Definition:Euclidean Metric/Real Vector Space

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

The Euclidean metric on $\R^n$ is defined as:
 * $\displaystyle d_2 \left({x, y}\right) := \left({\sum_{i \mathop = 1}^n \left({x_i - y_i}\right)^2}\right)^{1 / 2}$

where $x = \left({x_1, x_2, \ldots, x_n}\right), y = \left({y_1, y_2, \ldots, y_n}\right) \in \R^n$.

Also known as
The Euclidean metric is sometimes also referred to as the usual metric.

Also see

 * Euclidean Metric on Real Vector Space is Metric


 * Metric Induces Topology, from which it follows that the Euclidean space is also a topological space.

In this context, the topology induced by the Euclidean metric is sometimes called the usual topology.

Bear in mind that Euclid himself did not in fact conceive of the Euclidean metric. 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.