Hilbert space HilbertSpace 2002-02-13 12:21:19 2004-12-07 16:51:37 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 \emph{Hilbert space} is an inner product space which is \PMlinkid{complete}{603} under the \PMlinkescapetext{induced} metric. In particular, a Hilbert space is a Banach space in the norm \PMlinkescapetext{induced} by the inner product, since the norm and the inner product both induce the same metric. Any finite-dimensional inner product space is a Hilbert space, but it is worth mentioning that some authors require the space to be infinite dimensional for it to be called a Hilbert space.