Editing Uniform convergence (section)

== Definition ==

We first define uniform convergence for [[Real-valued function|real-valued functions]], although the concept is readily generalized to functions mapping to [[Metric space|metric spaces]] and, more generally, [[Uniform space|uniform spaces]] (see [[Uniform convergence#Generalizations|below]]). 

Suppose <math>E</math> is a [[Set (mathematics)|set]] and <math>(f_n)_{n \in \N}</math> is a sequence of real-valued functions on it. We say the sequence <math>(f_n)_{n \in \N}</math> is '''uniformly convergent''' on <math>E</math> with limit <math>f: E \to \R</math> if for every <math>\varepsilon > 0,</math> there exists a natural number <math>N</math> such that for all <math>n \geq N</math> and for all <math>x \in E</math>

:<math>|f_n(x)-f(x)|<\varepsilon.</math>

The notation for uniform convergence of <math>f_n</math> to <math>f</math> is not quite standardized and different authors have used a variety of symbols, including (in roughly decreasing order of popularity): 

:<math>f_n\rightrightarrows f, \quad \underset{n\to\infty}{\mathrm{unif\ lim}}f_n = f, \quad f_n \overset{\mathrm{unif.}}{\longrightarrow} f, \quad f=\mathrm{u}\!\!-\!\!\!\lim_{n\to\infty} f_n .</math>

Frequently, no special symbol is used, and authors simply write 

:<math>f_n\to f \quad \mathrm{uniformly}</math> 

to indicate that convergence is uniform. (In contrast, the expression <math>f_n\to f</math> on <math>E</math> without an adverb is taken to mean [[pointwise convergence]] on <math>E</math>: for all <math> x \in E </math>, <math>f_n(x)\to f(x)</math> as <math>n\to\infty</math>.)

Since <math>\R</math> is a [[complete metric space]], the [[Cauchy sequence|Cauchy criterion]] can be used to give an equivalent alternative formulation for uniform convergence: <math>(f_n)_{n\in\N}</math> converges uniformly on <math>E</math> (in the previous sense) if and only if for every <math> \varepsilon > 0 </math>, there exists a natural number <math>N</math> such that

:<math>x\in E, m,n\geq N \implies |f_m(x)-f_n(x)|<\varepsilon</math>.

In yet another equivalent formulation, if we define 

:<math> d_n = \sup_{x\in E} |f_n(x) - f(x) |,</math>

then <math> f_n </math> converges to <math>f</math> uniformly if and only if <math>d_n\to 0</math> as <math>n\to\infty</math>. Thus, we can characterize uniform convergence of <math>(f_n)_{n \in \N}</math> on <math>E</math> as (simple) convergence of <math>(f_n)_{n \in \N}</math> in the [[function space]] <math>\R^E</math> with respect to the ''[[Uniform norm|uniform metric]]'' (also called the [[supremum]] metric), defined by 

:<math>d(f,g)=\sup_{x\in E} |f(x)-g(x)|.</math>

Symbolically, 

:<math>f_n\rightrightarrows f\iff d(f_n,f) \to 0</math>.

The sequence <math>(f_n)_{n \in \N}</math> is said to be '''locally uniformly convergent''' with limit <math>f</math> if <math>E </math> is a [[metric space]] and for every <math>x\in E</math>, there exists an <math>r > 0</math> such that <math>(f_n)</math> converges uniformly on <math>B(x,r)\cap E.</math> It is clear that uniform convergence implies local uniform convergence, which implies pointwise convergence.

=== Notes ===

Intuitively, a sequence of functions <math>f_n</math> converges uniformly to <math>f</math> if, given an arbitrarily small <math>\varepsilon>0</math>, we can find an <math>N\in\N</math> so that the functions <math>f_n</math> with <math>n>N</math> all fall within a "tube" of width <math>2\varepsilon</math> centered around <math>f</math> (i.e., between <math>f(x)-\varepsilon</math> and <math>f(x)+\varepsilon</math>) for the ''entire domain'' of the function.

Note that interchanging the order of quantifiers in the definition of uniform convergence by moving "for all <math>x\in E</math>" in front of "there exists a natural number <math>N</math>" results in a definition of [[pointwise convergence]] of the sequence. To make this difference explicit, in the case of uniform convergence, <math>N=N(\varepsilon)</math> can only depend on <math>\varepsilon</math>, and the choice of <math>N</math> has to work for all <math>x\in E</math>, for a specific value of <math>\varepsilon</math> that is given. In contrast, in the case of pointwise convergence, <math>N=N(\varepsilon,x)</math> may depend on both <math>\varepsilon</math> and <math>x</math>, and the choice of <math>N</math> only has to work for the specific values of <math>\varepsilon</math> and <math>x</math> that are given. Thus uniform convergence implies pointwise convergence, however the converse is not true, as the example in the section below illustrates.

=== Generalizations ===

One may straightforwardly extend the concept to functions ''E'' → ''M'', where (''M'', ''d'') is a [[metric space]], by replacing <math>|f_n(x)-f(x)|</math> with <math>d(f_n(x),f(x))</math>.

The most general setting is the uniform convergence of [[net (mathematics)|net]]s of functions ''E'' → ''X'', where ''X'' is a [[uniform space]]. We say that the net <math>(f_\alpha)</math> ''converges uniformly'' with limit ''f'' : ''E'' → ''X'' if and only if for every [[entourage (topology)|entourage]] ''V'' in ''X'', there exists an <math>\alpha_0</math>, such that for every ''x'' in ''E'' and every <math>\alpha\geq \alpha_0</math>, <math>(f_\alpha(x),f(x))</math> is in ''V''. In this situation, uniform limit of continuous functions remains continuous.

===Definition in a hyperreal setting===
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 hyperreal ''x'' in the domain of <math>f^*</math> 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).  In contrast, pointwise continuity requires this only for real ''x''.