Uniform convergence

In the mathematical field of analysis, uniform convergence is a type of convergence stronger than pointwise convergence. 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.
The concept is important because several properties of the functions f_{n}, such as continuity and Riemann integrability, are transferred to the limit f if the convergence is uniform.
Contents
History
Augustin Louis Cauchy in 1821 published the false statement, but with a purported proof, that the pointwise limit of a sequence of continuous functions is always continuous. Joseph Fourier and Niels Henrik Abel found counterexamples to this statement in the context of Fourier series, showing that Cauchy's proof had to be incorrect. Dirichlet then analyzed the proof and found the mistake: from the hypothesis of pointwise convergence it was assumed to follow that the sequence had the property of uniform convergence, not realizing that this is a strictly stronger condition.
The term uniform convergence was probably first used by Christoph Gudermann, in an 1838 paper on elliptic functions, where he employed the phrase "convergence in a uniform way" when the "mode of convergence" of a series is independent of the variables ϕ and ψ. While he thought it a "remarkable fact" when a series converged in this way, he did not give a formal definition, nor use the property in any of his proofs.^{[1]}
Later Gudermann's pupil Karl Weierstrass, who attended his course on elliptic functions in 1839–1840, coined the term gleichmäßig konvergent (German: uniformly convergent) which he used in his 1841 paper Zur Theorie der Potenzreihen, published in 1894. Independently a similar concept was used by Philipp Ludwig von Seidel^{[2]} and George Gabriel Stokes but without having any major impact on further development. G. H. Hardy compares the three definitions in his paper Sir George Stokes and the concept of uniform convergence and remarks: Weierstrass's discovery was the earliest, and he alone fully realized its farreaching importance as one of the fundamental ideas of analysis.
Under the influence of Weierstrass and Bernhard Riemann this concept and related questions were intensely studied at the end of the 19th century by Hermann Hankel, Paul du BoisReymond, Ulisse Dini, Cesare Arzelà and others.
Definition
Suppose S is a set and f_{n} : S → R is a realvalued function for every natural number n. We say that the sequence (f_{n})_{n∈N} is uniformly convergent with limit f : S → R if for every ε > 0, there exists a natural number N such that for all x ∈ S and all n ≥ N we have f_{n}(x) − f(x) < ε.
Consider the sequence α_{n} = sup_{x} f_{n}(x) − f(x) where the supremum is taken over all x ∈ S. Clearly f_{n} converges to f uniformly if and only if α_{n} tends to 0.
The sequence (f_{n})_{n∈N} is said to be locally uniformly convergent with limit f if for every x in some metric space S, there exists an r > 0 such that (f_{n}) converges uniformly on B(x,r) ∩ S.
Notes
Note that interchanging the order of "there exists N" and "for all x" in the definition above results in a statement equivalent to the pointwise convergence of the sequence. That notion can be defined as follows: the sequence (f_{n}) converges pointwise with limit f : S → R if and only if
 for every x ∈ S and every ε > 0, there exists a natural number N such that for all n ≥ N one has f_{n}(x) − f(x) < ε.
Here the order of the universal quantifiers for x and for ε is not important, but the order of the former and the existential quantifier for N is.
In the case of uniform convergence, N can only depend on ε, while in the case of pointwise convergence N may depend on both 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. This is because solving for n gives n > log ε / log x. This depends on x as well as on ε. Also note that it is impossible to find a suitable bound for n that does not depend on x because for any nonzero value of ε, log ε / log x grows without bounds as x tends to 1.
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 if and only if
 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.
Examples
Given a topological space X, we can equip the space of bounded real or complexvalued functions over X with the uniform norm topology. Then uniform convergence simply means convergence in the uniform norm topology.
The sequence with f_{n}(x): = x^{n} converges pointwise but not uniformly:
In this example one can easily see that pointwise convergence does not preserve differentiability or continuity. While each function of the sequence is smooth, that is to say that for all n, , the limit is not even continuous.
Exponential function
The series expansion of the exponential function can be shown to be uniformly convergent on any bounded subset S of using the Weierstrass Mtest.
Here is the series:
Any bounded subset is a subset of some disc D_{R} of radius R, centered on the origin in the complex plane. The Weierstrass Mtest requires us to find an upper bound M_{n} on the terms of the series, with M_{n} independent of the position in the disc:
This is trivial:
If is convergent, then the Mtest asserts that the original series is uniformly convergent.
The ratio test can be used here:
which means the series over M_{n} is convergent. Thus the original series converges uniformly for all , and since , the series is also uniformly convergent on S.
Properties
 Every uniformly convergent sequence is locally uniformly convergent.
 Every locally uniformly convergent sequence is compactly convergent.
 For locally compact spaces local uniform convergence and compact convergence coincide.
 A sequence of continuous functions on metric spaces, with the image metric space being complete, is uniformly convergent if and only if it is uniformly Cauchy.
Applications
To continuity
See also: Uniform limit theoremIf is a real interval (or indeed any topological space), we can talk about the continuity of the functions and . The following is the more important result about uniform convergence:
 Uniform convergence theorem. If is a sequence of continuous functions which converges uniformly towards the function on an interval , then is continuous on as well.
This theorem is proved by the " trick", and is the archetypal example of this trick: to prove a given inequality (), one uses the definitions of continuity and uniform convergence to produce 3 inequalities (), and then combines them via the triangle inequality to produce the desired inequality.
This theorem is important, since pointwise convergence of continuous functions is not enough to guarantee continuity of the limit function as the image illustrates.
More precisely, this theorem states that the uniform limit of uniformly continuous functions is uniformly continuous; for a locally compact space, continuity is equivalent to local uniform continuity, and thus the uniform limit of continuous functions is continuous.
To differentiability
If is an interval and all the functions are differentiable and converge to a limit , it is often desirable to differentiate the limit function by taking the limit of the derivatives of . 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 with uniform limit 0, but the derivatives do not approach 0. The precise statement covering this situation is as follows:
 If converges uniformly to , and if all the are differentiable, and if the derivatives converge uniformly to g, then is differentiable and its derivative is g.
To integrability
Similarly, one often wants to exchange integrals and limit processes. For the Riemann integral, this can be done if uniform convergence is assumed:
 If is a sequence of Riemann integrable functions which uniformly converge with limit , then is Riemann integrable and its integral can be computed as the limit of the integrals of the .
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 is a compact interval (or in general a compact topological space), and is a monotone increasing sequence (meaning for all n and x) of continuous functions with a pointwise limit which is also continuous, then the convergence is necessarily uniform (Dini's theorem). Uniform convergence is also guaranteed if is a compact interval and is an equicontinuous sequence that converges pointwise.
Almost uniform convergence
If the domain of the functions is a measure space then the related notion of almost uniform convergence can be defined. We say a sequence of functions converges almost uniformly on E if there is a measurable subset F of E with arbitrarily small measure such that the sequence converges uniformly on the complement E \ F.
Note that almost uniform convergence of a sequence does not mean that the sequence converges uniformly almost everywhere as might be inferred from the name.
Egorov's theorem guarantees that on a finite measure space, a sequence of functions that converges almost everywhere also converges almost uniformly on the same set.
Almost uniform convergence implies almost everywhere convergence and convergence in measure.
See also
Notes
 ^ Jahnke, Hans Niels (2003). "6.7 The Foundation of Analysis in the 19th Century: Weierstrass". A history of analysis. AMS Bookstore. ISBN 978 0 82182623 2, p. 184.
 ^ Lakatos, Imre (1976). Proofs and Refutations. Cambridge University Press. pp. 141. ISBN 052121078X.
References
 Konrad Knopp, Theory and Application of Infinite Series; Blackie and Son, London, 1954, reprinted by Dover Publications, ISBN 0486661652.
 G. H. Hardy, Sir George Stokes and the concept of uniform convergence; Proceedings of the Cambridge Philosophical Society, 19, pp. 148–156 (1918)
 Bourbaki; Elements of Mathematics: General Topology. Chapters 5–10 (Paperback); ISBN 038719374X
 Walter Rudin, Principles of Mathematical Analysis, 3rd ed., McGraw–Hill, 1976.
 Gerald Folland, Real Analysis: Modern Techniques and Their Applications, Second Edition, John Wiley & Sons, Inc., 1999, ISBN 0471317160.
External links
 Uniform convergence on PlanetMath
 Limit point of function on PlanetMath
 Converges uniformly on PlanetMath
 Convergent series on PlanetMath
 Graphic examples of uniform convergence of Fourier series from the University of Colorado
Categories: Calculus
 Mathematical series
 Topology of function spaces
 Convergence (mathematics)
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