# 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 $f_n: S \to \mathbb K$.

### Definition 1

The infinite product $\displaystyle \prod_{n \mathop = 1}^\infty f_n$ **converges locally uniformly** if and only if every point of $T$ has a compact neighborhood on which it converges uniformly.

### Definition 2

The infinite product $\displaystyle \prod_{n \mathop = 1}^\infty f_n$ **converges locally uniformly** if and only if it converges uniformly on every compact subspace of $T$.

## 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.