AllNormsAreNotEquivalent 2005-12-06 04:37:48 2006-07-29 08:36:43 Example equivalent norms % 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} Let $V$ be the vector space of continuous functions $[-1,1]\to \R$ that are differentiable at $0$. Then we can define norms $$ \Vert f \Vert = \max_{x\in [-1,1]} |f|, $$ and $$ \Vert f \Vert' = \Vert f \Vert+|f'(0)|. $$ It is not difficult to find a sequence of functions $f_1, f_2, \ldots$ in $V$ such that \begin{enumerate} \item $f_k'(0)=k$ for $k=1,2,\ldots$, \item $\Vert f_k\Vert = 1$. \end{enumerate} Then $\Vert f_k \Vert = 1$, and $\Vert f_k \Vert'=1+k$, so there is no $C>1$ such that $$ \Vert f \Vert' \le C \Vert f \Vert \quad f\in V, $$ and $\Vert\cdot \Vert$ and $\Vert\cdot \Vert'$ cannot be \PMlinkescapetext{equivalent}.