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 if, roughly speaking, they uniformly approximate over the whole domain, meaning that all but finitely many of the functions of the sequence lie in a uniform error bar 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 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).

is a set is uniformly convergent there exists a natural number

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

without an adverb is taken to mean pointwise convergence.)

is a complete metric space, the Cauchy criterion such that

1. .

In yet another equivalent formulation (derived by using the definition of limit and supremum), if we define

in the function space with respect to the uniform metric (also called the supremum metric), defined by

Symbolically,

1. .

is said to be locally uniformly convergent is a metric space 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

) for the entire domain of the function.

" results in a definition of pointwise convergence 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.

The most general setting is the uniform convergence of netss of functions E → X, where X is a uniform space converges uniformly with limit f : E → X if and only if for every entourage V in X, such that for every x in E 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 converges to f uniformly if for all hyperreal x and all infinite n (see microcontinuity for a similar definition of uniform continuity). In contrast, pointwise continuity requires this only for real x.

Examples

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:

1. .

, given by

Pointwise convergence: (here the upper square brackets indicate rounding up, see ceiling function approaches 1. These observations preclude the possibility of uniform convergence.

Non-uniformity of convergence:. Since

. In fact, it is easy to see that

.

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 is not even continuous.

Exponential function

The series expansion of the exponential function using the Weierstrass M-test.

Theorem (Weierstrass M-test). .

The complex exponential function can be expressed as the series:

centered on the origin in the complex plane independent of the position in the disc:

To do this, we notice

is convergent, then the M-test asserts that the original series is uniformly convergent.

The ratio test can be used here:

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.

• is a compact is a monotone increasing for all n and x) of continuous which is also continuous, then the convergence is necessarily uniform (Dini's theorem is an equicontinuous sequence that converges pointwise.

Applications

To continuity

are topological spaces, then it makes sense to talk about the continuity is a metric space 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.

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.

To differentiability

are differentiable. 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 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:

1. .

To integrability

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

1. is a sequence of Riemann integrable functions defined on a compact:

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 is within of the graph 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.

To analyticity

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).

To series

converges:

With this definition comes the following result: Let x0 be contained in the set E and each fn be continuous at x0 converges uniformly on E then f is continuous at x0 in E and each fn is integrable on E converges uniformly on E then f is integrable on E and the series of integrals of fn is equal to integral of the series of fn.

Function spaces

Uniform convergence is the norm topology on several important function spaces denote the space of real-valued or complex-valued continuous functions on the unit interval. With the supremum norm this is a Banach space.

.

is a compact Hausdorff space. In that case one often uses the topology of uniform convergence on compact subsets, also called the compact-open topology is a locally compact, σ-compact Hausdorff space, this topology is generated by the seminorms

into a Fréchet space.

is a closed subspace.

of essentially bounded measurable functions, together with the essential supremum norm.

Almost uniform convergence

If the domain of the functions is a measure space E then the related notion of almost uniform convergence converges almost uniformly on E. 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.

See also

• Uniform convergence in probability

• Modes of convergence (annotated index)

• Dini's theorem

• Arzelà–Ascoli theorem

• Compact-open topology

Notes

References

• Konrad Knopp, Theory and Application of Infinite Series; Blackie and Son, London, 1954, reprinted by Dover Publications, .

• 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);

• 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, .

• William Wade, An Introduction to Analysis, 3rd ed., Pearson, 2005

External links

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