nLab
uniform convergence

Contents

Definition

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

ϵ>0N(ϵ)nN(ϵ)xX|f n(x)f(x)|<ϵ. \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 us uniform about this convergence is that the bound N(ϵ)N(\epsilon) is required to work for all xXx \in X (hence uniformly over xx). This is in contrast to pointwise convergence where one allows a different bound NN to exist for each ϵ\epsilon and each point xXx \in X separately. Since for non-finite XX the maximum of all such local choices of NN in general does not exist, uniform convergence is a stronger condition than pointwise convergence.

Examples

Proposition

Let

  1. XX be a set;

  2. YY a complete metric space.

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

If XX 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)

References

References

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

Last revised on July 5, 2017 at 05:33:45. See the history of this page for a list of all contributions to it.