Uniform convergence

In 1821 Augustin-Louis Cauchy published a proof that a convergent sum of continuous functions is always continuous, to which Niels Henrik Abel in 1826 found purported counterexamples in the context of Fourier series, 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.{{cite journal | doi=10.1016/j.hm.2004.11.010 | volume=32 | issue=4 | title=Exceptions and counterexamples: Understanding Abel's comment on Cauchy's Theorem | journal=Historia Mathematica | pages=453–480| year=2005 | last1=Sørensen | first1=Henrik Kragh | doi-access= }}

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 $\backslash sum\_\{n=1\}^\backslash infty\; f\_n(x,\backslash phi,\backslash psi)$ 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.{{Cite book

|title=A history of analysis

|first=Hans Niels

|last=Jahnke

|publisher=AMS Bookstore

|year=2003

|isbn=978-0-8218-2623-2

|chapter=6.7 The Foundation of Analysis in the 19th Century: Weierstrass

|chapter-url=https://books.google.com/books?id=CVRZEXFVsZkC&pg=PA184

|page=184

}}

Later Gudermann's pupil Karl Weierstrass, who attended his course on elliptic functions in 1839–1840, coined the term gleichmäßig konvergent ({{langx|de|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{{cite book |last=Lakatos |first=Imre |author-link=Imre Lakatos |title=Proofs and Refutations|year=1976|publisher=Cambridge University Press |pages=[https://archive.org/details/proofsrefutation0000laka/page/141 141] |isbn=978-0-521-21078-2|title-link=Proofs and Refutations }} and George Gabriel Stokes. 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 far-reaching 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 Bois-Reymond, Ulisse Dini, Cesare Arzelà and others.

We first define uniform convergence for real-valued functions, although the concept is readily generalized to functions mapping to metric spaces and, more generally, uniform spaces (see below).

Suppose $E$ is a set and $(f\_n)\_\{n\; \backslash in\; \backslash N\}$ is a sequence of real-valued functions on it. We say the sequence $(f\_n)\_\{n\; \backslash in\; \backslash N\}$ 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 for all $x\; \backslash in\; E$

:

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):

:$f\_n\backslash rightrightarrows\; f,\; \backslash quad\; \backslash underset\{n\backslash to\backslash infty\}\{\backslash mathrm\{unif\backslash \; lim\}\}f\_n\; =\; f,\; \backslash quad\; f\_n\; \backslash overset\{\backslash mathrm\{unif.\}\}\{\backslash longrightarrow\}\; f,\; \backslash quad\; f=\backslash mathrm\{u\}\backslash !\backslash !-\backslash !\backslash !\backslash !\backslash lim\_\{n\backslash to\backslash infty\}\; f\_n\; .$

Frequently, no special symbol is used, and authors simply write

:$f\_n\backslash to\; f\; \backslash quad\; \backslash mathrm\{uniformly\}$

to indicate that convergence is uniform. (In contrast, the expression $f\_n\backslash to\; f$ on $E$ without an adverb is taken to mean pointwise convergence 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 criterion can be used to give an equivalent alternative formulation for uniform convergence: $(f\_n)\_\{n\backslash in\backslash N\}$ 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

:.

In yet another equivalent formulation, if we define

:$d\_n\; =\; \backslash sup\_\{x\backslash in\; E\}\; |f\_n(x)\; -\; f(x)\; |,$

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 $(f\_n)\_\{n\; \backslash in\; \backslash N\}$ on $E$ as (simple) convergence of $(f\_n)\_\{n\; \backslash in\; \backslash N\}$ in the function space $\backslash R^E$ with respect to the uniform metric (also called the supremum metric), defined by

:$d(f,g)=\backslash sup\_\{x\backslash in\; E\}\; |f(x)-g(x)|.$

Symbolically,

:$f\_n\backslash rightrightarrows\; f\backslash iff\; d(f\_n,f)\; \backslash to\; 0$.

The sequence $(f\_n)\_\{n\; \backslash in\; \backslash N\}$ 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.

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 of pointwise convergence 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.

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 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 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.

Uniform convergence admits a simplified definition in a hyperreal setting. Thus, a sequence $f\_n$ converges to f uniformly if for all hyperreal 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). In contrast, pointwise continuity requires this only for real x.

For $x\; \backslash in\; [0,1)$, a basic example of uniform convergence can be illustrated as follows: the sequence $(1/2)^\{x+n\}$ converges uniformly, while $x^n$ does not. Specifically, assume $\backslash epsilon=1/4$. Each function $(1/2)^\{x+n\}$ 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 real or complex-valued functions over X with the uniform norm topology, with the uniform metric defined by

:$d(f,g)=\backslash |f-g\backslash |\_\{\backslash infty\}=\backslash sup\_\{x\backslash in\; X\}\; |f(x)-g(x)|.$

Then uniform convergence simply means convergence in the uniform norm topology:

:$\backslash lim\_\{n\backslash to\backslash infty\}\backslash |f\_n-f\backslash |\_\{\backslash infty\}=0$.

The sequence of functions $(f\_n)$

:$\backslash begin\{cases\}\; f\_n:[0,1]\backslash to\; [0,1]\; \backslash \backslash \; f\_n(x)=x^n\; \backslash end\{cases\}$

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

:

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 whenever $n\backslash geq\; N$ by choosing $N\; =\; \backslash lceil\backslash log\backslash epsilon/\backslash log\; x\backslash rceil$, which is the minimum integer exponent of $x$ that allows it to reach or dip below $\backslash epsilon$ (here the upper square brackets indicate rounding up, see 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 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 $x\_0\; =\; (1/2)^\{1/N\}$. Since

:$\backslash left|f\_N(x\_0)\; -\; f(x\_0)\backslash right|\; =\; \backslash left|\; \backslash left[\; \backslash left(\backslash frac\{1\}\{2\}\backslash right)^\{\backslash frac\{1\}\{N\}\}\; \backslash right]^N\; -\; 0\; \backslash right|\; =\; \backslash frac\{1\}\{2\}\; >\; \backslash frac\{1\}\{4\}\; =\; \backslash epsilon,$

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

:$\backslash lim\_\{n\backslash to\backslash infty\}\backslash |f\_n-f\backslash |\_\{\backslash infty\}=1,$

contrary to the requirement that $\backslash |f\_n-f\backslash |\_\{\backslash infty\}\backslash to\; 0$ 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, $f\_n\backslash in\; C^\{\backslash infty\}([0,1])$, the limit $\backslash lim\_\{n\backslash to\; \backslash infty\}f\_n$ is not even continuous.

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 absolutely and uniformly on $E$.

The complex exponential function can be expressed as the series:

:$\backslash sum\_\{n=0\}^\{\backslash infty\}\backslash frac\{z^n\}\{n!\}.$

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:

:$\backslash left|\; \backslash frac\{z^n\}\{n!\}\; \backslash right|\backslash le\; M\_n,\; \backslash forall\; z\backslash in\; D\_R.$

To do this, we notice

:$\backslash left|\; \backslash frac\{z^n\}\{n!\}\backslash right|\; \backslash le\; \backslash frac\{|z|^n\}\{n!\}\; \backslash le\; \backslash frac\{R^n\}\{n!\}$

and take $M\_n=\backslash tfrac\{R^n\}\{n!\}.$

If $\backslash sum\_\{n=0\}^\{\backslash infty\}M\_n$ is convergent, then the M-test asserts that the original series is uniformly convergent.

The ratio test can be used here:

:$\backslash lim\_\{n\; \backslash to\; \backslash infty\}\backslash frac\{M\_\{n+1\}\}\{M\_n\}=\backslash lim\_\{n\; \backslash to\; \backslash infty\}\backslash frac\{R^\{n+1\}\}\{R^n\}\backslash frac\{n!\}\{(n+1)!\}=\backslash lim\_\{n\; \backslash to\; \backslash infty\}\backslash frac\{R\}\{n+1\}=0$

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 $S.$

• 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.
• If $S$ is a compact interval (or in general a compact topological space), and $(f\_n)$ is a monotone increasing sequence (meaning $f\_n(x)\; \backslash leq\; f\_\{n+1\}(x)$ 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.

{{Main|Uniform limit theorem}}

Image:Drini nonuniformconvergence SVG.svg

If $E$ and $M$ are topological spaces, then it makes sense to talk about the 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:

{{math theorem | name = Uniform limit theorem | math_statement = 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 "{{math|ε/3}} trick", and is the archetypal example of this trick: to prove a given inequality ({{math|ε}}), one uses the definitions of continuity and uniform convergence to produce 3 inequalities ({{math|ε/3}}), and then combines them via the triangle inequality to produce the desired inequality.

{{Proof|Let $x\_0\backslash in\; E$ be an arbitrary point. We will prove that $f$ is continuous at $x\_0$. Let $\backslash varepsilon\; >0$. By uniform convergence, there exists a natural number $N$ such that

$\backslash forall\; x\; \backslash in\; E\backslash quad\; d(f\_N(x),f(x))\backslash leq\; \backslash tfrac\{\backslash varepsilon\}\{3\}$

(uniform convergence shows that the above statement is true for all $n\backslash geq\; N$, but we will only use it for one function of the sequence, namely $f\_N$).

It follows from the continuity of $f\_N$ at $x\_0\backslash in\; E$ that there exists an open set $U$ containing $x\_0$ such that

$\backslash forall\; x\backslash in\; U\backslash quad\; d(f\_N(x),f\_N(x\_0))\backslash leq\backslash tfrac\{\backslash varepsilon\}\{3\}$.

Hence, using the triangle inequality,

$\backslash forall\; x\backslash in\; U\backslash quad\; d(f(x),\; f(x\_0))\backslash leq\; d(f(x),f\_N(x))+d(f\_N(x),f\_N(x\_0))+d(f\_N(x\_0),f(x\_0))\backslash leq\backslash varepsilon$,

which gives us the continuity of $f$ at $x\_0$.$\backslash quad\backslash square$}}

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 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.

If $S$ is an interval and all the functions $f\_n$ are 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 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 $f\_n(x)\; =\; n^\{-1/2\}\{\backslash sin(nx)\}$ 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 $f\text{'}\_n(x)=n^\{1/2\}\backslash cos\; nx,$ 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). 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 $\backslash lim\_\{n\backslash to\backslash infty\}\; f\_n(x\_0)$ 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 $f\text{'}(x)\; =\; \backslash lim\_\{n\backslash to\; \backslash infty\}\; f\text{'}\_n(x)$ for $x\; \backslash in\; [a,\; b]$.

Similarly, one often wants to exchange integrals and limit processes. For the Riemann integral, this can be done if uniform convergence is assumed:

: If $(f\_n)\_\{n=1\}^\backslash infty$ is a sequence of Riemann integrable functions defined on a 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 $f\_n$: $$\backslash int\_I\; f\; =\; \backslash lim\_\{n\backslash to\backslash infty\}\backslash int\_I\; f\_n.$$

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 {{math|ε}} 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 integral instead.

Using Morera's Theorem, one can show that if a sequence of 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).

We say that $\backslash sum\_\{n=1\}^\backslash infty\; f\_n$ converges:

{{ordered list | list-style-type=lower-roman

| pointwise on E if and only if the sequence of partial sums $s\_n(x)=\backslash sum\_\{j=1\}^\{n\}\; f\_j(x)$ converges for every $x\backslash in\; E$.

| uniformly on E if and only if s_n converges uniformly as $n\backslash to\backslash infty$.

| absolutely on E if and only if $\backslash sum\_\{n=1\}^\backslash infty\; |f\_n|$ converges for every $x\; \backslash in\; E$.

}}

With this definition comes the following result:

Let x₀ be contained in the set E and each f_n be continuous at x₀. If $f\; =\; \backslash sum\_\{n=1\}^\backslash infty\; f\_n$ converges uniformly on E then f is continuous at x₀ in E. Suppose that $E\; =\; [a,\; b]$ and each f_n is integrable on E. If $\backslash sum\_\{n=1\}^\backslash infty\; f\_n$ 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.

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 almost everywhere also converges almost uniformly on the same set.

Almost uniform convergence implies almost everywhere convergence and convergence in measure.

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

{{reflist}}

• Konrad Knopp, Theory and Application of Infinite Series; Blackie and Son, London, 1954, reprinted by Dover Publications, {{isbn|0-486-66165-2}}.
• 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|0-387-19374-X}}
• 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|0-471-31716-0}}.
• William Wade, An Introduction to Analysis, 3rd ed., Pearson, 2005

{{springer|title=Uniform convergence|id=p/u095230}}

• [https://web.archive.org/web/20070302194212/http://amath.colorado.edu/courses/5350/2002fall/uniform.html Graphic examples of uniform convergence of Fourier series] from the University of Colorado

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Category:Convergence (mathematics)