|
|
|
Viewing Version
8
of
'uniform convergence'
|
[ view 'uniform convergence'
|
back to history
]
| Title of object: |
uniform convergence |
| Canonical Name: |
UniformConvergence |
| Type: |
Definition |
| Created on: |
2002-12-09 06:02:22 |
| Modified on: |
2002-12-11 12:12:15 |
| Classification: |
msc:40A30 |
| Defines: |
uniformly convergent |
Revision comment (for changes between this and next version):
| Changes for correction #6198 ('quote marks should look like ``this'', because ``this" doesn't work with latex2html'). |
Preamble:
% this is the default PlanetMath preamble. as your knowledge
% of TeX increases, you will probably want to edit this, but
% it should be fine as is for beginners.
% almost certainly you want these
\usepackage{amssymb}
\usepackage{amsmath}
\usepackage{amsfonts}
% used for TeXing text within eps files
%\usepackage{psfrag}
% need this for including graphics (\includegraphics)
%\usepackage{graphicx}
% for neatly defining theorems and propositions
%\usepackage{amsthm}
% making logically defined graphics
%\usepackage{xypic}
% there are many more packages, add them here as you need them
% define commands here |
Content:
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
\emph{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 $f_n\rightrightarrows f$. |
|
|
|
|
|
