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.
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