uniform convergence

Let $X$ be any set, and let $(Y,d)$ be a metric space. A sequence $f_{1},f_{2},\dots$ of functions mapping $X$ to $Y$ is said to be uniformly convergent to another function $f$ if, for each $\varepsilon>0$ , there exists $N$ such that, for all $x$ and all $n>N$ , we have $d(f_{n}(x),f(x))<\varepsilon$ . This is denoted by $f_{n}\xrightarrow[]{u}f$ , or “ $f_{n}\rightarrow f$ uniformly” or, less frequently, by $f_{n}\rightrightarrows f$ .

Title	uniform convergence
Canonical name	UniformConvergence
Date of creation	2013-03-22 13:13:49
Last modified on	2013-03-22 13:13:49
Owner	Koro (127)
Last modified by	Koro (127)
Numerical id	14
Author	Koro (127)
Entry type	Definition
Classification	msc 40A30
Related topic	CompactOpenTopology
Related topic	ConvergesUniformly
Defines	uniformly convergent