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# Uniform convergence

In the mathematical field of analysis, uniform convergence is a type of convergence stronger than pointwise convergence. A sequence {fn} of functions converges uniformly to a limiting function f if the speed of convergence of fn(x) to f(x) does not depend on x.

The concept is important because several properties of the functions fn, such as continuity and Riemann integrability, are transferred to the limit f if the convergence is uniform.

## History

Augustin Louis Cauchy in 1821 published the false statement, but with a purported proof, that the pointwise limit of a sequence of continuous functions is always continuous. Joseph Fourier and Niels Henrik Abel found counterexamples to this statement in the context of Fourier series, showing that Cauchy's proof had to be incorrect. Dirichlet then analyzed the proof and found the mistake: from the hypothesis of pointwise convergence it was assumed to follow that the sequence had the property of uniform convergence, not realizing that this is a strictly stronger condition.

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 $\textstyle{\sum_{n=1}^\infty f_n(x,\phi,\psi)}$ is independent of the variables ϕ and ψ. 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 (German: uniformly convergent) which he used in his 1841 paper Zur Theorie der Potenzreihen, published in 1894. Independently a similar concept was used by Philipp Ludwig von Seidel and George Gabriel Stokes but without having any major impact on further development. 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

Suppose S is a set and fn : SR is a real-valued function for every natural number n. We say that the sequence (fn)nN is uniformly convergent with limit f : SR if for every ε > 0, there exists a natural number N such that for all xS and all nN we have |fn(x) − f(x)| < ε.

Consider the sequence αn = supx |fn(x) − f(x)| where the supremum is taken over all xS. Clearly fn converges to f uniformly if and only if αn tends to 0.

The sequence (fn)nN is said to be locally uniformly convergent with limit f if for every x in some metric space S, there exists an r > 0 such that (fn) converges uniformly on B(x,r) ∩ S.

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