nLab uniform convergence

Definition

In analysis, uniform convergence refers to a type of convergence of sequences $(f_n)_{n \in \mathbb{N}}$ of functions $f_n \colon X \to \mathbb{R}$ into the real numbers. In terms of epsilontic analysis such a sequence converges uniformly to some function $f \colon X \to \mathbb{R}$ if for each positive real number $\epsilon \gt 0$ there exists a natural number $N_\epsilon \in \mathbb{N}$ such that if $n \geq N(\epsilon)$ then for all points $x \in X$ the difference (in absolute value) between the value of $f_n$ at that point and that of $f$ at that point is smaller than $\epsilon$:

$$\underset{\epsilon \gt 0}{\forall} \; \underset{N(\epsilon) \in \mathbb{N}}{\exists} \; \underset{n \geq N(\epsilon)}{\forall} \; \underset{x \in X}{\forall} \; \, \vert f_n(x) - f(x) \vert \lt \epsilon \,.$$

What is uniform about this convergence is that the bound $N(\epsilon)$ is required to work for all $x \in X$ (hence uniformly over $x$). This is in contrast to pointwise convergence where one allows a different bound $N$ to exist for each $\epsilon$ and each point $x \in X$ separately. Since for non-finite $X$ the maximum of all such local choices of $N$ in general does not exist, uniform convergence is a stronger condition than pointwise convergence.

Examples

Proposition

Let

1. $X$ be a set;

2. $Y$ a complete metric space.

Consider the set $F(X,Y)$ of functions $X \to Y$ as a metric space via the supremum norm. Then this is again complete: every Cauchy sequence of functions converges uniformly.

If $X$ is equipped with the structure of a topological space and if the Cauchy sequence of functions consist of continuous functions, then also the limit function is continuous.

(e.g. Gamelin-Greene 83, theorem I 2.5 and II 3.5)

Related concepts

• pointwise convergence

• uniformly continuous map

• uniform property

• topology of uniform convergence

References

• Wikipedia, Uniform convergence

References

• Theodore Gamelin, Robert Greene, Introduction to Topology, Dover (1983, 199)