QuotientNorm 2007-07-07 18:25:47 2007-07-07 18:28:34 Definition 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} % 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 $V$ be a normed vector space with norm $\| \cdot \|$. Let $M$ be a closed subspace of $V$ and $V/M$ the quotient vector space. The norm $\| \cdot \|$ induces a norm $\| \cdot \|_{V/M}$ in $V/M$, called the {\bf quotient norm}, given by \begin{displaymath} \| v+M \|_{V/M}:= \inf_{u \in v+M} \|u\| = \inf_{m \in M} \|v+m\| \end{displaymath} {\bf Theorem -} $\| \cdot \|_{V/M}$ is a norm in $V/M$ iff $M$ is closed in $V$.