Definition:Euclidean Metric/Real Vector Space

Definition
Consider the Euclidean space $\left({\R^n, d}\right)$.

The metric $d$ on such a space, defined as:
 * $\displaystyle d \left({x, y}\right) = \left({\sum_{i=1}^n \left({x_i - y_i}\right)^2}\right)^{1 / 2}$

is called the Euclidean metric.

This is sometimes also referred to as the usual metric.

Real Number Line
On the real number line, it can be seen that this definition degenerates to:
 * $d \left({x, y}\right) = \sqrt {\left({x - y}\right)^2} = \left|{x - y}\right|$

See absolute value.