which it converges uniformly.