Definition:Euclidean Space

Real Vector Space
For a positive integer $n$, Euclidean $n$-space is defined as the set $\R^n$ considered with the following:


 * addition and scalar multiplication, so that $\R^n$ is considered as a real vector space
 * the dot product of two vectors in $\R^n$, so that $\R^n$ is considered as an inner product space
 * the Euclidean norm induced by the dot product
 * the Euclidean metric induced by the Euclidean norm
 * the Euclidean topology (sometimes called the usual topology) induced by the Euclidean metric

Any vector space for which a metric is defined that is precisely equivalent, for any two points, to the Euclidean metric is called a Euclidean space.

Rational Euclidean Space
Let $\Q^n$ be an $n$-dimensional vector space of rational numbers.

From Rational Numbers form Metric Space it follows from the above definition it follows that $\Q^n$ is also a Euclidean $n$-space.

Complex Euclidean Space
The set of complex numbers $\C$ is also a metric space, as is proved here.

Also see
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.