Uniform convergence

In mathematical analysis, a sequence { f_n } of functions converges uniformly to a limiting function f if the speed of convergence of f_n(x) to f(x) does not depend on x. This notion is used because several important properties of the functions f_n, such as continuity, differentiability and Riemann integrability, are only transferred to the limit f if the convergence is uniform.

Definition and comparison with pointwise convergence

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 if

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.

The sequence (f_n) is said to converge locally uniformly with limit f if for every x in S, if there exists an r > 0 such that (f_n) converges uniformly on B(x,r) ∩ S.

Compare uniform convergence to the concept of pointwise convergence: The sequence (f_n) converges pointwise with limit f : S → R if and only if

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.

In the case of uniform convergence, N can only depend on ε, while in the case of pointwise convergence N may depend on ε and x. It is therefore plain that uniform convergence implies pointwise convergence. The converse is not true, as the following example shows: take S to be the unit interval ^* and define f_n(x) = xⁿ for every natural number n. Then (f_n) converges pointwise to the function f defined by f(x) = 0 if x < 1 and f(1) = 1. This convergence is not uniform: for instance for ε = 1/4, there exists no N as required by the definition.

Topological reformulation

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

Theorems

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:

Uniform convergence theorem. If (f_n) is a sequence of continuous functions which converges uniformly towards the function f, then f is continuous as well. The former theorem is important, since pointwise convergence of continuous functions is not enough to guarantee continuity of the limit function as the image illustrates.

If S is an interval and all the functions f_n are differentiable and converge to a limit f, it is often desirable to differentiate the limit function f by taking the limit of the derivatives of f_n. This is however in general not possible: even if the convergence is uniform, the limit function need not be differentiable, and even if it is differentiable, the derivative of the limit function need not be equal to the limit of the derivatives. Consider for instance f_n(x) = 1/n sin(nx) with uniform limit 0, but the derivatives do not approach 0. The precise statement covering this situation is as follows:

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.

Similarly, one often wants to exchange integrals and limit processes. For the Riemann integral, one needs to require uniform convergence:

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.

Much stronger theorems in this respect, which require not much more than pointwise convergence, can be obtained if one abandons the Riemann integral and uses the Lebesgue integral instead.

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"). Uniform convergence is also guaranteed if S is a compact interval and (f_n) is an equicontinuous sequence that converges pointwise.

Generalizations

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 nets 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 α₀, such that for every x in I and every α≥α₀: (f_α(x), f(x)) is in V.

The above mentioned theorem, stating that the uniform limit of continuous functions is continuous, remains correct in these settings.

History

Augustin Louis Cauchy in 1821 published a faulty proof of the false statement that the pointwise limit of a sequence of continuous functions is always continuous. Joseph Fourier and Niels Henrik 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.

Reference

Theory and Application of Infinite Series, Konrad Knopp, Blackie and Son, London, 1954, reprinted by Dover Publications, ISBN 0486661652.