Definition:Locally Uniform Convergence of Product/Definition 2

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 $\ds \prod_{n \mathop = 1}^\infty f_n$ converges locally uniformly it converges uniformly on every compact subspace of $T$.