EvenLattice 2009-01-12 01:27:55 2009-01-12 01:27:55 Definition % 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 A lattice $L$ in $\mathbb{R}^n$ is \PMlinkescapetext{even} if the norm of every vector in $L$ is an even integer. Note that because of the polarization identity $$(x,y) = \frac{(x+y,x+y) - (x,x) - (y,y)}{2}$$ it follows that all scalar products in an even lattice are \PMlinkescapetext{integral}.