Definition:Locally Uniform Convergence of Product

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 $f_n: S \to \mathbb K$.

Remark
As with uniform convergence, the notion of locally uniform convergence of a product is delicate, which is why one usually restricts to locally bounded mappings.

Also see

 * Equivalence of Definitions of Locally Uniform Convergence of Product
 * Definition:Locally Uniform Absolute Convergence of Product