| Version 10 |
Version 9 |
| \paragraph{Definition} |
\paragraph{Definition} |
| We call a metric space $(\En, d)$ Euclidean $n$-space if the group of |
We call a metric space $(\En, d)$ Euclidean $n$-space if the group of |
| translation isometries of $\En$ is transitive on $\En$ and is |
translation isometries of $\En$ is transitive on $\En$ and is |
| isomorphic to an $n$-dimensional, real inner product space compatible |
isomorphic to an $n$-dimensional, real inner product space compatible |
| with the metric $d$. To be more precise, we are saying that there |
with the metric $d$. To be more precise, we are saying that there |
| exists an $n$-dimensional inner product space $(\V,\langle \cdot,\cdot\rangle)$ and a mapping |
exists an $n$-dimensional inner product space $(\V,\langle \cdot,\cdot\rangle)$ and a mapping |
| \[ +: \En\times\V\to\En \] |
\[ +: \En\times\V\to\En \] |
| such that |
such that |
| \begin{enumerate} |
\begin{enumerate} |
| \item for all $x,y\in \En$ there exists a unique $u\in \V$ satisfying |
\item for all $x,y\in \En$ there exists a unique $u\in \V$ satisfying |
| \[ y=x+u,\quad d(x,y)^2=\langle u,u\rangle,\] |
\[ y=x+u,\quad d(x,y)^2=\langle u,u\rangle,\] |
| \item such that for all $x,y\in\En$ and all $u\in \V$ we have |
\item such that for all $x,y\in\En$ and all $u\in \V$ we have |
| \[ d(x+u,y+u)=d(x,y),\] |
\[ d(x+u,y+u)=d(x,y),\] |
| \item and such that |
\item and such that |
| \[ (x+u)+v=x+(u+v),\quad x\in \En,\; u,v\in \V.\] |
\[ (x+u)+v=x+(u+v),\quad x\in \En,\; u,v\in \V.\] |
| \end{enumerate} |
\end{enumerate} |
|
|
| \paragraph{Remarks.} |
\paragraph{Remarks.} |
| \begin{itemize} |
\begin{itemize} |
| \item |
\item |
| Alternatively, we can consider Euclidean space as an inner product |
Alternatively, we can consider Euclidean space as an inner product |
| space that has forgotten which point is its origin. |
space that has forgotten which point is its origin. |
|
|
| \item |
\item |
| It is common to refer to 2-dimensional Euclidean space as the |
It is common to refer to 2-dimensional Euclidean space as the |
| \emph{Euclidean plane}. |
\emph{Euclidean plane}. |
|
\item The term \emph{Euclidean vector space} is just another way to refer |
|
to a positive-definite inner product space. |
|
\end{itemize} |
