In the

Graphic examples of uniform convergence of Fourier series

from the University of Colorado {{series (mathematics) Calculus Mathematical series Topology of function spaces Convergence (mathematics)

mathematical
Mathematics (from Greek
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Greece
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field of analysis
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, uniform convergence is a mode
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** Imperative mood
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of convergence
Convergence may refer to:
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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 sequence
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of functions
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$(f\_n)$ converges uniformly to a limiting function $f$ on a set $E$ if, given any arbitrarily small positive number $\backslash epsilon$, a number $N$ can be found such that each of the functions $f\_N,\; f\_,f\_,\backslash ldots$ differ from $f$ by no more than $\backslash epsilon$ ''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 $\backslash epsilon$ of $f(x)$, we do not need to know the value of $x\backslash in\; E$ in question — there can be found a single value of $N=N(\backslash epsilon)$ ''independent of $x$'', such that choosing $n\backslash geq\; N$ will ensure that $f\_n(x)$ is within $\backslash epsilon$ of $f(x)$ ''for all $x\backslash in\; E$''. In contrast, pointwise convergence of $f\_n$ to $f$ merely guarantees that for any $x\backslash in\; E$ given in advance, we can find $N=N(\backslash epsilon,\; x)$ ($N$ can depend on the value of ''$x$'') so that, ''for that particular'' ''$x$'', $f\_n(x)$ falls within $\backslash epsilon$ of $f(x)$ whenever $n\backslash geq\; N$.
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 ...

, 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 limit
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$f$ if the convergence is uniform, but not necessarily if the convergence is not uniform.
History

In 1821Augustin-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 ...

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 ...

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 $\backslash textstyle$ is independent of the variables $\backslash phi$ and $\backslash psi.$ 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 ...

, 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 ...

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
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. 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 $f:\; E\; \backslash to\; \backslash R$ if for every $\backslash epsilon\; >\; 0,$ there exists a natural number $N$ such that for all $n\; \backslash geq\; N$ and $x\; \backslash in\; E$ :$,\; f\_n(x)-f(x),\; <\backslash epsilon.$ 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): :$f\_n\backslash rightrightarrows\; f,\; \backslash quad\; \backslash undersetf\_n\; =\; f,\; \backslash quad\; f\_n\; \backslash overset\; f.$ Frequently, no special symbol is used, and authors simply write :$f\_n\backslash to\; f\; \backslash quad\; \backslash mathrm$ to indicate that convergence is uniform. (In contrast, the expression $f\_n\backslash to\; f$ on $E$ without an adverb is taken to meanpointwise 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 $x\; \backslash in\; E$, $f\_n(x)\backslash to\; f(x)$ as $n\backslash to\backslash infty$.)
Since $\backslash R$ 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 $\backslash epsilon\; >\; 0$, there exists a natural number $N$ such that
:$x\backslash in\; E,\; m,n\backslash geq\; N\; \backslash implies\; ,\; f\_m(x)-f\_n(x),\; <\backslash epsilon$.
In yet another equivalent formulation, if we define
:$d\_n\; =\; \backslash sup\_\; ,\; f\_n(x)\; -\; f(x)\; ,\; ,$
then $f\_n$ converges to $f$ uniformly if and only if $d\_n\backslash to\; 0$ as $n\backslash to\backslash infty$. Thus, we can characterize uniform convergence of $(f\_n)\_$ on $E$ as (simple) convergence of $(f\_n)\_$ in the function space $\backslash R^E$ with respect to the ''Uniform norm, uniform metric'' (also called the supremum metric), defined by
:$d(f,g)=\backslash sup\_\; ,\; f(x)-g(x),\; .$
Symbolically,
:$f\_n\backslash rightrightarrows\; f\backslash iff\; \backslash lim\_\; d(f\_n,f)=\; 0$.
The sequence $(f\_n)\_$ is said to be locally uniformly convergent with limit $f$ if $E$ is a metric space and for every $x\backslash in\; E$, there exists an $r\; >\; 0$ such that $(f\_n)$ converges uniformly on $B(x,r)\backslash cap\; E.$ 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 $\backslash epsilon>0$, we can find an $N\backslash in\backslash N$ so that the functions $f\_n$ with $n>N$ all fall within a "tube" of width $2\backslash epsilon$ centered around $f$ (i.e., between $f(x)-\backslash epsilon$ and $f(x)+\backslash epsilon$) for the ''entire domain'' of the function. Note that interchanging the order of quantifiers in the definition of uniform convergence by moving "for all $x\backslash in\; E$" in front of "there exists a natural number $N$" results in a definition ofpointwise 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, $N=N(\backslash epsilon)$ can only depend on $\backslash epsilon$, and the choice of $N$ has to work for all $x\backslash in\; E$, for a specific value of $\backslash epsilon$ that is given. In contrast, in the case of pointwise convergence, $N=N(\backslash epsilon,x)$ may depend on both $\backslash epsilon$ and $x$, and the choice of $N$ only has to work for the specific values of $\backslash epsilon$ 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 $(f\_\backslash alpha)$ ''converges uniformly'' with limit ''f'' : ''E'' → ''X'' if and only if for every entourage (topology), entourage ''V'' in ''X'', there exists an $\backslash alpha\_0$, such that for every ''x'' in ''E'' and every $\backslash alpha\backslash geq\; \backslash alpha\_0$, $(f\_\backslash alpha(x),f(x))$ 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 $x\; \backslash in\; [0,1)$, a basic example of uniform convergence can be illustrated as follows: the sequence $(1/2)^$ converges uniformly, while $x^n$ does not. Specifically, assume $\backslash epsilon=1/4$. Each function $(1/2)^$ is less than or equal to $1/4$ when $n\; \backslash geq\; 2$, 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 :$d(f,g)=\backslash ,\; f-g\backslash ,\; \_=\backslash sup\_\; ,\; f(x)-g(x),\; .$ Then uniform convergence simply meansconvergence
Convergence may refer to:
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*Convergence (book series), ''Convergence'' (book series), edited by Ruth Nanda Anshen
*Convergence (comics), "Convergence" (comics), two separate story lines published by DC Comics:
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in the uniform norm topology:
:$\backslash lim\_\backslash ,\; f\_n-f\backslash ,\; \_=0$.
The sequence of functions $(f\_n)$
:$\backslash begin\; f\_n:[0,1]\backslash to\; [0,1]\; \backslash \backslash \; f\_n(x)=x^n\; \backslash end$
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 $n\backslash to\backslash infty$ is the function $f$, given by
: $f(x)\; =\; \backslash lim\_\; f\_n(x)\; =\; \backslash begin\; 0,\; \&\; x\; \backslash in\; [0,1);\; \backslash \backslash \; 1,\; \&\; x=1.\; \backslash end$
''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 $x\; \backslash in\; (0,1)$ and given $\backslash epsilon>0$, we can ensure that $,\; f\_n(x)-f(x),\; <\backslash epsilon$ whenever $n\backslash geq\; N$ by choosing $N\; =\; \backslash lceil\backslash log\backslash epsilon/\backslash log\; x\backslash rceil$ (here the upper square brackets indicate rounding up, see Floor and ceiling functions, ceiling function). Hence, $f\_n\backslash to\; f$ pointwise for all $x\backslash in[0,1]$. Note that the choice of $N$ depends on the value of $\backslash epsilon$ and $x$. Moreover, for a fixed choice of $\backslash epsilon$, $N$ (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 $\backslash epsilon>0$ so that no matter how large we choose $N,$ there will be values of $x\; \backslash in\; [0,1]$ and $n\; \backslash geq\; N$ such that $,\; f\_n(x)-f(x),\; \backslash geq\backslash epsilon.$ To see this, first observe that regardless of how large $n$ becomes, there is always an $x\_0\; \backslash in\; [0,1)$ such that $f\_n(x\_0)=1/2.$ Thus, if we choose $\backslash epsilon\; =\; 1/4,$ we can never find an $N$ such that $,\; f\_n(x)-f(x),\; <\backslash epsilon$ for all $x\backslash in[0,1]$ and $n\backslash geq\; N$. Explicitly, whatever candidate we choose for $N$, consider the value of $f\_N$ at $x\_0\; =\; (1/2)^$. Since
:$\backslash left,\; f\_N(x\_0)\; -\; f(x\_0)\backslash \; =\; \backslash left,\; \backslash left[\; \backslash left(\backslash frac\backslash right)^\; \backslash right]^N\; -\; 0\; \backslash \; =\; \backslash frac\; >\; \backslash frac\; =\; \backslash epsilon,$
the candidate fails because we have found an example of an $x\backslash in[0,1]$ that "escaped" our attempt to "confine" each $f\_n\backslash \; (n\backslash geq\; N)$ to within $\backslash epsilon$ of $f$ for all $x\backslash in[0,1]$. In fact, it is easy to see that
:$\backslash lim\_\backslash ,\; f\_n-f\backslash ,\; \_=1,$
contrary to the requirement that $\backslash ,\; f\_n-f\backslash ,\; \_\backslash to\; 0$ if $f\_n\; \backslash rightrightarrows\; f$.
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'', $f\_n\backslash in\; C^([0,1])$, the limit $\backslash lim\_f\_n$ is not even continuous.
Exponential function

The series expansion of the exponential function can be shown to be uniformly convergent on any bounded subset $S\; \backslash subset\; \backslash C$ using the Weierstrass M-test. Theorem (Weierstrass M-test). ''Let $(f\_n)$ be a sequence of functions $f\_n:E\backslash to\; \backslash C$ and let $M\_n$ be a sequence of positive real numbers such that $,\; f\_n(x),\; \backslash le\; M\_n$ for all $x\backslash in\; E$ and $n=1,2,\; 3,\; \backslash ldots$ If $\backslash sum\_n\; M\_n$ converges, then $\backslash sum\_n\; f\_n$ converges uniformly on $E$.'' The complex exponential function can be expressed as the series: :$\backslash sum\_^\backslash frac.$ 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: :$\backslash left,\; \backslash frac\; \backslash \backslash le\; M\_n,\; \backslash forall\; z\backslash in\; D\_R.$ To do this, we notice :$\backslash left,\; \backslash frac\backslash \; \backslash le\; \backslash frac\; \backslash le\; \backslash frac$ and take $M\_n=\backslash tfrac.$ If $\backslash sum\_^M\_n$ is convergent, then the M-test asserts that the original series is uniformly convergent. The ratio test can be used here: :$\backslash lim\_\backslash frac=\backslash lim\_\backslash frac\backslash frac=\backslash lim\_\backslash frac=0$ which means the series over $M\_n$ is convergent. Thus the original series converges uniformly for all $z\backslash in\; D\_R,$ and since $S\backslash subset\; D\_R$, 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 $f\_n(x)\; \backslash leq\; f\_(x)$ 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 $f\_n,f:E\backslash to\; M$. 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 $f\_n:E\backslash to\; M$. If $f\_n\backslash rightrightarrows\; f$ 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 aFourier 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 $f\text{'}$ by taking the limit of the sequence $f\text{'}\_n$. 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 $f\_n\backslash rightrightarrows\; f\backslash equiv\; 0$. Clearly, $f\text{'}$ is also identically zero. However, the derivatives of the sequence of functions are given by $f\text{'}\_n(x)=n^\backslash cos\; nx,$ and the sequence $f\text{'}\_n$ does not converge to $f\text{'},$ 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 $\backslash lim\_\; f\_n(x\_0)$ exists (and is finite) for some $x\_0\backslash in[a,b]$ and the sequence $(f\text{'}\_n)$ converges uniformly on $[a,b]$, then $f\_n$ converges uniformly to a function $f$ on $[a,b]$, and $f\text{'}(x)\; =\; \backslash lim\_\; f\text{'}\_n(x)$ for $x\; \backslash in\; [a,\; b]$.''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 $(f\_n)\_^\backslash infty$ 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 $f\_n$:'' :: ''$\backslash int\_I\; f\; =\; \backslash lim\_\backslash int\_I\; f\_n.$'' 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 $\backslash varepsilon\; ,\; I,$ 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 $\backslash textstyle\backslash sum\_^\backslash infty\; f\_n$ 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 $\backslash textstyle\; f\; =\; \backslash sum\_^\backslash infty\; f\_n$ 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 $\backslash textstyle\backslash sum\_^\backslash infty\; f\_n$ 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 f_{n}.

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 $\backslash delta\; >\; 0$ there exists a measurable set $E\_\backslash delta$ with measure less than $\backslash delta$ such that the sequence of functions $(f\_n)$ converges uniformly on $E\; \backslash setminus\; E\_\backslash delta$. 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 theoremNotes

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)