ScalingOfTheOpenBallInANormedVectorSpace 2005-10-29 13:56:35 2009-04-19 18:21:50 Theorem normed 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} \usepackage{amsthm} \usepackage{mathrsfs} % used for TeXing text within eps files %\usepackage{psfrag} % need this for including graphics (\includegraphics) %\usepackage{graphicx} % for neatly defining theorems and propositions % % 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}} \usepackage{bbm} \newcommand{\Z}{\mathbbmss{Z}} \newcommand{\C}{\mathbbmss{C}} \newcommand{\F}{\mathbbmss{F}} \newcommand{\R}{\mathbbmss{R}} \newcommand{\Q}{\mathbbmss{Q}} \newcommand*{\norm}[1]{\lVert #1 \rVert} \newcommand*{\abs}[1]{| #1 |} \newtheorem{thm}{Theorem} \newtheorem{defn}{Definition} \newtheorem{prop}{Proposition} \newtheorem{lemma}{Lemma} \newtheorem{cor}{Corollary} % This is written from scratch with no reference Let $V$ be a vector space over a field $F$ (real or complex), and let $\Vert\cdot \Vert$ be a norm on $V$. Further, for $r>0$, $v\in V$, let $$ B_r(v) = \{ w\in V: \Vert w-v\Vert < r \}. $$ Then for any non-zero $\lambda\in F$, we have $$ \lambda B_r(v) = B_{|\lambda| r}(\lambda v). $$ The claim is clear for $\lambda =0$, so we can assume that $\lambda \neq 0$. Then \begin{eqnarray*} \lambda B_r(v) &=& \{ z\in V: \Vert w-v\Vert < r\ \mbox{and}\ z=\lambda w \} \\ &=& \{ z\in V: \Vert \frac{z}{\lambda}-v\Vert < r \} \\ &=& \{ z\in V: \Vert z-\lambda v\Vert < |\lambda| r \} \\ &=& B_{|\lambda| r}(\lambda v). \end{eqnarray*}