Math 2.0

Sequences, series, summability Definition

Uniform convergence

Builds on Pointwise convergence

Definition

Let SS be a set, and let fn ⁣:S→ℝf_n \colon S \to \mathbb{R} for n≥1n \ge 1 and f ⁣:S→ℝf \colon S \to \mathbb{R} be functions. The sequence (fn)(f_n) converges uniformly to ff if for every ε>0\varepsilon > 0 there is NN such that ∣fn(x)−f(x)∣<εfor all n≥N and all x∈S.|f_n(x) - f(x)| < \varepsilon \quad \text{for all } n \ge N \text{ and all } x \in S.

Remarks

Here NN depends only on ε\varepsilon, not on xx. Uniform convergence implies pointwise convergence. Equivalently, sup⁡x∈S∣fn(x)−f(x)∣→0\sup_{x \in S} |f_n(x) - f(x)| \to 0 as n→∞n \to \infty. For example, sin⁡(nx)n→0\frac{\sin(n x)}{n} \to 0 uniformly on ℝ\mathbb{R}. Mathlib defines it as TendstoUniformly.

Reading path

To understand this page from the ground up, read these 5 pages in order. The foundations come first.

Show the path as a graph
Sequence, definition, no evidence levelSequencedefinitionReal sequence, definition, no evidence levelReal sequencedefinitionLimit of a sequence, definition, no evidence levelLimit of a sequencedefinitionPointwise convergence, definition, no evidence levelPointwise convergencedefinitionUniform convergence, definition, no evidence levelUniform convergencedefinition
Each page uses the pages above it. The colour of the bar on a box shows how strong the evidence is, and the box also gives it as text. The list has the same pages.
  • no evidence level
  1. Sequence definition
  2. Real sequence definition
  3. Limit of a sequence definition
  4. Pointwise convergence definition
  5. Uniform convergence definition

Where this leads

84 pages build on this one.

Further reading

  1. Web https://en.wikipedia.org/wiki/Uniform_convergence, Definition
About this page
Tag08FPABDQ4C
URIm2://08FPABDQ4C
Kinddefinition
WritingW1 stated
MSC40A30
DefinesUniform convergence
UsesPointwise convergence

Markdown · JSON · Edit on GitHub

What this page needs for W2

  • edge instance_of in: one example (0 of 1)
  • edge not_instance_of in: one non-example that fails one clause (0 of 1)

The writing levels say how complete the text of a page is. They do not measure its truth.

What links here

Used by