KreinMilmanTheorem 2004-06-15 16:50:45 2005-03-07 05:07:31 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 \theoremstyle{theorem} \newtheorem*{thm}{Theorem} \newtheorem*{lemma}{Lemma} \newtheorem*{conj}{Conjecture} \newtheorem*{cor}{Corollary} \newtheorem*{example}{Example} \theoremstyle{definition} \newtheorem*{defn}{Definition} \begin{thm} Let $X$ be a locally convex topological vector space, and let $K \subset X$ be a compact \PMlinkname{convex subset}{ConvexSet}. Then $K$ is the closed convex hull of its extreme points. \end{thm} % FIXME: this needs to go into the convex hull entry or some such The closed convex hull above is defined as the intersection of all closed convex subsets of $X$ that contain $K$. This turns out to be the same as the closure of the convex hull in a topological vector space. \begin{thebibliography}{9} \bibitem{royden} H.\@ L.\@ Royden. \emph{\PMlinkescapetext{Real Analysis}}. Prentice-Hall, Englewood Cliffs, New Jersey, 1988 \end{thebibliography}