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 {\displaystyle (f_{n})}(f_{n}) converges uniformly to a limiting function {\displaystyle f}f on a set {\displaystyle E}E if, given any arbitrarily small positive number {\displaystyle \epsilon }\epsilon , a number {\displaystyle N}N can be found such that each of the functions {\displaystyle f_{N},f_{N+1},f_{N+2},\ldots }{\displaystyle f_{N},f_{N+1},f_{N+2},\ldots } differs from {\displaystyle f}f by no more than {\displaystyle \epsilon }\epsilon at every point {\displaystyle x}x in {\displaystyle E}E. Described in an informal way, if {\displaystyle f_{n}}f_{n} converges to {\displaystyle f}f uniformly, then the rate at which {\displaystyle f_{n}(x)}f_{n}(x) approaches {\displaystyle f(x)}f(x) is “uniform” throughout its domain in the following sense: in order to guarantee that {\displaystyle f_{n}(x)}f_{n}(x) falls within a certain distance {\displaystyle \epsilon }\epsilon of {\displaystyle f(x)}f(x), we do not need to know the value of {\displaystyle x\in E}x\in E in question — there can be found a single value of {\displaystyle N=N(\epsilon )}{\displaystyle N=N(\epsilon )} independent of {\displaystyle x}x, such that choosing {\displaystyle n\geq N}n\geq N will ensure that {\displaystyle f_{n}(x)}f_{n}(x) is within {\displaystyle \epsilon }\epsilon of {\displaystyle f(x)}f(x) for all {\displaystyle x\in E}x\in E. In contrast, pointwise convergence of {\displaystyle f_{n}}f_{n} to {\displaystyle f}f merely guarantees that for any {\displaystyle x\in E}x\in E given in advance, we can find {\displaystyle N=N(\epsilon ,x)}{\displaystyle N=N(\epsilon ,x)} ({\displaystyle N}N can depend on the value of {\displaystyle x}x) so that, for that particular {\displaystyle x}x, {\displaystyle f_{n}(x)}f_{n}(x) falls within {\displaystyle \epsilon }\epsilon of {\displaystyle f(x)}f(x) whenever {\displaystyle n\geq N}n\geq N.

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 {\displaystyle f_{n}}f_{n}, such as continuity, Riemann integrability, and, with additional hypotheses, differentiability, are transferred to the limit {\displaystyle f}f if the convergence is uniform, but not necessarily if the convergence is not uniform.

1 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.[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 )}{\textstyle \sum {n=1}^{\infty }f{n}(x,\phi ,\psi )} is independent of the variables {\displaystyle \phi }\phi and {\displaystyle \psi .}\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.

2 Definition

2.1 Notes

2.2 Generalizations

2.3 Definition in a hyperreal setting

3 Examples

3.1 Exponential function

4 Properties

5 Applications

5.1 To continuity

5.2 To differentiability

5.3 To integrability

5.4 To analyticity

5.5 To series

6 Almost uniform convergence

7 See also