Hyberplane 2005-05-10 13:38:13 2008-09-02 02:47:34 Definition real hyperplane complex hyperplane % 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 $E$ be a linear space over a field $k$. A hyperplane $H$ in $E$ is defined as the set of the form $$H=\{x\in E:f(x)=a\}$$ where $a \in k$ and $f$ is a nonzero linear functional, $f \colon E \to k$. If $k=\mathbb{R}$ or $\mathbb{C}$, then $H$ is called a \emph{real hyperplane} or \emph{complex hyperplane} respectively. \textbf{Remark}. When $k=\mathbb{C}$, the word ``hyperplane'' also has a more restrictive meaning: it is the zero set of a complex linear functional (by setting $a=0$ above).