Euclidean distanceEuclideanDistance2002-01-05 02:22:342007-06-17 21:13:35Definition%\usepackage{graphicx}
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\newcommand{\mathbb}[1]{\mathbbmss{#1}}If $u=(x_1,y_1)$ and $v=(x_2,y_2)$ are two points on the plane, their \emph{Euclidean distance} is given by
\begin{equation}\label{equno}
\sqrt{(x_1-x_2)^2 + (y_1-y_2)^2}.
\end{equation}
Geometrically, it's the length of the segment joining $u$ and $v$, and also the norm of the difference vector (considering $\R^n$ as vector space).
This distance induces a metric (and therefore a topology) on $\mathbb{R}^2$, called \emph{Euclidean metric (on $\R^2$)} or \emph{standard metric (on $\mathbb{R}^2)$}. The topology so induced is called \emph{standard topology} or \emph{usual topology on $\R^2$} and one basis can be obtained considering the set of all the open balls.
If $a=(x_1,x_2,\ldots,x_n)$ and $b=(y_1,y_2,\ldots,y_n)$, then formula \ref{equno} can be generalized to $\R^n$ by defining the Euclidean distance from $a$ to $b$ as
\begin{equation}d(a,b)=\sqrt{(x_1-y_1)^2+(x_2-y_2)^2+\cdots+(x_n-y_n)^2}.\end{equation}
Notice that this distance coincides with absolute value when $n=1$.
Euclidean distance on $\mathbb{R}^n$ is also a metric (Euclidean or standard metric), and therefore we can give $\mathbb{R}^n$ a topology, which is called the standard (canonical, usual, etc) topology of $\mathbb{R}^n$. The resulting (topological and vectorial) space is known as \emph{Euclidean space}.
This can also be done for $\C^n$ since as set $\C=\R^2$ and thus the metric on $\C$ is the same given to $\R^2$, and in general, $\C^n$ gets the same metric as $R^{2n}$.