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| In the [[mathematics|mathematical]] field of [[mathematical analysis|analysis]], '''uniform convergence''' is a type of [[Limit of a sequence|convergence]] stronger than [[pointwise convergence]]. A [[sequence]] {''f''<sub>''n''</sub>} of [[function (mathematics)|functions]] '''converges uniformly''' to a limiting function ''f'' if the speed of convergence of ''f''<sub>''n''</sub>(''x'') to ''f''(''x'') does not depend on ''x''.
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| The concept is important because several properties of the functions ''f''<sub>''n''</sub>, such as [[continuous function|continuity]] and [[Riemann integral|Riemann integrability]], are transferred to the [[limit of a function|limit]] ''f'' if the convergence is uniform.
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| Uniform convergence to a function on a given interval can be defined in terms of the [[uniform norm]].
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| == History ==
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| Some historians claim{{who?|date=January 2014}} that [[Augustin Louis Cauchy]] in 1821 published a false statement, but with a purported proof, that the pointwise limit of a series of continuous functions is always continuous; however, [[Imre Lakatos | Lakatos]] offers a re-assessment{{cn|date=January 2014}} of Cauchy's approach. [[Niels Henrik Abel]] in 1826 found purported counterexamples to this statement in the context of [[Fourier series]], arguing that Cauchy's proof had to be incorrect.{{cn|date=January 2014}} Cauchy ultimately responded in 1853 with a clarification of his 1821 formulation.{{cn|date=January 2014}}
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| 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>\textstyle{\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
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| |first=Hans Niels
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| |last=Jahnke
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| |publisher=AMS Bookstore
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| |year=2003
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| |isbn=978-0-8218-2623-2
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| |chapter=6.7 The Foundation of Analysis in the 19th Century: Weierstrass
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| |postscript=, [http://books.google.com/books?id=CVRZEXFVsZkC&pg=PA184 p. 184].
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| }}</ref>
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| Later Gudermann's pupil [[Karl Weierstrass]], who attended his course on elliptic functions in 1839–1840, coined the term ''gleichmäßig konvergent'' ({{lang-de|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]]<ref>{{cite book |last=Lakatos |first=Imre |authorlink=Imre Lakatos |title=[[Proofs and Refutations]]|year=1976|publisher=Cambridge University Press |pages=141 |isbn=0-521-21078-X}}</ref> and [[George Gabriel Stokes]] but without having any major impact on further development.{{Citation needed|date=December 2011}} [[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."
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| 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.
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| == Definition ==
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| Suppose {{math|''S''}} is a [[Set (mathematics)|set]] and {{math|''f''<sub>''n''</sub> : ''S'' → '''R'''}} is a [[real number|real]]-valued function for every [[natural number]] {{math|''n''}}. We say that the sequence {{math|(''f''<sub>''n''</sub>)<sub>''n''∈'''N'''</sub>}} is '''uniformly convergent''' with limit {{math|''f'' : ''S'' → '''R'''}} if for every {{math|ε > 0}}, there exists a natural number {{math|''N''}} such that for all {{math|''x'' ∈ ''S''}} and all {{math|''n'' ≥ ''N''}} we have {{math|{{!}}''f''<sub>''n''</sub>(''x'') − ''f''(''x''){{!}} < ε}}.
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| Consider the sequence {{math|''α''<sub>''n''</sub> {{=}} sup<sub>''x''</sub> {{!}}''f''<sub>''n''</sub>(''x'') − ''f''(''x''){{!}}}} where the [[supremum]] is taken over all {{math|''x'' ∈ ''S''}}. Then {{math|''f''<sub>''n''</sub>}} converges to {{math|''f''}} uniformly [[if and only if]] {{math|''α''<sub>''n''</sub>}} tends to 0.
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| The sequence {{math|(''f''<sub>''n''</sub>)<sub>''n''∈'''N'''</sub>}} is said to be '''locally uniformly convergent''' with limit {{math|''f''}} if for every {{math|''x''}} in some metric space {{math|''S''}}, there exists an {{math|''r'' > 0}} such that {{math|(''f''<sub>''n''</sub>)}} converges uniformly on {{math|''B''(''x'',''r'') ∩ ''S''}}.
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| === Notes ===
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| Note that interchanging the order of "there exists {{math|''N''}}" and "for all {{math|''x''}}" in the definition above results in a statement equivalent to the [[pointwise convergence]] of the sequence. That notion can be defined as follows: the sequence (''f''<sub>''n''</sub>) converges pointwise with limit {{math|''f'' : ''S'' → '''R'''}} if and only if
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| :for every {{math|''x'' ∈ ''S''}} and every {{math|ε > 0}}, there exists a natural number ''N'' such that for all {{math|''n'' ≥ ''N''}} one has {{math|{{!}}''f''<sub>''n''</sub>(''x'') − ''f''(''x''){{!}} < ε}}.
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| Here the order of the [[universal quantifier]]s for {{math|''x''}} and for {{math|ε}} is not important, but the order of the former and the [[existential quantifier]] for {{math|''N''}} is.
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| In the case of uniform convergence, {{math|''N''}} can only depend on {{math|ε}}, while in the case of pointwise convergence {{math|''N''}} may depend on both {{math|ε}} and {{math|''x''}}. It is therefore plain that uniform convergence implies pointwise convergence. The converse is not true, as the following example shows: take {{math|''S''}} to be the [[unit interval]] [0,1] and define {{math|''f''<sub>''n''</sub>(''x'') {{=}} ''x''<sup>''n''</sup>}} for every natural number {{math|''n''}}. Then {{math|(''f''<sub>''n''</sub>)}} converges pointwise to the function {{math|''f''}} defined by {{math|''f''(''x'') {{=}} 0}} if {{math|''x'' < 1}} and {{math|''f''(1) {{=}} 1}}. This convergence is not uniform: for instance for {{math|ε {{=}} 1/4}}, there exists no {{math|''N''}} as required by the definition. This is because solving for {{math|''n''}} gives {{math|''n'' > log ε / log ''x''}}. This depends on {{math|''x''}} as well as on {{math|ε}}. Also note that it is impossible to find a suitable bound for {{math|''n''}} that does not depend on {{math|''x''}} because for any nonzero value of {{math|ε}}, {{math|log ε / log ''x''}} grows without bounds as {{math|''x''}} tends to 1.
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| === Generalizations ===
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| One may straightforwardly extend the concept to functions ''S'' → ''M'', where (''M'', ''d'') is a [[metric space]], by replacing |''f''<sub>''n''</sub>(''x'') − ''f''(''x'')| with ''d''(''f''<sub>''n''</sub>(''x''), ''f''(''x'')).
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| The most general setting is the uniform convergence of [[net (mathematics)|net]]s of functions ''S'' → ''X'', where ''X'' is a [[uniform space]]. We say that the net (''f''<sub>α</sub>) ''converges uniformly'' with limit ''f'' : ''S'' → ''X'' if and only if
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| :for every [[entourage (topology)|entourage]] ''V'' in ''X'', there exists an α<sub>0</sub>, such that for every ''x'' in ''S'' and every α ≥ α<sub>0</sub>: (''f''<sub>α</sub>(''x''), ''f''(''x'')) is in ''V''.
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| The above mentioned theorem, stating that the uniform limit of continuous functions is continuous, remains correct in these settings.
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| ===Definition in a hyperreal setting===
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| Uniform convergence admits a simplified definition in a [[hyperreal number|hyperreal]] setting. Thus, a sequence <math>f_n</math> converges to ''f'' uniformly if for all ''x'' in the domain of ''f*'' and all infinite ''n'', <math>f_n^*(x)</math> is infinitely close to <math>f^*(x)</math> (see [[microcontinuity]] for a similar definition of uniform continuity).
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| == Examples ==
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| Given a [[topological space]] ''X'', we can equip the space of [[bounded function|bounded]] [[real number|real]] or [[complex number|complex]]-valued functions over ''X'' with the [[uniform norm]] topology. Then uniform convergence simply means [[Limit of a sequence|convergence]] in the [[uniform norm]] topology.
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| The sequence <math> f_n:[0,1]\rightarrow [0,1] </math> with <math> f_n(x):=x^n </math> converges pointwise but not uniformly:
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| : <math>\lim_{n\rightarrow \infty}f_n(x) = \begin{cases} 0, & x \in [0,1) \\ 1, & x=1. \end{cases} </math>
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| 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'', <math>f_n\in C^{\infty}([0,1])</math>, the limit <math>\lim_{n\rightarrow \infty}f_n</math> is not even continuous.
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| === Exponential function ===
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| The series expansion of the [[exponential function]] can be shown to be uniformly convergent on any bounded subset S of <math>\mathbb{C}</math> using the [[Weierstrass M-test]].
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| Here is the series:
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| ::<math>\sum_{n=0}^{\infty}\frac{z^n}{n!}.</math>
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| Any bounded subset is a subset of some disc <math>D_R</math> of radius R, centered on the origin in the [[complex plane]]. The Weierstrass M-test requires us to find an upper bound <math>M_n</math> on the terms of the series, with <math>M_n</math> independent of the position in the disc:
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| ::<math>\left| \frac{z^n}{n!}\right|\le M_n , \forall z\in D_R.</math>
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| This is trivial:
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| ::<math>\left| \frac{z^n}{n!}\right| \le \frac{\left| z\right|^n}{n!} \le \frac{R^n}{n!}</math>
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| ::<math>\Rightarrow M_n=\frac{R^n}{n!}.</math>
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| If <math>\sum_{n=0}^{\infty}M_n</math> is convergent, then the M-test asserts that the original series is uniformly convergent.
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| The [[ratio test]] can be used here:
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| ::<math>\lim_{n \to \infty}\frac{M_{n+1}}{M_n}=\lim_{n \to \infty}\frac{R^{n+1}}{R^n}\frac{n!}{(n+1)!}=\lim_{n \to \infty}\frac{R}{n+1}=0</math>
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| which means the series over <math>M_n</math> is convergent.
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| Thus the original series converges uniformly for all <math>z\in D_R</math>, and since <math>S\subset D_R</math>, the series is also uniformly convergent on S.
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| == Properties ==
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| * Every uniformly convergent sequence is locally uniformly convergent.
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| * Every locally uniformly convergent sequence is [[compactly convergent]].
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| * For [[locally compact space]]s local uniform convergence and compact convergence coincide.
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| * 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 sequence|uniformly Cauchy]].
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| == Applications ==
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| ===To continuity===
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| {{see also|Uniform limit theorem}}
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| [[Image:Drini nonuniformconvergence SVG.svg|thumb|350px|right|Counterexample to a strengthening of the uniform convergence theorem, in which pointwise convergence, rather than uniform convergence, is assumed. The continuous green functions <math>\scriptstyle \scriptstyle\sin^n(x)</math> converge to the non-continuous red function. This can happen only if convergence is not uniform.]]
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| If <math>\scriptstyle S</math> is a real [[interval (mathematics)|interval]] (or indeed any [[topological space]]), we can talk about the [[continuous function (topology)|continuity]] of the functions <math>\scriptstyle f_n</math> and <math>\scriptstyle f</math>. The following is the more important result about uniform convergence:
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| : '''Uniform convergence theorem'''. If <math>\scriptstyle (f_n)_n</math> is a sequence of continuous functions which converges ''uniformly'' towards the function <math>\scriptstyle f</math> on an interval <math>\scriptstyle S</math>, then <math>\scriptstyle f</math> is continuous on <math>\scriptstyle S</math> as well.
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| This theorem is proved by the "<math>\epsilon/3</math> trick", and is the archetypal example of this trick: to prove a given inequality (<math><\epsilon</math>), one uses the definitions of continuity and uniform convergence to produce 3 inequalities (<math><\epsilon/3</math>), and then combines them via the [[triangle inequality]] to produce the desired inequality.
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| This theorem is important, since pointwise convergence of continuous functions is not enough to guarantee continuity of the limit function as the image illustrates.
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| 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.
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| ===To differentiability===
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| If <math>\scriptstyle S</math> is an interval and all the functions <math>\scriptstyle f_n</math> are [[derivative|differentiable]] and converge to a limit <math>\scriptstyle f</math>, it is often desirable to differentiate the limit function <math>\scriptstyle f</math> by taking the limit of the derivatives of <math>\scriptstyle f_n</math>. This is however in general not possible: even if the convergence is uniform, the limit function need not be differentiable, and even if it is differentiable, the derivative of the limit function need not be equal to the limit of the derivatives. Consider for instance <math>\scriptstyle f_n(x) = \frac1n \sin(nx)</math> with uniform limit 0, but the derivatives do not approach 0. The precise statement covering this situation is as follows:
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| : If <math>\scriptstyle f_n</math> converges pointwise to <math>\scriptstyle f</math>, and if all the <math>\scriptstyle f_n</math> are differentiable, and if the derivatives <math>\scriptstyle f '_n</math> converge uniformly to ''g'', then <math>\scriptstyle f</math> is differentiable and its derivative is ''g''.
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| ===To integrability===
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| Similarly, one often wants to exchange integrals and limit processes. For the [[Riemann integral]], this can be done if uniform convergence is assumed:
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| : If <math>\scriptstyle (f_n)_{n=1}^\infty</math> is a sequence of Riemann integrable functions which uniformly converge with limit <math>\scriptstyle f</math>, then <math>\scriptstyle f</math> is Riemann integrable and its integral can be computed as the limit of the integrals of the <math>\scriptstyle f_n</math>.
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| 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 integration|Lebesgue integral]] instead.
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| : If <math>\scriptstyle S</math> is a [[compact space|compact]] interval (or in general a compact topological space), and <math>\scriptstyle (f_n)</math> is a [[monotonic|monotone increasing]] sequence (meaning <math>\scriptstyle f_n(x) \leq f_{n+1}(x)</math> for all ''n'' and ''x'') of ''continuous'' functions with a pointwise limit <math>\scriptstyle f</math> which is also continuous, then the convergence is necessarily uniform ([[Dini's theorem]]). Uniform convergence is also guaranteed if <math>\scriptstyle S</math> is a compact interval and <math>\scriptstyle(f_n)</math> is an [[equicontinuity|equicontinuous]] sequence that converges pointwise.
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| ===To Analyticity===
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| If a sequence of [[Analytic_function|analytic]] functions converges uniformly in a region S of the complex plane, then the limit is analytic in S. This demonstrates an example 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.
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| ==Almost uniform convergence==
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| If the domain of the functions is a [[measure space]] then the related notion of '''almost uniform convergence''' can be defined. We say a sequence of functions converges almost uniformly on ''E'' if there is a measurable subset ''F'' of ''E'' with arbitrarily small measure such that the sequence converges uniformly on the complement ''E'' \ ''F''.
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| 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.
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| [[Egorov's theorem]] guarantees that on a finite measure space, a sequence of functions that converges [[Pointwise_convergence#Almost everywhere convergence|almost everywhere]] also converges almost uniformly on the same set.
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| Almost uniform convergence implies [[almost everywhere convergence]] and [[convergence in measure]].
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| ==See also==
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| *[[Modes of convergence (annotated index)]]
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| *[[Dini's theorem]]
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| ==Notes==
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| {{reflist}}
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| ==References==
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| * [[Konrad Knopp]], <cite>Theory and Application of Infinite Series</cite>; Blackie and Son, London, 1954, reprinted by Dover Publications, ISBN 0-486-66165-2.
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| * [[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)
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| * [[Bourbaki]]; <cite>Elements of Mathematics: General Topology. Chapters 5–10 (Paperback)</cite>; ISBN 0-387-19374-X
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| * Walter Rudin, <cite>Principles of Mathematical Analysis</cite>, 3rd ed., McGraw–Hill, 1976.
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| * [[Gerald Folland]], Real Analysis: Modern Techniques and Their Applications, Second Edition, John Wiley & Sons, Inc., 1999, ISBN 0-471-31716-0.
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| ==External links==
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| * {{springer|title=Uniform convergence|id=p/u095230}}
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| * [http://www.encyclopediaofmath.org/index.php/Cauchy_criteria Cauchy criteria] at [http://www.encyclopediaofmath.org/ Encyclopedia of Mathematics]
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| * {{planetmath reference|id=3700|title=Uniform convergence}}
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| * {{planetmath reference|id=6209|title=Limit point of function}}
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| * {{planetmath reference|id=4986|title=Converges uniformly}}
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| * {{planetmath reference|id=2311|title=Convergent series}}
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| * [http://amath.colorado.edu/courses/5350/2002fall/uniform.html Graphic examples of uniform convergence of Fourier series] from the University of Colorado
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| [[Category:Calculus]]
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| [[Category:Mathematical series]]
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| [[Category:Topology of function spaces]]
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| [[Category:Convergence (mathematics)]]
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Tires for big vans tackle specialised roles, performing totally different jobs relying on their mounting place. The entrance pair preserve steering control; the trailer tires are devoted to shouldering the payload. The drive-axle tires, those mounted at the rear of the tractor unit, should carry heavy hundreds whereas transmitting the torque of an enormous diesel engine to the asphalt. On-line The web is a superb software that can put you in contact with wholesale tire dealers in your space in a short time. Run a easy search for "New York tire distributors" (obviously utilizing your own location) to seek out the big players in your area. Order on-line at this time or name us with any questions and speak to one among our knowledgeable sales individuals. How do I Discover an ATV Snow Plow on Craigslist?
Teen needs to be carry out the Boat Trailer Tires search accordance with all of your needs make use of the web to accomplish the thought, this will significantly help you in evaluating the auto tires with distinct brands and choose the the auto tires that may handiest fits to your wants. Now, numerous the auto tires producers which have the endorsed web page on the net, there seemed to be as well as a variety of them in addition have an online store to advertise their particular products and options. That is certainly obviously very helpful as a result of reality during these web sites it's attainable to easily get hold of the info in addition to total descriptions to the Trailer Tires About the Author
On the surface, the ADVAN A048 molds a race-ready compound that provides dry grip over a wide range of temperatures right into a 6/32-inch deep, "single block" tread design that increases tread stiffness and includes a large center space that stays in continuous contact with the track to reinforce cornering stability, on-center feel and steering response. As the tire 's outer shoulder tread wears from onerous cornering, the Yokohama A048's directional tread design and symmetric inner building allows worn tires to be remounted "inside-out" on their wheels to assist delay the lifetime of the Yokohama A048 tires We suggest shaving ADVAN A048 tires to between three/32" to 4/32" of tread depth for competitive road racing.
A nonpetroleum oil — this another generally discovered within the kitchen — has also develop into an ingredient in Michelin's tires. The Primacy MXM4 all-climate tire, equipped as original equipment on a number of Mercedes-Benz and Infiniti fashions, uses sunflower oil in its formulation for improved traction at low temperatures and shorter braking distances within the moist. Turning heads and projecting an appealing illustration of urban magnificence, American Racing Baja Wheels and Rims are the right Wheels and Rims for an individual trying to be a logo of favor. Providing the perfect combo of fashion and energy these Wheels and Rims can remodel your car into the baddest automobile burning up the roads and put everybody's eyes in your journey. learn more Nissan Tire Issues
Silica filler is a expensive, sophisticated material that requires numerous vitality to produce. The Oregon crew efficiently changed as much as 12 p.c of the silica utilized in low-rolling-resistance tires without degrading performance. Extra analysis is required to confirm the lengthy-term durability of those tire formulations. Considerably extra subtle is the impact of recent manufacturing machines that can maintain a lot closer manufacturing tolerances, Mr. Herzlich mentioned. As tire crops retool, new tools lets them fabricate tires with a lot greater precision, which saves weight as a result of only the required materials are included within the tire. In case you answered sure to all of these, chances are you'll wish to begin selling online. What Does P245 Stand for on My Tire Size? Skid Steer tires
On-line car outlets and automobile part stores not only help customers save money and time but in addition arranges for delivery the parts right to their properties. Their smart websites and specially designed online procuring portals are serving to clients select the precise product, have instantaneous session with automobile specialists. With such myriads of benefits of on-line auto shops and high quality of Canadian brands has led to rapid growth of Canada auto market in current decade. If you cherished this article and you also would like to be given more info relating to purchase tires generously visit our own site. About the Creator
Discover The Greatest Articles at Related information on Bikes Best strategies for choosing essentially the most applicable motorbike saddlebag For the file, I personally refuse to provide "the look" to different drivers, particularly after the snow flies, as a result of I determine if they might transfer their vehicle then they would transfer their vehicle. They aren't spinning their tires in an try to wreck my day and make me late for work. Let's get again to the listener comments about winter tires and finally winter driving and you may see why I've realized that I'm not going to win this season on the subject of avoiding "the look." You may have an inventory of 1,000,000 folks, but when they are all freebie seekers and tire kickers then you'll make no gross sales and don't have any enterprise.