# 3.1. Introduction#

## 3.1.1. Distance Functions#

Definition 3.1 (Distance function/Metric)

Let $$X$$ be a nonempty set. A function $$d : X \times X \to \RR$$ is called a distance function or a metric if it satisfies the following properties for any elements $$x,y,z \in X$$:

1. Non-negativity: $$d(x, y) \geq 0$$

2. Identity of indiscernibles: $$d(x, y) = 0 \iff x = y$$

3. Symmetry: $$d(x, y) = d(y, x)$$

4. Triangle inequality: $$d(x,y) \leq d(x, z) + d(z, y)$$

• It is customary to call the elements of a set $$X$$ associated with a distance function as points.

• Distance functions are real valued.

• Distance functions map an ordered pair of points in $$X$$ to a real number.

• Distance between two points in the set $$X$$ can only be non-negative.

• Distance of a point with itself is 0. In other words, if the distance between two points is 0, then the points are identical. i.e. the distance function works as a discriminator between the points of the set $$X$$.

• Symmetry means that the distance from a point $$x$$ to another point $$y$$ is same as the distance from $$y$$ to $$x$$.

• Triangle inequality says that the direct distance between two points can never be longer than the distance covered through an intermediate point.

## 3.1.2. Metric Spaces#

Definition 3.2 (Metric space)

Let $$d$$ be a distance function on a set $$X$$. Then we say that $$(X, d)$$ is a metric space. The elements of $$X$$ are called points.

• In general, a set $$X$$ can be associated with different metrics (distance functions) say $$d_1$$ and $$d_2$$. In that case, the corresponding metric spaces $$(X, d_1)$$ and $$(X, d_2)$$ are different.

• When a set $$X$$ is equipped with a metric $$d$$ to create a metric space $$(X, d)$$, we say that $$X$$ has been metrized.

• If the metric $$d$$ associated with a set $$X$$ is obvious from the context, we will denote the corresponding metric space $$(X,d)$$ by simply $$X$$. E.g., $$|x-y|$$ is the standard distance function on the set $$\RR$$.

• When we say that let $$Y$$ be a subset of a metric space $$(X,d)$$, we mean that $$Y \subset X$$.

• Similarly, a point in a metric space $$(X,d)$$ means the point in the underlying set $$X$$.

Note

Some authors prefer the notation $$d : X \times X \to \RR_+$$. With this notation, the non-negativity property is embedded in the type signature of the function (i.e. the codomain specification) and doesn’t need to be stated explicitly.

## 3.1.3. Properties of Metrics#

Proposition 3.1 (Triangle inequality alternate form)

Let $$(X, d)$$ be a metric space. Let $$x,y,z \in X$$.

$|d (x, z) - d(y, z)| \leq d(x,y).$

Proof. From triangle inequality:

$d (x, z) \leq d(x, y) + d (y, z) \implies d (x, z) - d(y, z) \leq d (x, y).$

Interchanging $$x$$ and $$y$$ gives:

$d (y, z) - d (x, z) \leq d (y, x) = d (x, y).$

Combining the two, we get:

$|d (x, z) - d(y, z)| \leq d(x,y).$

## 3.1.4. Metric Subspaces#

Definition 3.3 (Metric subspace)

Let $$(X, d)$$ be a metric space. Let $$Y \subset X$$ be a nonempty subset of $$X$$. Then, $$Y$$ can be viewed as a metric space in its own right with the distance function $$d$$ restricted to $$Y \times Y$$, denoted as $$d|_{Y \times Y}$$. We then say that $$(Y, d|_{Y \times Y})$$ or simply $$Y$$ is a metric subspace of $$X$$.

It is customary to drop the subscript $$Y \times Y$$ from the restriction of $$d$$ and write the subspace simply as $$(Y, d)$$.

Example 3.1

$$[0,1]$$ is a metric subspace of $$\RR$$ with the standard metric $$d(x, y) = |x -y|$$ restricted to $$[0,1]$$. In other words, the distance between any two points $$x, y \in [0, 1]$$ is calculated by viewing $$x,y$$ as points in $$\RR$$ and using the standard metric for $$\RR$$.

## 3.1.5. Examples#

Example 3.2 ($$\RR^n$$ p-distance)

For some $$1 \leq p \lt \infty$$, the function $$d_p : \RR^n \times \RR^n \to \RR$$:

$d_p (x, y) \triangleq \left ( \sum_{i=1}^n |x_i - y_i|^p \right )^{\frac{1}{p}}$

is a metric and $$(\RR^n, d_p)$$ is a metric space.

Example 3.3 ($$\RR^n$$ Euclidean space)

The $$d_2$$ metric over $$\RR^n$$:

$d_2 (x, y) \triangleq \left ( \sum_{i=1}^n |x_i - y_i|^2 \right )^{\frac{1}{2}}$

is known as the Euclidean distance and the metric space $$(\RR^n, d_2)$$ is known as the n-dimensional Euclidean (metric) space.

The standard metric for $$\RR^n$$ is the Euclidean metric.

Example 3.4 (Discrete metric)

Let $$X$$ be a nonempty set:

Define:

$\begin{split} d(x,y) = \begin{cases} 0 & x = y \\ 1 & x \neq y \end{cases}. \end{split}$

$$(X, d)$$ is a metric space. This distance is called discrete distance and the metric space is called a discrete metric space.

Discrete metric spaces are discussed in depth in Discrete Metric Space. They help clarify many subtle issues in the theory of metric spaces.

Example 3.5 ($$\ERL$$ A metric space for the extended real line)

Consider the mapping $$\varphi : \ERL \to [-1, 1]$$ given by:

$\begin{split} \varphi(x) = \begin{cases} \frac{t}{1 + |t|} & x \in \RR \\ -1 & x = -\infty \\ 1 & x = \infty \end{cases}. \end{split}$

$$\varphi$$ is a bijection from $$\ERL$$ onto $$[-1, 1]$$.

$$[-1, 1]$$ is a metric space with the standard metric for the real line $$d_{\RR}(x, y) = |x - y|$$ restricted to $$[-1, 1]$$.

Consider a function $$d: \ERL \times \ERL \to \RR$$ defined as

$d (s, t) = | \varphi(s) - \varphi(t)|.$

The function $$d$$ satisfies all the requirements of a metric. It is the standard metric on $$\ERL$$.

Example 3.6 ($$\ell^p$$ Real sequences)

For any $$1 \leq p < \infty$$, we define:

$\ell^p = \left \{ \{ a_n \} \in \RR^{\Nat} \ST \sum_{i=1}^{\infty} |a_i|^p \right \}$

as the set of real sequences $$\{ a_n \}$$ such that the series $$\sum a_n^p$$ is absolutely summable.

It can be shown that the set $$\ell^p$$ is closed under sequence addition.

Define a map $$d_p : \ell^p \times \ell^p \to \RR$$ as

$d_p (\{a_n \}, \{ b_n \}) = \sum_{i=1}^{\infty} |a_i - b_i|^p.$

$$d_p$$ is a valid distance function over $$\ell^p$$. We metrize $$\ell^p$$ with $$d_p$$ as the standard metric.

## 3.1.6. Products of Metric Spaces#

Definition 3.4 (Finite products of metric spaces)

Let $$(X_1, d_1), (X_2, d_2), \dots, (X_n, d_n)$$ be $$n$$ metric spaces.

Let $$X = X_1 \times X_2 \times \dots \times X_n$$. Define a map $$\rho : X \times X \to \RR$$ as:

$\rho ((a_1, a_2, \dots, a_n), (b_1, b_2, \dots, b_n)) = \sum_{i=1}^n d_i (a_i, b_i).$

$$\rho$$ is a distance function on $$X$$. The metric space $$(X, \rho)$$ is called the product of metric spaces $$(X_i, d_i)$$.

## 3.1.7. Distance between Sets and Points#

Definition 3.5 (Distance between a point and a set)

The distance between a nonempty set $$A \subseteq X$$ and a point $$x\in X$$ is defined as:

$d(x, A) \triangleq \inf \{ d(x,a) \Forall a \in A \}.$
• Since $$A$$ is nonempty, hence the set $$D = \{ d(x,a) \Forall a \in A \}$$ is not empty.

• $$D$$ is bounded from below since $$d(x, a) \geq 0$$.

• Since $$D$$ is bounded from below, hence it does have an infimum.

• Thus, $$d(x, A)$$ is well-defined and finite.

• Since $$A$$ is non-empty, hence there exists $$a \in A$$.

• $$d(x, a) \in D$$.

• Thus, $$D$$ is bounded from above too.

• Thus, $$0 \leq d(x, A) \leq d(x, a)$$.

• If $$x \in A$$, then $$d(x, A) = 0$$.

Theorem 3.1

If $$x \in A$$, then $$d(x, A) = 0$$.

Example 3.7

1. Let $$X = \RR$$ and $$A = (0, 1)$$.

2. Let $$x = 0$$.

3. Then $$d(x, A) = 0$$.

4. However, $$x \notin A$$.

5. Thus, $$d(x,A) = 0$$ doesn’t imply that $$x \in A$$.

Distance of a set with its accumulation points is 0. See Theorem 3.21.

## 3.1.8. Distance between Sets#

Definition 3.6 (Distance between sets)

The distance between two nonempty sets $$A,B \subseteq X$$ is defined as:

$d(A, B) \triangleq \inf \{ d(a,b) \ST a \in A, b \in B \}.$