Difference between revisions of "Uniform convergence"
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− | A sequence of | + | 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]] | + | 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) = |
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+ | 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. | ||
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[[Category:Analysis]] | [[Category:Analysis]] |
Revision as of 09:35, 3 March 2010
A sequence of functions is said to uniformly converge to a function if for every positive real number , then there exists such that for all positive integers , we have . (More generally, we can replace with any metric space .)
Every uniformly convergent sequence converges pointwise, but the converse is not necessarily true. For example, the sequence of functions defined by for converges pointwise to the function , 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.
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