BanachKreinSmulianTheorem 2005-05-08 17:52:28 2005-07-05 10:34:35 Theorem Banach space convex set weak-* topology % 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 Let $E$ be a Banach space. A convex subset $C$ of the dual space $E^*$ is closed in the weak-$*$ topology if and only if the intersection of $C$ with the ball $B_{r}(0)$ is weak-$*$ closed for every $r>0$. \begin{thebibliography}{1} \bibitem{cite:DS} Dunford, N., and J. T. Schwartz, \emph{Linear Operators}, Part I, Interscience Publishers, 1967. \end{thebibliography}