# Definition:Locally Uniform Convergence of Product

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

### Definition 1

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

## Warning

As with **uniform convergence**, the notion of **locally uniform convergence** of a product is delicate.

Hence one usually restricts its definition to locally bounded mappings.

## Also see

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

- Results about
**locally uniform convergence of product**can be found**here**.