In the mathematical field of analysis, uniform convergence is a mode of convergence of functions stronger than pointwise convergence. A sequence of functions converges uniformly to a limiting function if, roughly speaking, they uniformly approximate the function over the whole domain, meaning that all but finitely many of the functions of the sequence lie in a uniform error bar of the original function. Graphically this means that, given any thin band around the graph of , the graphs of all but finitely many of the functions lie within that thin band. This is in contrast to pointwise convergence, in which all but finitely many of the functions lie in a thin band at each point, but the finite set of functions which must be excluded in order for that to be the case varies from point to point.
The strength of uniform convergence makes it ideal in many applications, where pointwise convergence is not sufficient. For example, the uniform limit of a sequence of continuous functions is automatically continuous; the uniform limit of Riemann integrable functions is automatically Riemann integrable. With additional hypotheses, differentiability can be transferred to the limit function as well. 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 was first formalized by Karl Weierstrass.
History
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.
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 is independent of the variables and 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, who attended his course on elliptic functions in 1839–1840, coined the term gleichmäßig konvergent () which he used in his 1841 paper Zur Theorie der Potenzreihen, published in 1894. Independently, similar concepts were articulated by Philipp Ludwig von Seidel 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.
Definition
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 is a set and is a sequence of real-valued functions defined on it. We say the sequence is uniformly convergent on with limit if for every there exists a natural number such that for all and for all
The notation for uniform convergence of to is not quite standardized and different authors have used a variety of symbols, including (in roughly decreasing order of popularity):
Frequently, no special symbol is used, and authors simply write
to indicate that convergence is uniform. (In contrast, the expression on without an adverb is taken to mean pointwise convergence on : for all , as .)
Since is a complete metric space, the Cauchy criterion can be used to give an equivalent alternative formulation for uniform convergence: converges uniformly on (in the previous sense) if and only if for every , there exists a natural number such that
.
In yet another equivalent formulation (derived by using the definition of limit and supremum), if we define
then converges to uniformly if and only if as . Thus, we can characterize uniform convergence of on as (simple) convergence of in the function space with respect to the uniform metric (also called the supremum metric), defined by
Symbolically,
.
The sequence is said to be locally uniformly convergent with limit if is a metric space and for every , there exists an such that converges uniformly on where is a ball centered at x with the radius r. It is clear that uniform convergence implies local uniform convergence, which implies pointwise convergence.
Notes
Intuitively, a sequence of functions converges uniformly to if, given an arbitrarily small , we can find an so that the functions with all fall within a "tube" of width centered around (i.e., between and ) for the entire domain of the function.
Note that interchanging the order of quantifiers in the definition of uniform convergence by moving "for all " in front of "there exists a natural number " results in a definition of pointwise convergence of the sequence. To make this difference explicit, in the case of uniform convergence, can only depend on , and the choice of has to work for all , for a specific value of that is given. In contrast, in the case of pointwise convergence, may depend on both and , and the choice of only has to work for the specific values of and 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 with .
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 converges uniformly with limit f : E → X if and only if for every entourage V in X, there exists an , such that for every x in E and every , 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 setting. Thus, a sequence converges to f uniformly if for all hyperreal x in the domain of and all infinite n, is infinitely close to (see microcontinuity for a similar definition of uniform continuity). In contrast, pointwise continuity requires this only for real x.
Examples For , a basic example of uniform convergence can be illustrated as follows: the sequence converges uniformly, while does not. Specifically, assume . Each function is less than or equal to when , regardless of the value of . On the other hand, is only less than or equal to at ever increasing values of when values of 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
Then uniform convergence simply means convergence in the uniform norm topology:
.
The sequence of functions
is a classic example of a sequence of functions that converges to a function pointwise but not uniformly. To show this, we first observe that the pointwise limit of as is the function , given by
Pointwise convergence: Convergence is trivial for and , since and , for all . For and given , we can ensure that whenever by choosing , which is the minimum integer exponent of that allows it to reach or dip below (here the upper square brackets indicate rounding up, see ceiling function). Hence, pointwise for all . Note that the choice of depends on the value of and . Moreover, for a fixed choice of , (which cannot be defined to be smaller) grows without bound as approaches 1. These observations preclude the possibility of uniform convergence.
Non-uniformity of convergence: The convergence is not uniform, because we can find an so that no matter how large we choose there will be values of and such that To see this, first observe that regardless of how large becomes, there is always an such that Thus, if we choose we can never find an such that for all and . Explicitly, whatever candidate we choose for , consider the value of at . Since
the candidate fails because we have found an example of an that "escaped" our attempt to "confine" each to within of for all . In fact, it is easy to see that
contrary to the requirement that if .
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, , the limit is not even continuous.
Exponential function The series expansion of the exponential function can be shown to be uniformly convergent on any bounded subset using the Weierstrass M-test.
Theorem (Weierstrass M-test). Let be a sequence of functions and let be a sequence of positive real numbers such that for all and If converges, then converges absolutely and uniformly on .
The complex exponential function can be expressed as the series:
Any bounded subset is a subset of some disc of radius centered on the origin in the complex plane. The Weierstrass M-test requires us to find an upper bound on the terms of the series, with independent of the position in the disc:
To do this, we notice
and take
If is convergent, then the M-test asserts that the original series is uniformly convergent.
The ratio test can be used here:
which means the series over is convergent. Thus the original series converges uniformly for all and since , the series is also uniformly convergent on
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. If is a compact interval (or in general a compact topological space), and is a monotone increasing sequence (meaning for all n and x) of continuous functions with a pointwise limit which is also continuous, then the convergence is necessarily uniform (Dini's theorem). Uniform convergence is also guaranteed if is a compact interval and is an equicontinuous sequence that converges pointwise.
Applications
To continuity
If and are topological spaces, then it makes sense to talk about the continuity of the functions . If we further assume that is a metric space, then (uniform) convergence of the to is also well defined. The following result states that continuity is preserved by uniform convergence:
This theorem is proved by the " trick", and is the archetypal example of this trick: to prove a given inequality (that a desired quantity is less than ), one uses the definitions of continuity and uniform convergence to produce 3 inequalities (demonstrating three separate quantities are each less than ), and then combines them via the triangle inequality to produce the desired inequality.