PolarizationIdentity3 2007-11-15 17:55:45 2007-11-15 17:55:45 Theorem % 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 {\bf Theorem} [polarization identity] {\bf -} Let $X$ be an inner product space over $\mathbb{R}$. The following identity holds for every $x, y \in X$: \begin{displaymath} \langle x, y \rangle = \frac{1}{4}(\|x +y \|^2 - \|x-y\|^2) \end{displaymath} If $X$ is an inner product space over $\mathbb{C}$ instead, the identity becomes \begin{displaymath} \langle x, y \rangle = \frac{1}{4}(\|x +y \|^2 - \|x-y\|^2) + \frac{1}{4}i(\|x+iy\|^2-\|x-iy\|^2) \end{displaymath} {\bf Remark -} This result shows that the inner product of $X$ is determined by the norm. Moreover, it can be shown that if a normed space $V$ \PMlinkescapetext{satisfies} the parallelogram law, the above formulas define an inner product compatible with the norm of $V$.