|
|
|
Revision difference : Euclidean vector space |
| Version 2 |
Version 1 |
| \section{Definition} |
\section{Definition} |
| A Euclidean vector space is a synonym for ''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 |
A Euclidean vector space is a synonym for ''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 |
| $$ d(u,v) = \sqrt{\left< u-v, u-v \right>},\quad u,v\in V.$$ |
$$ d(u,v) = \sqrt{\left< u-v, u-v \right>},\quad u,v\in V.$$ |
|
|
| \section{Remarks.} |
\section{Remarks.} |
| \begin{itemize} |
\begin{itemize} |
| \item An analoguos object with complex numbers as the base field is called a unitary space. |
\item An analoguos 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 Hilbert spaces. |
\item Dropping the assumption of finite-dimensionality we arrive at the class of real 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. |
\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} |
\end{itemize} |
|
|
|
|
