LocallyConvexTopologicalVectorSpace 2003-07-05 12:36:36 2006-02-17 11:16:35 Definition topological vector space % 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 \newcommand{\sR}[0]{\mathbb{R}} \newcommand{\sC}[0]{\mathbb{C}} \newcommand{\sN}[0]{\mathbb{N}} \newcommand{\sZ}[0]{\mathbb{Z}} %\renewcommand{\bibname}{References} % The below lines should work as the command % \renewcommand{\bibname}{References} % without creating havoc when rendering an entry in % the page-image mode. \makeatletter \@ifundefined{bibname}{}{\renewcommand{\bibname}{References}} \makeatother {\bf Definition} Let $V$ be a topological vector space over a subfield of the complex numbers (usually taken to be $\mathbb{R}$ or $\mathbb{C}$). If the topology of $V$ has a basis where each member is a convex set, then $V$ is a \emph{locally convex topological vector space} \cite{folland}. Though most vector spaces occurring in practice are locally convex, the spaces $L^p$ for $0<p<1$ are examples of spaces which are not locally convex. \begin{thebibliography}{9} \bibitem{folland} G.B. Folland, \emph{Real Analysis: Modern Techniques and Their Applications}, 2nd ed, John Wiley \& Sons, Inc., 1999. \end{thebibliography}