uniform convergence
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field of
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, uniform convergence is a
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of
convergence Convergence may refer to: Arts and media Literature *Convergence (book series), ''Convergence'' (book series), edited by Ruth Nanda Anshen *Convergence (comics), "Convergence" (comics), two separate story lines published by DC Comics: **A four-par ...
of functions stronger than
pointwise convergence In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no gener ...
. A
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sequence
of
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(f_n) converges uniformly to a limiting function f on a set E if, given any arbitrarily small positive number \epsilon, a number N can be found such that each of the functions f_N, f_,f_,\ldots differ from f by no more than \epsilon ''at every point'' x ''in'' E. Described in an informal way, if f_n converges to f uniformly, then the rate at which f_n(x) approaches f(x) is "uniform" throughout its domain in the following sense: in order to guarantee that f_n(x) falls within a certain distance \epsilon of f(x), we do not need to know the value of x\in E in question — there can be found a single value of N=N(\epsilon) ''independent of x'', such that choosing n\geq N will ensure that f_n(x) is within \epsilon of f(x) ''for all x\in E''. In contrast, pointwise convergence of f_n to f merely guarantees that for any x\in E given in advance, we can find N=N(\epsilon, x) (N can depend on the value of ''x'') so that, ''for that particular'' ''x'', f_n(x) falls within \epsilon of f(x) whenever n\geq N. The difference between uniform convergence and pointwise convergence was not fully appreciated early in the history of calculus, leading to instances of faulty reasoning. The concept, which was first formalized by
Karl Weierstrass Karl Theodor Wilhelm Weierstrass (german: link=no, Weierstraß ; 31 October 1815 – 19 February 1897) was a German mathematics, mathematician often cited as the "father of modern mathematical analysis, analysis". Despite leaving university withou ...

Karl Weierstrass
, is important because several properties of the functions f_n, such as continuity, Riemann integrability, and, with additional hypotheses,
differentiability In calculus Calculus, originally called infinitesimal calculus or "the calculus of infinitesimal In mathematics, infinitesimals or infinitesimal numbers are quantities that are closer to zero than any standard real number, but are not zero. T ...
, are transferred to the
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limit
f if the convergence is uniform, but not necessarily if the convergence is not uniform.


History

In 1821
Augustin-Louis Cauchy Baron Baron is a rank of nobility or title of honour, often hereditary, in various European countries, either current or historical. The female equivalent is baroness. Typically, the title denotes an aristocrat who ranks higher than a lord ...

Augustin-Louis Cauchy
published a proof that a convergent sum of continuous functions is always continuous, to which
Niels Henrik Abel Niels Henrik Abel (; ; 5 August 1802 – 6 April 1829) was a Norwegian mathematician A mathematician is someone who uses an extensive knowledge of mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics a ...

Niels Henrik Abel
in 1826 found purported counterexamples in the context of
Fourier series In mathematics, a Fourier series () is a periodic function composed of harmonically related Sine wave, sinusoids combined by a weighted summation. With appropriate weights, one cycle (or ''period'') of the summation can be made to approximate an ...
, arguing that Cauchy's proof had to be incorrect. Completely standard notions of convergence did not exist at the time, and Cauchy handled convergence using infinitesimal methods. When put into the modern language, what Cauchy proved is that a uniformly convergent sequence of continuous functions has a continuous limit. The failure of a merely pointwise-convergent limit of continuous functions to converge to a continuous function illustrates the importance of distinguishing between different types of convergence when handling sequences of functions. The term uniform convergence was probably first used by
Christoph Gudermann Christoph Gudermann (25 March 1798 – 25 September 1852) was a German mathematician noted for introducing the Gudermannian function of the Gudermannian function The Gudermannian function, named after Christoph Gudermann (1798–1852), relate ...
, in an 1838 paper on
elliptic functions In the mathematical field of complex analysis elliptic functions are a special kind of Meromorphic function, meromorphic functions, that satisfy two periodicity conditions. They are named elliptic functions because they come from elliptic integrals. ...
, where he employed the phrase "convergence in a uniform way" when the "mode of convergence" of a series \textstyle is independent of the variables \phi and \psi. 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. Later Gudermann's pupil
Karl Weierstrass Karl Theodor Wilhelm Weierstrass (german: link=no, Weierstraß ; 31 October 1815 – 19 February 1897) was a German mathematics, mathematician often cited as the "father of modern mathematical analysis, analysis". Despite leaving university withou ...

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, similar concepts were articulated by
Philipp Ludwig von Seidel Philipp Ludwig von Seidel (; 24 October 1821 in Zweibrücken, Germany ) , image_map = , map_caption = , map_width = 250px , capital = Berlin , coordinates = , largest_city = capital , languages_type = Official language , language ...

Philipp Ludwig von Seidel
and
George Gabriel Stokes Sir George Gabriel Stokes, 1st Baronet, (; 13 August 1819 – 1 February 1903) was an Anglo-Irish physicist A physicist is a scientist A scientist is a person who conducts Scientific method, scientific research to advance knowledge in a ...

George Gabriel Stokes
.
G. H. Hardy Godfrey Harold Hardy (7 February 1877 – 1 December 1947) was an English mathematics, mathematician, known for his achievements in number theory and mathematical analysis. In biology, he is known for the Hardy–Weinberg principle, a basic prin ...
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 far-reaching importance as one of the fundamental ideas of analysis." Under the influence of Weierstrass and
Bernhard Riemann Georg Friedrich Bernhard Riemann (; 17 September 1826 – 20 July 1866) was a German mathematician A mathematician is someone who uses an extensive knowledge of mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of ...
this concept and related questions were intensely studied at the end of the 19th century by
Hermann Hankel Hermann Hankel (14 February 1839 – 29 August 1873) was a German mathematician A mathematician is someone who uses an extensive knowledge of mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quant ...
, Paul du Bois-Reymond, Ulisse Dini, Cesare Arzelà and others.


Definition

We first define uniform convergence for Real-valued function, real-valued functions, although the concept is readily generalized to functions mapping to Metric space, metric spaces and, more generally, Uniform space, uniform spaces (see Uniform convergence#Generalizations, below). Suppose E is a Set (mathematics), set and (f_n)_ is a sequence of real-valued functions on it. We say the sequence (f_n)_ is uniformly convergent on E with limit f: E \to \R if for every \epsilon > 0, there exists a natural number N such that for all n \geq N and x \in E :, f_n(x)-f(x), <\epsilon. The notation for uniform convergence of f_n to f is not quite standardized and different authors have used a variety of symbols, including (in roughly decreasing order of popularity): :f_n\rightrightarrows f, \quad \undersetf_n = f, \quad f_n \overset f. Frequently, no special symbol is used, and authors simply write :f_n\to f \quad \mathrm to indicate that convergence is uniform. (In contrast, the expression f_n\to f on E without an adverb is taken to mean
pointwise convergence In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no gener ...
on E: for all x \in E , f_n(x)\to f(x) as n\to\infty.) Since \R is a complete metric space, the Cauchy sequence, Cauchy criterion can be used to give an equivalent alternative formulation for uniform convergence: (f_n)_ converges uniformly on E (in the previous sense) if and only if for every \epsilon > 0 , there exists a natural number N such that :x\in E, m,n\geq N \implies , f_m(x)-f_n(x), <\epsilon. In yet another equivalent formulation, if we define : d_n = \sup_ , f_n(x) - f(x) , , then f_n converges to f uniformly if and only if d_n\to 0 as n\to\infty. Thus, we can characterize uniform convergence of (f_n)_ on E as (simple) convergence of (f_n)_ in the function space \R^E with respect to the ''Uniform norm, uniform metric'' (also called the supremum metric), defined by :d(f,g)=\sup_ , f(x)-g(x), . Symbolically, :f_n\rightrightarrows f\iff \lim_ d(f_n,f)= 0. The sequence (f_n)_ is said to be locally uniformly convergent with limit f if E is a metric space and for every x\in E, there exists an r > 0 such that (f_n) converges uniformly on B(x,r)\cap E. It is clear that uniform convergence implies local uniform convergence, which implies pointwise convergence.


Notes

Intuitively, a sequence of functions f_n converges uniformly to f if, given an arbitrarily small \epsilon>0, we can find an N\in\N so that the functions f_n with n>N all fall within a "tube" of width 2\epsilon centered around f (i.e., between f(x)-\epsilon and f(x)+\epsilon) for the ''entire domain'' of the function. Note that interchanging the order of quantifiers in the definition of uniform convergence by moving "for all x\in E" in front of "there exists a natural number N" results in a definition of
pointwise convergence In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no gener ...
of the sequence. To make this difference explicit, in the case of uniform convergence, N=N(\epsilon) can only depend on \epsilon, and the choice of N has to work for all x\in E, for a specific value of \epsilon that is given. In contrast, in the case of pointwise convergence, N=N(\epsilon,x) may depend on both \epsilon and x, and the choice of N only has to work for the specific values of \epsilon and x that are given. Thus uniform convergence implies pointwise convergence, however the converse is not true, as the example in the section below illustrates.


Generalizations

One may straightforwardly extend the concept to functions ''E'' → ''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 net (mathematics), nets of functions ''E'' → ''X'', where ''X'' is a uniform space. We say that the net (f_\alpha) ''converges uniformly'' with limit ''f'' : ''E'' → ''X'' if and only if for every entourage (topology), entourage ''V'' in ''X'', there exists an \alpha_0, such that for every ''x'' in ''E'' and every \alpha\geq \alpha_0, (f_\alpha(x),f(x)) is in ''V''. In this situation, uniform limit of continuous functions remains continuous.


Definition in a hyperreal setting

Uniform convergence admits a simplified definition in a hyperreal number, hyperreal setting. Thus, a sequence f_n converges to ''f'' uniformly if for all ''x'' in the domain of f^* and all infinite ''n'', f_n^*(x) is infinitely close to f^*(x) (see microcontinuity for a similar definition of uniform continuity).


Examples

For x \in [0,1), a basic example of uniform convergence can be illustrated as follows: the sequence (1/2)^ converges uniformly, while x^n does not. Specifically, assume \epsilon=1/4. Each function (1/2)^ is less than or equal to 1/4 when n \geq 2, regardless of the value of x. On the other hand, x^n is only less than or equal to 1/4 at ever increasing values of n when values of x are selected closer and closer to 1 (explained more in depth further below). Given a topological space ''X'', we can equip the space of bounded function, bounded real number, real or complex number, complex-valued functions over ''X'' with the uniform norm topology, with the uniform metric defined by :d(f,g)=\, f-g\, _=\sup_ , f(x)-g(x), . Then uniform convergence simply means
convergence Convergence may refer to: Arts and media Literature *Convergence (book series), ''Convergence'' (book series), edited by Ruth Nanda Anshen *Convergence (comics), "Convergence" (comics), two separate story lines published by DC Comics: **A four-par ...
in the uniform norm topology: :\lim_\, f_n-f\, _=0. The sequence of functions (f_n) :\begin f_n:[0,1]\to [0,1] \\ f_n(x)=x^n \end is a classic example of a sequence of functions that converges to a function f pointwise but not uniformly. To show this, we first observe that the pointwise limit of (f_n) as n\to\infty is the function f, given by : f(x) = \lim_ f_n(x) = \begin 0, & x \in [0,1); \\ 1, & x=1. \end ''Pointwise convergence:'' Convergence is trivial for x=0 and x=1, since f_n(0)=f(0)=0 and f_n(1)=f(1)=1, for all n. For x \in (0,1) and given \epsilon>0, we can ensure that , f_n(x)-f(x), <\epsilon whenever n\geq N by choosing N = \lceil\log\epsilon/\log x\rceil (here the upper square brackets indicate rounding up, see Floor and ceiling functions, ceiling function). Hence, f_n\to f pointwise for all x\in[0,1]. Note that the choice of N depends on the value of \epsilon and x. Moreover, for a fixed choice of \epsilon, N (which cannot be defined to be smaller) grows without bound as x approaches 1. These observations preclude the possibility of uniform convergence. ''Non-uniformity of convergence:'' The convergence is not uniform, because we can find an \epsilon>0 so that no matter how large we choose N, there will be values of x \in [0,1] and n \geq N such that , f_n(x)-f(x), \geq\epsilon. To see this, first observe that regardless of how large n becomes, there is always an x_0 \in [0,1) such that f_n(x_0)=1/2. Thus, if we choose \epsilon = 1/4, we can never find an N such that , f_n(x)-f(x), <\epsilon for all x\in[0,1] and n\geq N. Explicitly, whatever candidate we choose for N, consider the value of f_N at x_0 = (1/2)^. Since :\left, f_N(x_0) - f(x_0)\ = \left, \left[ \left(\frac\right)^ \right]^N - 0 \ = \frac > \frac = \epsilon, the candidate fails because we have found an example of an x\in[0,1] that "escaped" our attempt to "confine" each f_n\ (n\geq N) to within \epsilon of f for all x\in[0,1]. In fact, it is easy to see that :\lim_\, f_n-f\, _=1, contrary to the requirement that \, f_n-f\, _\to 0 if f_n \rightrightarrows f. 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'', f_n\in C^([0,1]), the limit \lim_f_n 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 \subset \C using the Weierstrass M-test. Theorem (Weierstrass M-test). ''Let (f_n) be a sequence of functions f_n:E\to \C and let M_n be a sequence of positive real numbers such that , f_n(x), \le M_n for all x\in E and n=1,2, 3, \ldots If \sum_n M_n converges, then \sum_n f_n converges uniformly on E.'' The complex exponential function can be expressed as the series: :\sum_^\frac. Any bounded subset is a subset of some disc D_R of radius R, centered on the origin in the complex plane. The Weierstrass M-test 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: :\left, \frac \\le M_n, \forall z\in D_R. To do this, we notice :\left, \frac\ \le \frac \le \frac and take M_n=\tfrac. If \sum_^M_n is convergent, then the M-test asserts that the original series is uniformly convergent. The ratio test can be used here: :\lim_\frac=\lim_\frac\frac=\lim_\frac=0 which means the series over M_n is convergent. Thus the original series converges uniformly for all z\in D_R, and since S\subset D_R, 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 sequence, uniformly Cauchy. * If S is a compact space, compact interval (or in general a compact topological space), and (f_n) is a monotonic, monotone increasing sequence (meaning f_n(x) \leq f_(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 equicontinuity, equicontinuous sequence that converges pointwise.


Applications


To continuity

If E and M are topological space, topological spaces, then it makes sense to talk about the continuous function (topology), continuity of the functions f_n,f:E\to M. If we further assume that M is a metric space, then (uniform) convergence of the f_n to f is also well defined. The following result states that continuity is preserved by uniform convergence: : Uniform limit theorem. ''Suppose E is a topological space, M is a metric space, and (f_n) is a sequence of continuous functions f_n:E\to M. If f_n\rightrightarrows f on E, then f is also continuous.'' 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 an important one in the history of real and Fourier analysis, since many 18th century mathematicians had the intuitive understanding that a sequence of continuous functions always converges to a continuous function. The image above shows a counterexample, and many discontinuous functions could, in fact, be written as a
Fourier series In mathematics, a Fourier series () is a periodic function composed of harmonically related Sine wave, sinusoids combined by a weighted summation. With appropriate weights, one cycle (or ''period'') of the summation can be made to approximate an ...
of continuous functions. The erroneous claim that the pointwise limit of a sequence of continuous functions is continuous (originally stated in terms of convergent series of continuous functions) is infamously known as "Cauchy's wrong theorem". The uniform limit theorem shows that a stronger form of convergence, uniform convergence, is needed to ensure the preservation of continuity in the limit function. 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 S is an interval and all the functions f_n are derivative, differentiable and converge to a limit f, it is often desirable to determine the derivative function f' by taking the limit of the sequence f'_n. This is however in general not possible: even if the convergence is uniform, the limit function need not be differentiable (not even if the sequence consists of everywhere-analytic function, analytic functions, see Weierstrass function), 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) = n^ with uniform limit f_n\rightrightarrows f\equiv 0. Clearly, f' is also identically zero. However, the derivatives of the sequence of functions are given by f'_n(x)=n^\cos nx, and the sequence f'_n does not converge to f', or even to any function at all. In order to ensure a connection between the limit of a sequence of differentiable functions and the limit of the sequence of derivatives, the uniform convergence of the sequence of derivatives plus the convergence of the sequence of functions at at least one point is required:Rudin, Walter (1976). ''iarchive:PrinciplesOfMathematicalAnalysis, Principles of Mathematical Analysis'' 3rd edition, Theorem 7.17. McGraw-Hill: New York. : ''If (f_n) is a sequence of differentiable functions on [a,b] such that \lim_ f_n(x_0) exists (and is finite) for some x_0\in[a,b] and the sequence (f'_n) converges uniformly on [a,b], then f_n converges uniformly to a function f on [a,b], and f'(x) = \lim_ f'_n(x) for x \in [a, b].''


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 (f_n)_^\infty is a sequence of Riemann integrable functions defined on a compact space, compact interval I 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:'' :: ''\int_I f = \lim_\int_I f_n.'' In fact, for a uniformly convergent family of bounded functions on an interval, the upper and lower Riemann integrals converge to the upper and lower Riemann integrals of the limit function. This follows because, for ''n'' sufficiently large, the graph of f_n is within of the graph of ''f'', and so the upper sum and lower sum of f_n are each within \varepsilon , I, of the value of the upper and lower sums of f, respectively. 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 integration, Lebesgue integral instead.


To analyticity

Using Morera's Theorem, one can show that if a sequence of Analytic function, analytic functions converges uniformly in a region S of the complex plane, then the limit is analytic in S. This example demonstrates that complex functions are more well-behaved than real functions, since the uniform limit of analytic functions on a real interval need not even be differentiable (see Weierstrass function).


To series

We say that \textstyle\sum_^\infty f_n converges: With this definition comes the following result:
Let ''x''0 be contained in the set ''E'' and each ''f''''n'' be continuous at ''x''0. If \textstyle f = \sum_^\infty f_n converges uniformly on ''E'' then ''f'' is continuous at ''x''0 in ''E''. Suppose that E = [a, b] and each ''f''''n'' is integrable on ''E''. If \textstyle\sum_^\infty f_n converges uniformly on ''E'' then ''f'' is integrable on ''E'' and the series of integrals of ''f''''n'' is equal to integral of the series of fn.


Almost uniform convergence

If the domain of the functions is a measure space ''E'' then the related notion of almost uniform convergence can be defined. We say a sequence of functions (f_n) converges almost uniformly on ''E'' if for every \delta > 0 there exists a measurable set E_\delta with measure less than \delta such that the sequence of functions (f_n) converges uniformly on E \setminus E_\delta. In other words, almost uniform convergence means there are sets of arbitrarily small measure for which the sequence of functions converges uniformly on their complement. 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. However, Egorov's theorem does guarantee that on a finite measure space, a sequence of functions that converges Pointwise convergence#Almost everywhere convergence, almost everywhere also converges almost uniformly on the same set. Almost uniform convergence implies almost everywhere convergence and convergence in measure.


See also

*Uniform convergence in probability *Modes of convergence (annotated index) *Dini's theorem *Arzelà–Ascoli theorem


Notes


References

* Konrad Knopp, Theory and Application of Infinite Series; Blackie and Son, London, 1954, reprinted by Dover Publications, . *
G. H. Hardy Godfrey Harold Hardy (7 February 1877 – 1 December 1947) was an English mathematics, mathematician, known for his achievements in number theory and mathematical analysis. In biology, he is known for the Hardy–Weinberg principle, a basic prin ...
, Sir George Stokes and the concept of uniform convergence; Proceedings of the Cambridge Philosophical Society, 19, pp. 148–156 (1918) * Nicolas Bourbaki, Bourbaki; Elements of Mathematics: General Topology. Chapters 5–10 (paperback); * 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, . * William Wade, An Introduction to Analysis, 3rd ed., Pearson, 2005


External links

*
Graphic examples of uniform convergence of Fourier series
from the University of Colorado {{series (mathematics) Calculus Mathematical series Topology of function spaces Convergence (mathematics)