Loterre: Mathematics: uniform convergence

mathematical analysis > functional analysis > uniform convergence

mathematical analysis > calculus > series > uniform convergence

Preferred term

uniform convergence

Definition(s)

In the mathematical field of analysis, uniform convergence is a mode of convergence of functions stronger than pointwise convergence. A sequence of functions ${\\displaystyle (f_{n})}$ converges uniformly to a limiting function ${\\displaystyle f}$ on a set ${\\displaystyle E}$ as the function domain if, given any arbitrarily small positive number ${\\displaystyle \\epsilon }$ , a number ${\\displaystyle N}$ can be found such that each of the functions ${\\displaystyle f_{N},f_{N+1},f_{N+2},\\ldots }$ differs from ${\\displaystyle f}$ by no more than ${\\displaystyle \\epsilon }$ at every point ${\\displaystyle x}$ in ${\\displaystyle E}$ . Described in an informal way, if ${\\displaystyle f_{n}}$ converges to ${\\displaystyle f}$ uniformly, then the rate at which ${\\displaystyle f_{n}(x)}$ approaches ${\\displaystyle f(x)}$ is "uniform" throughout its domain in the following sense: in order to show that ${\\displaystyle f_{n}(x)}$ uniformly falls within a certain distance ${\\displaystyle \\epsilon }$ of ${\\displaystyle f(x)}$ , we do not need to know the value of ${\\displaystyle x\\in E}$ in question — there can be found a single value of ${\\displaystyle N=N(\\epsilon )}$ independent of ${\\displaystyle x}$ , such that choosing ${\\displaystyle n\\geq N}$ will ensure that ${\\displaystyle f_{n}(x)}$ is within ${\\displaystyle \\epsilon }$ of ${\\displaystyle f(x)}$ for all ${\\displaystyle x\\in E}$ . In contrast, pointwise convergence of ${\\displaystyle f_{n}}$ to ${\\displaystyle f}$ merely guarantees that for any ${\\displaystyle x\\in E}$ given in advance, we can find ${\\displaystyle N=N(\\epsilon ,x)}$ (i.e., ${\\displaystyle N}$ can depend on the value of ${\\displaystyle x}$ ) such that, for that particular ${\\displaystyle x}$ , ${\\displaystyle f_{n}(x)}$ falls within ${\\displaystyle \\epsilon }$ of ${\\displaystyle f(x)}$ whenever ${\\displaystyle n\\geq N}$ (a different ${\\displaystyle x}$ requiring a different ${\\displaystyle N}$ for pointwise convergence).
(Wikipedia, The Free Encyclopedia, https://en.wikipedia.org/wiki/Uniform_convergence)