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.