Difference between revisions of "Uniform convergence"

Revision as of 10:35, 3 March 2010

A sequence of functions $\{f_n\},\ f_n: X \to \mathbb{R}$ is said to uniformly converge to a function $f: X \to \mathbb{R}$ if for every positive real number $\varepsilon > 0$ , then there exists $N$ such that for all positive integers $n \ge N$ , we have $|f_n(x) - f(x)| < \varepsilon$ . (More generally, we can replace $\mathbb{R}$ with any metric space $Y$ .)

Every uniformly convergent sequence converges pointwise, but the converse is not necessarily true. For example, the sequence of functions defined by $f_n(x) = x^n$ for $x \in [0, 1]$ converges pointwise to the function $f(x) =\begin{cases} 1, & x = 1 \\0, & \text{otherwise}\end{cases}$ , but this convergence is not uniform.

Uniformly convergent sequences have a number of nice properties that pointwise convergent sequences do not necessarily have. A uniformly convergent sequence of continuous functions converges to a continuous function. A uniformly convergent sequence of differentiable functions defined on a closed interval converges to a differentiable function, and a sequence of Stieltjes-integrable functions converges to a Stieltjes-integrable function.

This article is a stub. Help us out by expanding it.

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-A sequence of functions <math>\{f_n\},\ f_n: X \to \mathbb{R}</math> is said to '''uniformly converge''' to a function <math>f: X \to Y</math> if for every positive real number <math>\varepsilon > 0</math>, then there exists <math>N</math> such that for all positive integers <math>n \ge N</math>, we have <math>|f_n(x) - f(x)| < \varepsilon</math>.
+A [[sequence]] of [[function]]s <math>\{f_n\},\ f_n: X \to \mathbb{R}</math> is said to '''uniformly converge''' to a function <math>f: X \to \mathbb{R}</math> if for every positive real number <math>\varepsilon > 0</math>, then there exists <math>N</math> such that for all positive integers <math>n \ge N</math>, we have <math>|f_n(x) - f(x)| < \varepsilon</math>.  (More generally, we can replace <math>\mathbb{R}</math> with any [[metric space]] <math>Y</math>.)
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-Uniformly convergent sequences have a number of nice properties that pointwise convergent sequences do not necessarily have. A sequence of [[continuous]] uniformly convergent functions converge to a continuous function. A sequence of differentiable uniformly convergent functions (on a [[closed set|closed interval]]) converge to a differentiable function, and a sequence of [[Riemann-Stieltjes Integral|Stieltjes-integrable]] functions converge to a Stieltjes-integrable function.
+Every uniformly convergent sequence converges [[pointwise convergence | pointwise]], but the [[converse]] is not necessarily true.  For example, the sequence of functions defined by <math>f_n(x) = x^n</math> for <math>x \in [0, 1]</math> converges pointwise to the function <math>f(x) =\begin{cases} 1, & x = 1 \\0, & \text{otherwise}\end{cases}</math>, but this convergence is ''not'' uniform.
+Uniformly convergent sequences have a number of nice properties that pointwise convergent sequences do not necessarily have.  A uniformly convergent sequence of [[continuous]] functions converges to a continuous function. A uniformly convergent sequence of differentiable functions defined on a [[closed set|closed interval]] converges to a differentiable function, and a sequence of [[Riemann-Stieltjes Integral|Stieltjes-integrable]] functions converges to a Stieltjes-integrable function.
 {{stub}}
 [[Category:Analysis]]

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Difference between revisions of "Uniform convergence"

Revision as of 10:35, 3 March 2010