Metric Space

Idea

A metric space is a generalization of vector spaces such as ${\mathbb{R}}^2$ with the standard Euclidian Metric, $\sqrt{(x_1-y_1{)}^2+(x_2-y_2{)}^2}$ to arbitrary underlying sets and metrics. It provides a sufficient way to talk about convergence in arbitrary setting, so long as a suitable metric exists.

Definition

A metric space is a set $X$ paired with a real-valued metric function $d:X\to X\to \mathbb{R}$, such that the following properties hold,

• non-negativity: $d(x,y)\ge 0$ for all $x,y\in X$
• identity of indiscernibles: $d(x,y)=0$ iff $x=y$ for all $x,y\in X$
• symmetry: $d(x,y)=d(y,x)$ for all $x,y\in X$
• triangle inequality: $d(x,z)\le d(x,y)+d(y,z)$ for all $x,y,z\in X$.

Examples

Euiclidian Metric:

$$d(x,y):=\sqrt{\sum _{i\in [\mathrm{0..}n)}(y_i-x_i{)}^2}$$