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.[1]

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 ${\textstyle \sum _{n=1}^{\infty }f_{n}(x,\phi ,\psi )}$ is independent of the variables ${\displaystyle \phi }$ 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 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[3] 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 ${\displaystyle E}$ is a set and ${\displaystyle (f_{n})_{n\in \mathbb {N} }}$ is a sequence of real-valued functions on it. We say the sequence ${\displaystyle (f_{n})_{n\in \mathbb {N} }}$ is uniformly 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 ${\displaystyle n\geq N}$ and ${\displaystyle x\in E}$

${\displaystyle |f_{n}(x)-f(x)|<\epsilon .}$

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):

${\displaystyle f_{n}\rightrightarrows f,\quad {\underset {n\to \infty }{\mathrm {unif\ lim} }}f_{n}=f,\quad f_{n}{\overset {\mathrm {unif.} }{\longrightarrow }}f,\quad f=u-\lim _{n\to \infty }f_{n}.}$

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

${\displaystyle f_{n}\to f\quad \mathrm {uniformly} }$

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

Since ${\displaystyle \mathbb {R} }$ is a complete metric space, the Cauchy 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

${\displaystyle x\in E,m,n\geq N\implies |f_{m}(x)-f_{n}(x)|<\epsilon }$.

In yet another equivalent formulation, if we define

${\displaystyle d_{n}=\sup _{x\in E}|f_{n}(x)-f(x)|,}$

then ${\displaystyle f_{n}}$ converges to ${\displaystyle f}$ uniformly if and only if ${\displaystyle d_{n}\to 0}$ as ${\displaystyle n\to \infty }$. 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 the function space ${\displaystyle \mathbb {R} ^{E}}$ with respect to the uniform metric (also called the supremum metric), defined by

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

Symbolically,

${\displaystyle f_{n}\rightrightarrows f\iff \lim _{n\to \infty }d(f_{n},f)=0}$.

The sequence ${\displaystyle (f_{n})_{n\in \mathbb {N} }}$ is said to be locally uniformly convergent with limit ${\displaystyle f}$ if ${\displaystyle E}$ is a metric 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.

For ${\displaystyle x\in [0,1)}$, a basic example of uniform convergence can be illustrated as follows: the sequence ${\displaystyle (1/2)^{x+n}}$ converges uniformly, while ${\displaystyle x^{n}}$ does not. Specifically, assume ${\displaystyle \epsilon =1/4}$. Each function ${\displaystyle (1/2)^{x+n}}$ is less than or equal to ${\displaystyle 1/4}$ when ${\displaystyle n\geq 2}$, regardless of the value of ${\displaystyle x}$. On the other hand, ${\displaystyle x^{n}}$ is only less than or equal to ${\displaystyle 1/4}$ 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 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

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

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

${\displaystyle \lim _{n\to \infty }\|f_{n}-f\|_{\infty }=0}$.

The sequence of functions ${\displaystyle (f_{n})}$

${\displaystyle {\begin{cases}f_{n}:[0,1]\to [0,1]\\f_{n}(x)=x^{n}\end{cases}}}$

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 ${\displaystyle n\to \infty }$ is the function ${\displaystyle f}$, given by

${\displaystyle f(x)=\lim _{n\to \infty }f_{n}(x)={\begin{cases}0,&x\in [0,1);\\1,&x=1.\end{cases}}}$

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 ${\displaystyle |f_{n}(x)-f(x)|<\epsilon }$ whenever ${\displaystyle n\geq N}$ by choosing ${\displaystyle N=\lceil \log \epsilon /\log x\rceil }$ (here the upper square brackets indicate rounding up, see ceiling 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 ${\displaystyle n\geq N}$ such that ${\displaystyle |f_{n}(x)-f(x)|\geq \epsilon .}$ 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 ${\displaystyle f_{n}(x_{0})=1/2.}$ Thus, if we choose ${\displaystyle \epsilon =1/4,}$ we can never find an ${\displaystyle N}$ such that ${\displaystyle |f_{n}(x)-f(x)|<\epsilon }$ for all ${\displaystyle x\in [0,1]}$ and ${\displaystyle n\geq N}$. Explicitly, whatever candidate we choose for ${\displaystyle N}$, consider the value of ${\displaystyle f_{N}}$ at ${\displaystyle x_{0}=(1/2)^{1/N}}$. Since

${\displaystyle \left|f_{N}(x_{0})-f(x_{0})\right|=\left|\left[\left({\frac {1}{2}}\right)^{\frac {1}{N}}\right]^{N}-0\right|={\frac {1}{2}}>{\frac {1}{4}}=\epsilon ,}$

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

${\displaystyle \lim _{n\to \infty }\|f_{n}-f\|_{\infty }=1,}$

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

• 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 ${\displaystyle S}$ is a compact interval (or in general a compact topological space), and ${\displaystyle (f_{n})}$ is a monotone increasing sequence (meaning ${\displaystyle f_{n}(x)\leq f_{n+1}(x)}$ for all n and x) of continuous 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 an equicontinuous sequence that converges pointwise.

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 ${\displaystyle (f_{n})}$ converges almost uniformly on E 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 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

• 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