Definition:Locally Uniform Convergence of Product

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.