Definition:Euclidean Space/Real

Definition
Let $\R^n$ be an $n$-dimensional real vector space.

Let $M = \left({\R^n, d}\right)$ where $\displaystyle d \left({x, y}\right) = \left({\sum_{i \mathop = 1}^n \left({x_i - y_i}\right)^2}\right)^{1 / 2}$.

Then $M = \left({\R^n, d}\right)$ is a metric space.

Such a space is called a Euclidean $n$-space.

The metric $d$ is called the Euclidean Metric.

Also see

 * Euclidean Metric is Metric

Bear in mind that Euclid himself did not in fact conceive of the Euclidean space as defined here. It is called that because the geometric space which it gives rise to is Euclidean in the sense that it is consistent with Euclid's fifth postulate.