Definition:Locally Uniform Convergence of Product

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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.


Also see