Editing Uniform convergence (section)

== 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.<ref>{{cite journal | doi=10.1016/j.hm.2004.11.010 | volume=32 | issue=4 | title=Exceptions and counterexamples: Understanding Abel's comment on Cauchy's Theorem | journal=Historia Mathematica | pages=453–480| year=2005 | last1=Sørensen | first1=Henrik Kragh | doi-access= }}</ref>

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.<ref>{{Cite book
|title=A history of analysis
|first=Hans Niels
|last=Jahnke
|publisher=AMS Bookstore
|year=2003
|isbn=978-0-8218-2623-2
|chapter=6.7 The Foundation of Analysis in the 19th Century: Weierstrass
|chapter-url=https://books.google.com/books?id=CVRZEXFVsZkC&pg=PA184
|page=184
}}</ref>

Later Gudermann's pupil [[Karl Weierstrass]], who attended his course on elliptic functions in 1839–1840, coined the term ''gleichmäßig konvergent'' ({{langx|de|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]]<ref>{{cite book |last=Lakatos |first=Imre |author-link=Imre Lakatos |title=Proofs and Refutations|year=1976|publisher=Cambridge University Press |pages=[https://archive.org/details/proofsrefutation0000laka/page/141 141] |isbn=978-0-521-21078-2|title-link=Proofs and Refutations }}</ref> 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.