Uniform convergence


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 on a set as the function domain if, given any arbitrarily small positive number, a number can be found such that each of the functions differs from by no more than at every point ''in. That is, the required value of only depends on not on any particular
By contrast, pointwise convergence of to merely guarantees that given any we can find (that is, possibly depending on as well as such that for the specific'' value of given, falls within of whenever A different may require a larger value of
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, is important because several properties of the functions, such as continuity, Riemann integrability, and, with additional hypotheses, differentiability, are transferred to the limit if the convergence is uniform, but not necessarily if the convergence is not uniform.

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.
Suppose is a set and is a sequence of real-valued functions 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 :
Frequently, no special symbol is used, and authors simply write
to indicate that convergence is uniform.
Since is a complete metric space, the Cauchy criterion can be used to give an equivalent alternative formulation for uniform convergence: converges uniformly on if and only if for every, there exists a natural number such that
In yet another equivalent formulation, if we define
then converges to uniformly if and only if as. Thus, we can characterize uniform convergence of on as convergence of in the function space with respect to the uniform 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 It is clear that uniform convergence implies local uniform convergence, which implies pointwise convergence.

Generalizations

One may straightforwardly extend the concept to functions E → M, where 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 . 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.
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 . Hence, pointwise for all. Note that the choice of depends on the value of and. Moreover, for a fixed choice of, 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. 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, and is a monotone increasing sequence of continuous functions with a pointwise limit which is also continuous, then the convergence is necessarily uniform. 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 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, 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 a Fourier series of continuous functions. The erroneous claim that the pointwise limit of a sequence of continuous functions is continuous 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.