Table of contents |

2 Topological reformulation 3 Theorems 4 Generalizations 5 History 6 Reference |

Suppose *S* is a set and *f*_{n} : *S* `->` **R** are real-valued functions for every natural number *n*. We say that the sequence (*f*_{n}) *converges uniformly* with limit *f* : *S* `->` **R** iff

- for every ε > 0, there exists a natural number
*N*, such that for all*x*in*S*and all*n*≥*N*: |*f*_{n}(*x*) −*f*(*x*)| < ε

- for every
*x*in*S*and every ε > 0, there exists a natural number*N*, such that for all*n*≥*N*: |*f*_{n}(*x*) −*f*(*x*)| < ε

Given a topological space X, we can equip the space of real/complex functions over X with the uniform norm topology. Then, **uniform convergence** simply means convergence in the uniform norm topology.

If *S* is a real interval (or indeed any topological space), we can talk about the continuity of the functions *f*_{n} and *f*. The following is the more important result about uniform continuity:

- If (
*f*_{n}) is a sequence of*continuous*functions which converges*uniformly*towards the function*f*, then*f*is continuous as well.

- If
*f*_{n}converges uniformly to*f*, and if all the*f*_{n}are differentiable, and if the derivatives`f`'_{n}converge uniformly to*g*, then*f*is differentiable and its derivative is*g*.

- if (
*f*_{n}) is a sequence of Riemann integrable functions which uniformly converge with limit*f*, then*f*is Riemann integrable and its integral can be computed as the limit of the integrals of the*f*_{n}.

If *S* is a compact interval (or in general a compact topological space), and (*f*_{n}) is a monotone increasing sequence (meaning *f*_{n}(*x*) ≤ *f*_{n+1}(*x*) for all *n* and *x*) of *continuous* functions with a pointwise limit *f* which is also continuous, then the convergence is necessarily uniform ("Dini's theorem").

One may straightforwardly extend the concept to functions *S* `->` *M*, where (*M*, *d*) is a metric space, by replacing |*f*_{n}(*x*) - *f*(*x*)| with *d*(*f*_{n}(*x*), *f*(*x*)).

The most general setting is the uniform convergence of netss of functions *S* `->` *X*, where *X* is a uniform space. We say that the net (*f*_{α}) *converges uniformly* with limit *f* : *S* `->` *X* iff

- for every entourage
*V*in*X*, there exists an α_{0}, such that for every*x*in*I*and every α => α_{0}: (*f*_{α}(*x*),*f*(*x*)) is in*V*.

Cauchy in 1821 published a faulty proof of the false statement that the pointwise limit of a sequence of continuous functions is always continuous. Fourier and Abel found counter examples in the context of Fourier series. Dirichlet then analyzed Cauchy's proof and found the mistake: the notion of pointwise convergence had to be replaced by uniform convergence.

Riemann pointed to the need for distinguishing between absolutely and conditionally convergent series by his Rearrangement Theorem. It shows that it is possible to rearrange the terms of a conditionally cnvergent series so that the derived series convergest to any desired limit