Definition:Usual Metric

Definition

Let $X$ be one of the standard number fields $\Q$, $\R$, $\C$.

Let $X^n$ be a cartesian space on $X$.

The usual metric on $X^n$ is the Euclidean metric on $X^n$:

Real Numbers

The Euclidean metric on $\R^n$ is defined as:

$\ds \map {d_2} {x, y} := \paren {\sum_{i \mathop = 1}^n \paren {x_i - y_i}^2}^{1 / 2}$

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

Complex Plane

The Euclidean metric on $\C$ is defined as:

$\d \forall z_1, z_2 \in \C: \map d {z_1, z_2} := \size {z_1 - z_2}$

where $\size {z_1 - z_2}$ denotes the modulus of $z_1 - z_2$.