Definition:Locally Uniform Convergence of Product

Definition
Let $X$ be a weakly locally compact topological space.

Let $\mathbb K$ be a field with absolute value $\left\vert{\, \cdot \,}\right\vert$.

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

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 or continuous mappings.

Note that Continuous Implies Locally Bounded.

Also see

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