Metric space

Metric space is important tool of mathematics and physics. The goal of this article is comparison of the definitions from various sources in order to choose and adjust the most usual system of notations.

In addition to the TORI axioms, the good notation should provide:

Compatibility with the mathematical autoritet sources (Publications in mathematical journals, Proofwiki, Citizendium etc.)

Compatibility with notations used by physicists in the scientific publications.

If possible, the allowance with the notations, used in schools, should be provided.

This section is adopted from http://www.proofwiki.org/wiki/Definition:Metric_Space

metric space $$M=(A,d)$$ is an ordered pair consisting of

a non-empty set $$A$$ and

a real-valued function $$d:A×A→ℝ$$ which acts on $$A$$, satisfying the metric space axioms:

(M1):  		   ∀x∈A:	 d(x,x)=0
(M2):  		   ∀x,y,z∈A:	 d(x,y)+d(y,z)≥d(x,z)
(M3):  		   ∀x,y∈A:	 d(x,y)=d(y,x)
(M4):  		   ∀x,y∈A:	 x≠y⟹d(x,y)>0


Definition from Citizendium

Let $$X\,$$ be an arbitrary set. A metric $$d\,$$ on $$X\,$$ is a function $$d: X \times X \rightarrow \mathbb{R}$$   with the following properties:

1. $$d(x_2,x_1)=d(x_1,x_2) \quad \forall x_1,x_2 \in X$$   (symmetry)
2. $$d(x_1,x_2)\leq d(x_1,x_3)+d(x_3,x_2) \quad \forall x_1,x_2,x_3 \in X$$   (triangular inequality)
3. $$d(x_1,x_2)=0\ \Leftrightarrow \ x_1=x_2\,$$

It follows from the above three axioms of a metric (also called distance function) that:

$d(x_1,x_2) \geq 0 \quad \forall x_1,x_2 \in X$   (non-negativity)

A metric space is an ordered pair $$(X,d)\,$$ where $$X\,$$ is a set and $$d\,$$ is a metric on $$X\,$$.

For shorthand, a metric space $$(X,d)\,$$ is usually written simply as $$X\,$$ once the metric $$d\,$$ has been defined or is understood.

Definition from Wikipedia

A metric space is an ordered pair $$(M,d)$$ where $$M$$ is a set and $$d$$ is a metric on $$M$$, i.e., a function

$d \colon M \times M \rightarrow \mathbb{R}$

such that for any $$x, y, z \in M$$, the following holds:

1. $$d(x,y) \ge 0$$     (non-negative),
2. $$d(x,y) = 0\,$$ iff $$x = y\,$$     (identity of indiscernibles),
3. $$d(x,y) = d(y,x)\,$$     (symmetry) and
4. $$d(x,z) \le d(x,y) + d(y,z)$$     (triangle inequality) .

The first condition follows from the other three, since: for any $$x, y \in M$$, $d(x,y) + d(y,x) \ge d(x,x)$ (by triangle equality) $d(x,y) + d(x,y) \ge d(x,x)$ (by symmetry) $2d(x,y) \ge 0$ (by identity of indiscernibles) $d(x,y) \ge 0$

The function $$d$$ is also called distance function or simply distance. Often, $$d$$ is omitted and one just writes $$M$$ for a metric space if it is clear from the context what metric is used.

Examples

For the set of complex numbers, the metrics is defined as module of the difference between the elements;

$$d(p,q)= |p-q|$$

In application, the most common is case of the Euclidean space; the metric appear as distance between elements.