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# Euclidean distance

If $u=(x_{1},y_{1})$ and $v=(x_{2},y_{2})$ are two points on the plane, their *Euclidean distance* is given by

$\sqrt{(x_{1}-x_{2})^{2}+(y_{1}-y_{2})^{2}}.$ | (1) |

Geometrically, it’s the length of the segment joining $u$ and $v$, and also the norm of the difference vector (considering $\mathbbmss{R}^{n}$ as vector space).

This distance induces a metric (and therefore a topology) on $\mathbbmss{R}^{2}$, called *Euclidean metric (on $\mathbbmss{R}^{2}$)* or *standard metric (on $\mathbbmss{R}^{2})$*. The topology so induced is called *standard topology* or *usual topology on $\mathbbmss{R}^{2}$* 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 1 can be generalized to $\mathbbmss{R}^{n}$ by defining the Euclidean distance from $a$ to $b$ as

$d(a,b)=\sqrt{(x_{1}-y_{1})^{2}+(x_{2}-y_{2})^{2}+\cdots+(x_{n}-y_{n})^{2}}.$ | (2) |

Notice that this distance coincides with absolute value when $n=1$.
Euclidean distance on $\mathbbmss{R}^{n}$ is also a metric (Euclidean or standard metric), and therefore we can give $\mathbbmss{R}^{n}$ a topology, which is called the standard (canonical, usual, etc) topology of $\mathbbmss{R}^{n}$. The resulting (topological and vectorial) space is known as *Euclidean space*.

This can also be done for $\mathbbmss{C}^{n}$ since as set $\mathbbmss{C}=\mathbbmss{R}^{2}$ and thus the metric on $\mathbbmss{C}$ is the same given to $\mathbbmss{R}^{2}$, and in general, $\mathbbmss{C}^{n}$ gets the same metric as $R^{{2n}}$.

## Mathematics Subject Classification

53A99*no label found*54E35

*no label found*

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