Education: Metric space definition

A metric space is a Cartesian pair

(X, d)

where

X

is a non-empty set and

d: X \times X \rightarrow [0, \infty)

, is a function which is called the metric which satisfies the requirement that for all

a, b, c \in X

Note that some authors do not require metric spaces to be non-empty. We annotate

(X,d)

when we talk of a metric space

X

with the metric

d

An important example is the discrete metric. It may be defined any non-empty set X as follows $d(x,y) = \left\{ \begin{array}{ll} 1 & \text{if } x \neq y\\ 0 & \text{if } x = y\\ \end{array} \right.$

On the set of real numbers $\mathbb{R}$ , define $d(x,y) = |x-y|\,$ (The absolute distance between x and y).
To prove that this is indeed a metric space, we must show that d is really a metric. To begin with, $d(x,y) = |x - y| \ge 0$ for any

real numbers

x

and

y

- $d(x,y) = |x - y| = 0 \iff (x - y) = 0 \or -(x - y) = 0 \iff x = y$ .
- $d(x,y) = |x - y| = |y - x| = d(y,x)\,$
- $d(x,z) = |x - z| = |x - y + y - z| = |(x - y) + (y - z)| \le |x - y| + |y - z| = d(x,y) + d(y,z)$

On the plane $\mathbb{R}^2$ as the space, and let $d((x_{1}, y_1),(x_{2}, y_2)) = \sqrt{(x_1-x_2)^2 + (y_1-y_2)^2}.$

This is the euclidean distance between

(x_{1}, y_1)

and

(x_{2}, y_{2})

We can generalize the two preceding examples. Let $V$ be a normed vector space (over $\mathbb{R}$ or $\mathbb{C}$ ). We can define the metric to be: $d(x, y) = || x - y ||\,$ . Thus every normed vector space is a metric space.

For the vector space we have an interesting norm. Let and two vectors of . We define the p-norm: . For each p-norm there is a metric based on it. Interesting cases of p are:
- $p = 1\,$ . The metric is $d(x,y) = ||x - y||_1 = \sum_{i=1}^n {|x_i-y_i|}$
- $p = 2\,$ . The metric is good-old Euclid metric $d(x,y) = ||x - y||_2 = \sqrt{\sum_{i=1}^n {(x_i-y_i)^2}}$
- $p = \infty$ . This is a bit surprising: $d(x,y) = ||x - y||_\infty = \max_{i=1\ldots n}\{|x_i-y_i|\}$
  As an exercise, you can prove that $\lim_{p\rightarrow \infty}{||x||_p} = \max_{i=1\ldots n}\{|x_i|\}$ thus justifying the definition of $||\cdot||_\infty$ .

The Hilbert space is a metric space on the space of infinite sequences {a_k} such that
$\sum_{i=1}^\infty a_k^2$
converges, with a metric d({a_i}, {b_i})= $\sqrt{\sum_{i=1}^\infty (a_i-b_i)^2}$ .

The concept of the Erdős number suggests a metric on the set of all mathematicians. Take x and y to be two mathematicians, and define d(x, y) as 0 if x and y are the same person; 1 if x and y have co-authored a paper; n if the shortest sequence $(\{x, a_1\}, \{a_1, a_2\}, ..., \{a_{n-1}, y\})$ , where each step pairs two people who have co-authored a paper, is of length n; or ∞ if x ≠ y and no such sequence exists.
This metric is easily generalized to any reflexive relation (or undirected graph, which is the same thing).

Note that if we instead defined d(x, y) as the sum of the Erdős numbers of x and y, then d would not be a metric, as it would not satisfy $d(x, y) = 0 \iff x = y$ . For example, if x = y = Stanisław Ulam, then d(x, y) = 2.

09/10/2014