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'Euclidean space'
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| Title of object: |
Euclidean space |
| Canonical Name: |
EuclideanVectorSpace |
| Type: |
Definition |
| Created on: |
2004-04-08 12:22:57 |
| Modified on: |
2005-09-04 15:18:49 |
| Classification: |
msc:15A03 |
| Defines: |
Euclidean plane, Euclidean vector space |
Preamble:
\usepackage{amsmath}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{bbm}
\newcommand{\En}{\mathbbmss{E}^n}
\newcommand{\V}{\mathbbmss{V}}
\newcommand{\R}{\mathbbmss{R}}
\newcommand{\T}{\operatorname{T}} |
Content:
\paragraph{Definition}
We call a metric space $(\En, d)$ Euclidean $n$-space if the group of
translation isometries of $\En$ is transitive on $\En$ and is
isomorphic to an $n$-dimensional, real inner product space compatible
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
\[ +: \En\times\V\to\En \]
such that
\begin{enumerate}
\item for all $x,y\in \En$ there exists a unique $u\in \V$ satisfying
\[ y=x+u,\quad d(x,y)^2=\langle u,u\rangle,\]
\item such that for all $x,y\in\En$ and all $u\in \V$ we have
\[ d(x+u,y+u)=d(x,y),\]
\item and such that
\[ (x+u)+v=x+(u+v),\quad x\in \En,\; u,v\in \V.\]
\end{enumerate}
\paragraph{Remarks.}
\begin{itemize}
\item
Alternatively, we can consider Euclidean space as an inner product
space that has forgotten which point is its origin.
\item
It is common to refer to 2-dimensional Euclidean space as the
\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} |
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