Uniform convergence

thumb|300x300px|A sequence of functions <math>(f_n)</math> converges uniformly to <math>f</math> when for arbitrary small <math>\varepsilon</math> there is an index <math>N</math> such that the graph of <math>f_n</math> is in the <math>\varepsilon</math>-tube around <math>f</math> whenever <math>n\ge N.</math>

[[File:Drini-nonuniformconvergence.png|thumb|300x300px|

The limit of a sequence of continuous functions does not have to be continuous: the sequence of functions <math>f_n(x)=\sin^n(x)</math> (marked in green and blue) converges pointwise over the entire domain, but the limit function is discontinuous (marked in red).]]

In the mathematical field of analysis, uniform convergence is a mode of convergence of functions stronger than pointwise convergence. A sequence of functions <math>(f_n)</math> converges uniformly to a limiting function <math>f</math> on a set <math>E</math> as the function domain if, given any arbitrarily small positive number <math>\varepsilon</math>, a number <math>N</math> can be found such that each of the functions <math>f_N, f_{N+1},f_{N+2},\ldots</math> differs from <math>f</math> by no more than <math>\varepsilon</math> at every point <math>x</math> in <math>E</math>. That is, the required value of <math>N</math> only depends on not on any particular

By contrast, pointwise convergence of <math>(f_n)</math> to <math>f</math> merely guarantees that given any we can find <math>N=N(\varepsilon, x)</math> (that is, possibly depending on <math>x</math> as well as such that for the specific value of <math>x</math> given, <math>f_n(x)</math> falls within <math>\varepsilon</math> of <math>f(x)</math> whenever <math>n\geq N.</math> A different <math>x</math> may require a larger value of <math>N.</math>

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 <math>f_n</math>, such as continuity, Riemann integrability, and, with additional hypotheses, differentiability, are transferred to the limit <math>f</math> 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 <math display="inline">\sum_{n=1}^\infty f_n(x,\phi,\psi)</math> is independent of the variables <math>\phi</math> and <math>\psi.</math> 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).

: If <math>(f_n)</math> is a sequence of differentiable functions on <math>[a,b]</math> such that <math>\lim_{n\to\infty} f_n(x_0)</math> exists (and is finite) for some <math>x_0\in[a,b]</math> and the sequence <math>(f'_n)</math> converges uniformly on <math>[a,b]</math>, then <math>f_n</math> converges uniformly to a function <math>f</math> on <math>[a,b]</math>, and <math> f'(x) = \lim_{n\to \infty} f'_n(x)</math> for <math>x \in [a, b]</math>.

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:

: If <math>{(f_n)}_{n=1}^\infty</math> is a sequence of Riemann integrable functions defined on a compact interval <math>I</math> which uniformly converge with limit <math> f</math>, then <math> f</math> is Riemann integrable and its integral can be computed as the limit of the integrals of the <math> f_n</math>: <math display="block">\int_I f = \lim_{n\to\infty}\int_I f_n.</math>

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 sufficiently large, the graph of <math>f_n</math> is within of the graph of f, and so the upper sum and lower sum of <math>f_n</math> are each within <math>\varepsilon |I|</math> of the value of the upper and lower sums of <math>f</math>, 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

We say that <math display="inline">\sum_{n=1}^\infty f_n</math> converges:

With this definition comes the following result:

<blockquote>Let x0 be contained in the set E and each fn be continuous at x0. If <math display="inline"> f = \sum_{n=1}^\infty f_n</math> converges uniformly on E then f is continuous at x0 in E. Suppose that <math>E = [a, b]</math> and each fn is integrable on E. If <math display="inline">\sum_{n=1}^\infty f_n</math> 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.</blockquote>

Function spaces

Uniform convergence is the norm topology on several important function spaces. For example, let <math>C([0,1])</math> denote the space of real-valued or complex-valued continuous functions on the unit interval. With the supremum norm

<math display="block">\|f\|_\infty=\sup_{x\in[0,1]} |f(x)|,</math>

this is a Banach space. A sequence <math>f_n</math> in <math>C([0,1])</math> converges to <math>f</math> in this norm precisely when

<math display="block">\|f_n-f\|_\infty\to 0,</math>

which is exactly uniform convergence on <math>[0,1]</math>.

The completeness of <math>C([0,1])</math> follows from the completeness of the real or complex numbers. If <math>(f_n)</math> is Cauchy in the supremum norm, then for each <math>x\in[0,1]</math> the sequence <math>(f_n(x))</math> is Cauchy, since

<math display="block">|f_n(x)-f_m(x)|\leq \|f_n-f_m\|_\infty.</math>

Thus <math>f_n(x)</math> converges pointwise to a function <math>f(x)</math>. Moreover, the convergence is uniform: for every <math>\varepsilon>0</math>, there is an <math>N</math> such that <math>\|f_n-f_m\|_\infty<\varepsilon</math> whenever <math>n,m\geq N</math>; letting <math>m\to\infty</math> gives

<math display="block">|f_n(x)-f(x)|\leq\varepsilon</math> for all <math>x\in[0,1]</math> and all <math>n\geq N</math>. Since a uniform limit of continuous functions is continuous, <math>f\in C([0,1])</math>.

More generally, if <math>X</math> is a compact Hausdorff space, then <math>C(X)</math>, with the supremum norm, is a Banach space. If <math>X</math> is not compact, the supremum norm need not be finite on all of <math>C(X)</math>. In that case one often uses the topology of uniform convergence on compact subsets, also called the compact-open topology in this setting. For example, if <math>X</math> is a locally compact, σ-compact Hausdorff space, this topology is generated by the seminorms

:<math display="block">p_K(f)=\sup_{x\in K}|f(x)|,</math>

where <math>K</math> ranges over compact subsets of <math>X</math>, making <math>C(X)</math> into a Fréchet space.

For any set <math>X</math>, the space <math>\ell^\infty(X)</math> of all bounded real-valued or complex-valued functions on <math>X</math>, equipped with the supremum norm

<math display="block">\|f\|_\infty=\sup_{x\in X}|f(x)|,</math>

is also a Banach space. Convergence in this norm is exactly uniform convergence on <math>X</math>. The sup norm on <math>C(X)</math> for a compact space <math>X</math> is inherited from the norm of <math>\ell^\infty</math>, so <math>C(X)\subset\ell^\infty(X)</math> is a closed subspace.

If <math>(X,\mu)</math> is a measure space, the space <math>L^\infty(X,\mu)</math> of essentially bounded measurable functions on <math>X</math>, together with the essential supremum norm <math>\|f\|=\operatorname{ess sup}_{x\in X}|f(x)|</math>. For many spaces of practical interest, such as compact subsets of Euclidean space and compact topological, the natural Borel measure embeds <math>C(X)</math> into a closed subspace of <math>L^\infty(X,\mu)</math>. In these cases the norm giving uniform convergence is the same as the <math>L^\infty</math> norm, and therefore called the <math>L^\infty</math> norm.

Almost uniform convergence

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 <math>(f_n)</math> converges almost uniformly on E if for every <math>\delta > 0</math> there exists a measurable set <math>E_\delta</math> with measure less than <math>\delta</math> such that the sequence of functions <math>(f_n)</math> converges uniformly on <math>E \setminus E_\delta</math>. 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.

Notes

References

Konrad Knopp, <cite>Theory and Application of Infinite Series</cite>; Blackie and Son, London, 1954, reprinted by Dover Publications, .
G. H. Hardy, <cite>Sir George Stokes and the concept of uniform convergence</cite>; Proceedings of the Cambridge Philosophical Society, 19, pp. 148–156 (1918)
Bourbaki; <cite>Elements of Mathematics: General Topology. Chapters 5–10 (paperback)</cite>;
Walter Rudin, <cite>Principles of Mathematical Analysis</cite>, 3rd ed., McGraw–Hill, 1976.
Gerald Folland, Real Analysis: Modern Techniques and Their Applications, Second Edition, John Wiley & Sons, Inc., 1999, .
William Wade, <cite> An Introduction to Analysis</cite>, 3rd ed., Pearson, 2005

External links

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