In themathematical field ofanalysis,uniform convergence is amode of convergence of functions stronger thanpointwise convergence. Asequence offunctions${\displaystyle (f_n)}$converges uniformly to a limiting function${\displaystyle f}$ on a set${\displaystyle E}$ as the function domain if, given any arbitrarily small positive number${\displaystyle \epsilon }$, a number${\displaystyle N}$ can be found such that each of the functions${\displaystyle f_N,f_{N+1},f_{N+2},\dots }$ differs from${\displaystyle f}$ by no more than${\displaystyle \epsilon }$at every point${\displaystyle x}$in${\displaystyle E}$. That is, the required value of${\displaystyle N}$ only depends on${\displaystyle \epsilon ,}$ not on any particular${\displaystyle x}$.

By contrast, pointwise convergence of${\displaystyle (f_n)}$ to${\displaystyle f}$ merely guarantees that given any${\displaystyle x\in E}$, we can find${\displaystyle N=N(\epsilon ,x)}$ (that is, possibly depending on${\displaystyle x}$ as well as${\displaystyle \epsilon }$) such that for thespecific value of${\displaystyle x}$ given,${\displaystyle f_n(x)}$ falls within${\displaystyle \epsilon }$ of${\displaystyle f(x)}$ whenever${\displaystyle n\ge N.}$ A different${\displaystyle x}$ may require a larger value of${\displaystyle N.}$

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 byKarl Weierstrass, is important because several properties of the functions${\displaystyle f_n}$, such ascontinuity,Riemann integrability, and, with additional hypotheses,differentiability, are transferred to thelimit${\displaystyle f}$ if the convergence is uniform, but not necessarily if the convergence is not uniform.

In 1821Augustin-Louis Cauchy published a proof that a convergent sum of continuous functions is always continuous, to whichNiels Henrik Abel in 1826 found purported counterexamples in the context ofFourier 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.^[1]

The term uniform convergence was probably first used byChristoph Gudermann, in an 1838 paper onelliptic functions, where he employed the phrase "convergence in a uniform way" when the "mode of convergence" of a series$\sum _{n=1}^{\mathrm{\infty }}{f}_n(x,\varphi ,\psi )$ is independent of the variables${\displaystyle \varphi }$ and${\displaystyle \psi .}$ 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.^[2]

Later Gudermann's pupil,Karl Weierstrass, who attended his course on elliptic functions in 1839–1840, coined the termgleichmäßig konvergent (German:uniformly convergent) which he used in his 1841 paperZur Theorie der Potenzreihen, published in 1894. Independently, similar concepts were articulated byPhilipp Ludwig von Seidel^[3] andGeorge 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 andBernhard Riemann this concept and related questions were intensely studied at the end of the 19th century byHermann Hankel,Paul du Bois-Reymond,Ulisse Dini,Cesare Arzelà and others.

We first define uniform convergence forreal-valued functions, although the concept is readily generalized to functions mapping tometric spaces and, more generally,uniform spaces (seebelow).

Suppose${\displaystyle E}$ is aset and${\displaystyle (f_n{)}_{n\in \mathbb{N}}}$ is a sequence of real-valued functions on it. We say the sequence${\displaystyle (f_n{)}_{n\in \mathbb{N}}}$ isuniformly convergent on${\displaystyle E}$ with limit${\displaystyle f:E\to \mathbb{R}}$ if for every${\displaystyle \epsilon >0,}$ there exists a natural number${\displaystyle N}$ such that for all${\displaystyle n\ge N}$ and for all${\displaystyle x\in E}$

The notation for uniform convergence of${\displaystyle f_n}$ to${\displaystyle f}$ is not quite standardized and different authors have used a variety of symbols, including (in roughly decreasing order of popularity):

${\displaystyle f_n\rightrightarrows f,\underset{n\to \mathrm{\infty }}{\mathrm{u}\mathrm{n}\mathrm{i}\mathrm{f}\mathrm{l}\mathrm{i}\mathrm{m}}{f}_n=f,f_n\stackrel{\mathrm{u}\mathrm{n}\mathrm{i}\mathrm{f}.}{⟶}f,f=\mathrm{u}-\underset{n\to \mathrm{\infty }}{lim}{f}_n.}$

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

${\displaystyle f_n\to f\mathrm{u}\mathrm{n}\mathrm{i}\mathrm{f}\mathrm{o}\mathrm{r}\mathrm{m}\mathrm{l}\mathrm{y}}$

to indicate that convergence is uniform. (In contrast, the expression${\displaystyle f_n\to f}$ on${\displaystyle E}$ without an adverb is taken to meanpointwise convergence on${\displaystyle E}$: for all${\displaystyle x\in E}$,${\displaystyle f_n(x)\to f(x)}$ as${\displaystyle n\to \mathrm{\infty }}$.)

Since${\displaystyle \mathbb{R}}$ is acomplete metric space, theCauchy criterion can be used to give an equivalent alternative formulation for uniform convergence:${\displaystyle (f_n{)}_{n\in \mathbb{N}}}$ converges uniformly on${\displaystyle E}$ (in the previous sense) if and only if for every${\displaystyle \epsilon >0}$, there exists a natural number${\displaystyle N}$ such that

.

In yet another equivalent formulation, if we define

${\displaystyle d_n=\underset{x\in E}{sup}|f_n(x)-f(x)|,}$

then${\displaystyle f_n}$ converges to${\displaystyle f}$ uniformly if and only if${\displaystyle d_n\to 0}$ as${\displaystyle n\to \mathrm{\infty }}$. Thus, we can characterize uniform convergence of${\displaystyle (f_n{)}_{n\in \mathbb{N}}}$ on${\displaystyle E}$ as (simple) convergence of${\displaystyle (f_n{)}_{n\in \mathbb{N}}}$ in thefunction space${\displaystyle {\mathbb{R}}^E}$ with respect to theuniform metric (also called thesupremum metric), defined by

${\displaystyle d(f,g)=\underset{x\in E}{sup}|f(x)-g(x)|.}$

Symbolically,

${\displaystyle f_n\rightrightarrows f⟺d(f_n,f)\to 0}$.

The sequence${\displaystyle (f_n{)}_{n\in \mathbb{N}}}$ is said to belocally uniformly convergent with limit${\displaystyle f}$ if${\displaystyle E}$ is ametric space and for every${\displaystyle x\in E}$, there exists an${\displaystyle r>0}$ such that${\displaystyle (f_n)}$ converges uniformly on${\displaystyle B(x,r)\cap E.}$ It is clear that uniform convergence implies local uniform convergence, which implies pointwise convergence.

Intuitively, a sequence of functions${\displaystyle f_n}$ converges uniformly to${\displaystyle f}$ if, given an arbitrarily small${\displaystyle \epsilon >0}$, we can find an${\displaystyle N\in \mathbb{N}}$ so that the functions${\displaystyle f_n}$ with${\displaystyle n>N}$ all fall within a "tube" of width${\displaystyle 2\epsilon }$ centered around${\displaystyle f}$ (i.e., between${\displaystyle f(x)-\epsilon }$ and${\displaystyle f(x)+\epsilon }$) for theentire domain of the function.

Note that interchanging the order of quantifiers in the definition of uniform convergence by moving "for all${\displaystyle x\in E}$" in front of "there exists a natural number${\displaystyle N}$" results in a definition ofpointwise convergence of the sequence. To make this difference explicit, in the case of uniform convergence,${\displaystyle N=N(\epsilon )}$ can only depend on${\displaystyle \epsilon }$, and the choice of${\displaystyle N}$ has to work for all${\displaystyle x\in E}$, for a specific value of${\displaystyle \epsilon }$ that is given. In contrast, in the case of pointwise convergence,${\displaystyle N=N(\epsilon ,x)}$ may depend on both${\displaystyle \epsilon }$ and${\displaystyle x}$, and the choice of${\displaystyle N}$ only has to work for the specific values of${\displaystyle \epsilon }$ and${\displaystyle x}$ 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 functionsE →M, where (M,d) is ametric space, by replacing${\displaystyle |f_n(x)-f(x)|}$ with${\displaystyle d(f_n(x),f(x))}$.

The most general setting is the uniform convergence ofnets of functionsE →X, whereX is auniform space. We say that the net${\displaystyle (f_{\alpha })}$converges uniformly with limitf :E →X if and only if for everyentourageV inX, there exists an${\displaystyle {\alpha }_0}$, such that for everyx inE and every${\displaystyle \alpha \ge {\alpha }_0}$,${\displaystyle (f_{\alpha }(x),f(x))}$ is inV. In this situation, uniform limit of continuous functions remains continuous.

Uniform convergence admits a simplified definition in ahyperreal setting. Thus, a sequence${\displaystyle f_n}$ converges tof uniformly if for all hyperrealx in the domain of${\displaystyle f^{\ast }}$ and all infiniten,${\displaystyle f_n^{\ast }(x)}$ is infinitely close to${\displaystyle f^{\ast }(x)}$ (seemicrocontinuity for a similar definition of uniform continuity). In contrast, pointwise continuity requires this only for realx.

For${\displaystyle x\in [0,1)}$, a basic example of uniform convergence can be illustrated as follows: the sequence${\displaystyle (\frac{1}{2}){\phantom{)}}^{x+n}}$ converges uniformly, while${\displaystyle x^n}$ does not. Specifically, assume${\displaystyle \epsilon =\frac{1}{4}}$. Each function${\displaystyle (\frac{1}{2}){\phantom{)}}^{x+n}}$ is less than or equal to${\displaystyle \frac{1}{4}}$ when${\displaystyle n\ge 2}$, regardless of the value of${\displaystyle x}$. On the other hand,${\displaystyle x^n}$ is only less than or equal to${\displaystyle \frac{1}{4}}$ at ever increasing values of${\displaystyle n}$ when values of${\displaystyle x}$ are selected closer and closer to 1 (explained more in depth further below).

Given atopological spaceX, we can equip the space ofboundedreal orcomplex-valued functions overX with theuniform norm topology, with the uniform metric defined by

${\displaystyle d(f,g)={\Vert f-g\Vert }_{\mathrm{\infty }}=\underset{x\in X}{sup}|f(x)-g(x)|.}$

Then uniform convergence simply meansconvergence in theuniform norm topology:

${\displaystyle \underset{n\to \mathrm{\infty }}{lim}{\Vert f_n-f\Vert }_{\mathrm{\infty }}=0}$.

The sequence of functions${\displaystyle (f_n)}$

${\displaystyle \{\begin{array}{l}{f}_n:[0,1]\to [0,1]\\ f_n(x)=x^n\end{array}}$

is a classic example of a sequence of functions that converges to a function${\displaystyle f}$ pointwise but not uniformly. To show this, we first observe that the pointwise limit of${\displaystyle (f_n)}$ as${\displaystyle n\to \mathrm{\infty }}$ is the function${\displaystyle f}$, given by

Pointwise convergence: Convergence is trivial for${\displaystyle x=0}$ and${\displaystyle x=1}$, since${\displaystyle f_n(0)=f(0)=0}$ and${\displaystyle f_n(1)=f(1)=1}$, for all${\displaystyle n}$. For${\displaystyle x\in (0,1)}$ and given${\displaystyle \epsilon >0}$, we can ensure that whenever${\displaystyle n\ge N}$ by choosing${\displaystyle N=\lceil \log \epsilon /\log x\rceil }$, which is the minimum integer exponent of${\displaystyle x}$ that allows it to reach or dip below${\displaystyle \epsilon }$ (here the upper square brackets indicate rounding up, seeceiling function). Hence,${\displaystyle f_n\to f}$ pointwise for all${\displaystyle x\in [0,1]}$. Note that the choice of${\displaystyle N}$ depends on the value of${\displaystyle \epsilon }$ and${\displaystyle x}$. Moreover, for a fixed choice of${\displaystyle \epsilon }$,${\displaystyle N}$ (which cannot be defined to be smaller) grows without bound as${\displaystyle x}$ approaches 1. These observations preclude the possibility of uniform convergence.

Non-uniformity of convergence: The convergence is not uniform, because we can find an${\displaystyle \epsilon >0}$ so that no matter how large we choose${\displaystyle N,}$ there will be values of${\displaystyle x\in [0,1]}$ and${\displaystyle n\ge N}$ such that${\displaystyle |f_n(x)-f(x)|\ge \epsilon .}$ To see this, first observe that regardless of how large${\displaystyle n}$ becomes, there is always an${\displaystyle x_0\in [0,1)}$ such that${\displaystyle f_n(x_0)=\frac{1}{2}.}$ Thus, if we choose${\displaystyle \epsilon =\frac{1}{4},}$ we can never find an${\displaystyle N}$ such that for all${\displaystyle x\in [0,1]}$ and${\displaystyle n\ge N}$. Explicitly, whatever candidate we choose for${\displaystyle N}$, consider the value of${\displaystyle f_N}$ at${\displaystyle x_0=(\frac{1}{2}){\phantom{)}}^{1/N}}$. Since

${\displaystyle |f_N(x_0)-f(x_0)|=\left|((\frac{1}{2}{)}^{1/N}){\phantom{|}}^N-0\right|=\frac{1}{2}>\frac{1}{4}=\epsilon ,}$

the candidate fails because we have found an example of an${\displaystyle x\in [0,1]}$ that "escaped" our attempt to "confine" each${\displaystyle f_n(n\ge N)}$ to within${\displaystyle \epsilon }$ of${\displaystyle f}$ for all${\displaystyle x\in [0,1]}$. In fact, it is easy to see that

${\displaystyle \underset{n\to \mathrm{\infty }}{lim}\Vert f_n-f{\Vert }_{\mathrm{\infty }}=1,}$

contrary to the requirement that${\displaystyle \Vert f_n-f{\Vert }_{\mathrm{\infty }}\to 0}$ if${\displaystyle f_n\rightrightarrows f}$.

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 alln,${\displaystyle f_n\in C^{\mathrm{\infty }}([0,1])}$, the limit${\displaystyle \underset{n\to \mathrm{\infty }}{lim}{f}_n}$ is not even continuous.

The series expansion of theexponential function can be shown to be uniformly convergent on any bounded subset${\displaystyle S\subset \mathbb{C}}$ using theWeierstrass M-test.

Theorem (Weierstrass M-test).Let${\displaystyle (f_n)}$ be a sequence of functions${\displaystyle f_n:E\to \mathbb{C}}$ and let${\displaystyle M_n}$ be a sequence of positive real numbers such that${\displaystyle |f_n(x)|\le M_n}$ for all${\displaystyle x\in E}$ and${\displaystyle n=1,2,3,\dots }$ If$\sum _n{M}_n$ converges, then$\sum _n{f}_n$ converges absolutely and uniformly on${\displaystyle E}$.

The complex exponential function can be expressed as the series:

${\displaystyle \sum _{n=0}^{\mathrm{\infty }}\frac{z^n}{n!}.}$

Any bounded subset is a subset of some disc${\displaystyle D_R}$ of radius${\displaystyle R,}$ centered on the origin in thecomplex plane. The Weierstrass M-test requires us to find an upper bound${\displaystyle M_n}$ on the terms of the series, with${\displaystyle M_n}$ independent of the position in the disc:

${\displaystyle \left|\frac{z^n}{n!}\right|\le M_n,\mathrm{\forall }z\in D_R.}$

To do this, we notice

${\displaystyle \left|\frac{z^n}{n!}\right|\le \frac{|z{|}^n}{n!}\le \frac{R^n}{n!}}$

and take${\displaystyle M_n=\frac{R^n}{n!}.}$

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

Theratio test can be used here:

${\displaystyle \underset{n\to \mathrm{\infty }}{lim}\frac{M_{n+1}}{M_n}=\underset{n\to \mathrm{\infty }}{lim}\frac{R^{n+1}}{R^n}\frac{n!}{(n+1)!}=\underset{n\to \mathrm{\infty }}{lim}\frac{R}{n+1}=0}$

which means the series over${\displaystyle M_n}$ is convergent. Thus the original series converges uniformly for all${\displaystyle z\in D_R,}$ and since${\displaystyle S\subset D_R}$, the series is also uniformly convergent on${\displaystyle S.}$

• Every uniformly convergent sequence is locally uniformly convergent.
• Every locally uniformly convergent sequence iscompactly convergent.
• Forlocally 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 isuniformly Cauchy.
• If${\displaystyle S}$ is acompact interval (or in general a compact topological space), and${\displaystyle (f_n)}$ is amonotone increasing sequence (meaning${\displaystyle f_n(x)\le f_{n+1}(x)}$ for alln andx) ofcontinuous functions with a pointwise limit${\displaystyle f}$ which is also continuous, then the convergence is necessarily uniform (Dini's theorem). Uniform convergence is also guaranteed if${\displaystyle S}$ is a compact interval and${\displaystyle (f_n)}$ is anequicontinuous sequence that converges pointwise.

If${\displaystyle E}$ and${\displaystyle M}$ aretopological spaces, then it makes sense to talk about thecontinuity of the functions${\displaystyle f_n,f:E\to M}$. If we further assume that${\displaystyle M}$ is ametric space, then (uniform) convergence of the${\displaystyle f_n}$ to${\displaystyle f}$ is also well defined. The following result states that continuity is preserved by uniform convergence:

This theorem is proved by the "⁠${\displaystyle \epsilon /3}$⁠ trick", and is the archetypal example of this trick: to prove a given inequality (that a desired quantity is less than⁠${\displaystyle \epsilon }$⁠), one uses the definitions of continuity and uniform convergence to produce 3 inequalities (demonstrating three separate quantities are each less than⁠${\displaystyle \epsilon /3}$⁠), and then combines them via thetriangle 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 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 ofuniformly continuous functions is uniformly continuous; for alocally compact space, continuity is equivalent to local uniform continuity, and thus the uniform limit of continuous functions is continuous.

If${\displaystyle S}$ is an interval and all the functions${\displaystyle f_n}$ aredifferentiable and converge to a limit${\displaystyle f}$, it is often desirable to determine the derivative function${\displaystyle f^{\prime }}$ by taking the limit of the sequence${\displaystyle f_n^{\prime }}$. 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, seeWeierstrass 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${\displaystyle f_n(x)=n^{-1/2}\sin (nx)}$ with uniform limit${\displaystyle f_n\rightrightarrows f\equiv 0}$. Clearly,${\displaystyle f^{\prime }}$ is also identically zero. However, the derivatives of the sequence of functions are given by${\displaystyle f_n^{\prime }(x)=n^{1/2}\cos nx,}$ and the sequence${\displaystyle f_n^{\prime }}$ does not converge to${\displaystyle f^{\prime },}$ 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:^[4]

If${\displaystyle (f_n)}$ is a sequence of differentiable functions on${\displaystyle [a,b]}$ such that${\displaystyle \underset{n\to \mathrm{\infty }}{lim}{f}_n(x_0)}$ exists (and is finite) for some${\displaystyle x_0\in [a,b]}$ and the sequence${\displaystyle (f_n^{\prime })}$ converges uniformly on${\displaystyle [a,b]}$, then${\displaystyle f_n}$ converges uniformly to a function${\displaystyle f}$ on${\displaystyle [a,b]}$, and${\displaystyle f^{\prime }(x)=\underset{n\to \mathrm{\infty }}{lim}{f}_n^{\prime }(x)}$ for${\displaystyle x\in [a,b]}$.

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

If${\displaystyle {(f_n)}_{n=1}^{\mathrm{\infty }}}$ is a sequence of Riemann integrable functions defined on acompact interval${\displaystyle I}$ which uniformly converge with limit${\displaystyle f}$, then${\displaystyle f}$ is Riemann integrable and its integral can be computed as the limit of the integrals of the${\displaystyle f_n}$:$${\displaystyle {\int }_{I}f=\underset{n\to \mathrm{\infty }}{lim}{\int }_I{f}_n.}$$

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, forn sufficiently large, the graph of${\displaystyle f_n}$ is withinε of the graph off, and so the upper sum and lower sum of${\displaystyle f_n}$ are each within${\displaystyle \epsilon |I|}$ of the value of the upper and lower sums of${\displaystyle f}$, 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 theLebesgue integral instead.

UsingMorera's Theorem, one can show that if a sequence ofanalytic 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 (seeWeierstrass function).

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

With this definition comes the following result:

Letx₀ be contained in the setE and eachf_n be continuous atx₀. If$f=\sum _{n=1}^{\mathrm{\infty }}{f}_n$ converges uniformly onE thenf is continuous atx₀ inE. Suppose that${\displaystyle E=[a,b]}$ and eachf_n is integrable onE. If$\sum _{n=1}^{\mathrm{\infty }}{f}_n$ converges uniformly onE thenf is integrable onE and the series of integrals off_n is equal to integral of the series of f_n.

If the domain of the functions is ameasure spaceE then the related notion ofalmost uniform convergence can be defined. We say a sequence of functions${\displaystyle (f_n)}$ converges almost uniformly onE if for every${\displaystyle \delta >0}$ there exists a measurable set${\displaystyle E_{\delta }}$ with measure less than${\displaystyle \delta }$ such that the sequence of functions${\displaystyle (f_n)}$ converges uniformly on${\displaystyle E\setminus E_{\delta }}$. 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 uniformlyalmost 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 convergesalmost everywhere also converges almost uniformly on the same set.

Almost uniform convergence impliesalmost everywhere convergence andconvergence in measure.

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

• 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

"Uniform convergence",Encyclopedia of Mathematics,EMS Press, 2001 [1994]

• Graphic examples of uniform convergence of Fourier series from the University of Colorado

• Calculus
• Series (mathematics)
• Topology of function spaces
• Convergence (mathematics)

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