Metric Space

analysis topology

Definition

Metric Space

A metric space $⟨M,d⟩$ is a set $M$ with a metric $d:M\times M\to \mathbb{R}$.

Examples

For vectors $x,y\in {\mathbb{R}}^n$, common metrics include the following.

$L_p$ metric

For $1\le p<\infty$, the $L_p$ metric is

$$d_p(x,y)={\left(\sum _{i=1}^n|x_i-y_i{|}^p\right)}^{1/p}.$$

Sum metric

The sum metric, or $L_1$ metric, is

$$d_1(x,y)=\sum _{i=1}^n|x_i-y_i|.$$

Euclidean metric

The euclidean metric, or $L_2$ metric, is

$$d_2(x,y)=\sqrt{\sum _{i=1}^n(x_i-y_i{)}^2}.$$

Max metric

The max metric, or $L_{\infty }$ metric, is

$$d_{\infty }(x,y)=\underset{1\le i\le n}{\max }|x_i-y_i|.$$

Hamming distance

The Hamming distance is

$$d_H(x,y)=|\{i\mid x_i\ne y_i\}|.$$