Definition:Locally Uniform Convergence of Product/Definition 1

Definition
Let $T = \struct {S, \tau}$ be a weakly locally compact topological space.

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

Let $\sequence {f_n}$ 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.