# Category:Uniform Convergence

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This category contains results about Uniform Convergence.

Definitions specific to this category can be found in Definitions/Uniform Convergence.

Let $S$ be a set.

Let $M = \struct {A, d}$ be a metric space.

Let $\sequence {f_n}$ be a sequence of mappings $f_n: S \to A$.

Let:

- $\forall \epsilon \in \R_{>0}: \exists N \in \R: \forall n \ge N, \forall x \in S: \map d {\map {f_n} x, \map f x} < \epsilon$

Then **$\sequence {f_n}$ converges to $f$ uniformly on $S$ as $n \to \infty$**.

## Subcategories

This category has the following 3 subcategories, out of 3 total.

## Pages in category "Uniform Convergence"

The following 29 pages are in this category, out of 29 total.

### C

### D

### L

### U

- Uniform Limit of Analytic Functions is Analytic
- Uniform Limit of Holomorphic Functions is Holomorphic
- Uniform Product of Continuous Functions is Continuous
- Uniformly Absolutely Convergent Product is Uniformly Convergent
- Uniformly Continuous Function Preserves Uniform Convergence
- Uniformly Convergent iff Difference Under Supremum Metric Vanishes
- Uniformly Convergent Product Satisfies Uniform Cauchy Criterion
- Uniformly Convergent Sequence Evaluated on Convergent Sequence
- Uniformly Convergent Sequence Multiplied with Function
- Uniformly Convergent Sequence Multiplied with Function/Corollary
- Uniformly Convergent Sequence of Continuous Functions Converges to Continuous Function
- Uniformly Convergent Sequence of Continuous Functions Converges to Continuous Function/Corollary
- Uniformly Convergent Sequence on Dense Subset
- Uniformly Convergent Series of Continuous Functions Converges to Continuous Function
- Uniformly Convergent Series of Continuous Functions Converges to Continuous Function/Corollary