\begin{definition}[Definition:Locally Uniform Convergence of Product/Definition 1]
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''' {{iff}} every point of $T$ has a compact neighborhood on which it converges uniformly.
Category:Definitions/Locally Uniform Convergence of Product
\end{definition}