Metric Space

Topics: Calculus

---

(definition)

Let $X$ be a set and $d:x\times x\to \mathbb{R}$ be a function. We say that $(X,d)$ is a metric space if:

1. $d(x,y)=0⟺x=y$
2. $d(x,y)=d(y,x)$, $\forall x,y\in X$
3. $d(x,z)\le d(x,y)+d(y,z)$, $\forall x,y,z\in X$

In such a case, we say that $d$ is a metric.

Concept of Distance

Notice that these properties are related to the concept of distance. The first one asks for any two same elements to be in the same place, the second one asks for the distance to be the same between any two elements regardless of order, and the third one is the triangle inequality.

Examples §

Distance in ${\mathbb{R}}^n$

In ${\mathbb{R}}^n$, we define the distance (metric) as:

$$d(x,y)=\Vert x-y\Vert$$

That is, the norm of the difference between $x$ and $y$.

Example 1

Let $X=\{a_1,a_2,a_3,a_4,a_5\}$ and $d:x\times x\to \mathbb{R}$ be given by:

$$d(a_i,a_j)=\{\begin{array}{l}1\text{ if }i\ne j\\ 0\text{ if }i=j\end{array}$$

This $d$ defines a metric (it satisfies the previous properties).

Example 2

$(\mathbb{R},d)$ where $d:\mathbb{R}\times \mathbb{R}\to \mathbb{R}$, $d(x,y)=|x-y|$.

The properties of a metric space are satisfied, since they follow the properties of the absolute value.

Example 3

Let $\mathcal{C}[0,1]=\{f:[0,1]\to \mathbb{R}\mid f\text{ is continuous}\}$ and $d:\mathcal{C}[0,1]\times \mathcal{C}[0,1]\to \mathbb{R}$ be given by:

$$d(f,g)=\sup \{|f(x)-g(x)||x\in [0,1]\}$$

$d$ is a metric.