| Version 6 |
Version 5 |
| \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 (1) transitive on $\En$ and (2) is |
translation isometries of $\En$ is (1) transitive on $\En$ and (2) 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 $\V$ and a mapping
|
| \[ +: \En\times\V\to\En \] |
\[ +: \En\times\V\to\En \] |
| such that (1) |
such that (1) |
| for all $x,y\in \En$ there exists a unique $u\in \V$ satisfying |
for all $x,y\in \En$ there exists a unique $u\in \V$ satisfying |
|
such that |
| \[ y=x+u,\] |
\[ y=x+u,\] |
| and such that (2) for all $x,y\in\En$ and all $u\in \V$ we have |
and such that (2) for all $x,y\in\En$ and all $u\in \V$ we have |
| \[ d(x+u,y+u)=d(x,y).\] |
\[ d(x+u,y+u)=d(x,y).\] |
|
|
|
|
| \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 |
\item The term \emph{Euclidean vector space} is just another way to refer |
| to a positive-definite inner product space. |
to a positive-definite inner product space. |
| \end{itemize} |
\end{itemize} |
