Category:Definitions/Locally Uniform Convergence of Product

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This category contains definitions related to Locally Uniform Convergence of Product.
Related results can be found in Category:Locally Uniform Convergence of Product.


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 $T$.