WeierstrassApproximationTheorem 2004-09-05 09:37:56 2006-03-01 08:01:01 Theorem % 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 If $ f $ is a continuous real-valued function on a interval $ [a,b] $ then for all $ \varepsilon>0 $ there exists a polynomial $ P $ which satisfies $ |f(x)-P(x)|<\varepsilon \quad \forall x\in [a,b] $ This theorem also holds for compact subsets of $ \Bbb{R}^n . $ The Stone-Weierstrass theorem is a generalization to even more general situations.