Euclidean vector spaceEuclideanVectorSpace22006-01-22 12:27:542006-01-24 12:21:31Definitioninner product space\usepackage{amsmath}
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The term \emph{Euclidean vector space} is synonymous with \emph{finite-dimensional, real, positive definite, inner product space}. The canonical example is $\reals^n$, equipped with the usual dot product. Indeed, every Euclidean vector space $V$ is isomorphic to $\reals^n$, up to a choice of orthonormal basis of $V$. As well, every Euclidean vector space $V$ carries a natural metric space structure given by
$$ d(u,v) = \sqrt{\left< u-v, u-v \right>},\quad u,v\in V.$$
\section{Remarks.}
\begin{itemize}
\item An analogous object with complex numbers as the base field is called a unitary space.
\item Dropping the assumption of finite-dimensionality we arrive at the class of real pre-Hilbert spaces.
\item If we drop the inner product and the vector space structure, but retain the metric space structure, we arrive at the notion of a Euclidean space.
\end{itemize}