In
mathematical analysis, the concept of
uniform convergence is used to describe a situation where a
sequence of
functions (
f_{n}) converges to a limiting function
f in such a way that 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.
Suppose S is a set and f_{n} : I -> 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
Compare this to the concept of
pointwise convergence: The sequence (
f_{n}) converges pointwise with limit
f :
S -> R iff
- 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 [0,1] and define
f_{n}(
x) =
x^{n} 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.
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
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 integal instead.
If S is a compact interval (or in general a compact topological space), and (f_{n}) is an 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 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 α_{0}, such that for every x in I and every α => α_{0}: (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.
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.
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