A metric space is a set X equipped with a distance structure. Here we will study various properties of the metric space X that are well-behaved with respect to this distance structure. For example, the set ℝ of real numbers is a metric space with distance function d(x,y) = |x-y| for x, y ∈ ℝ.

Table of Contents

## Definition of Metric Space

Let X be a non-empty set. A function d: X × X →ℝ is said to be a **metric** on X if for all x, y, z ∈ X the following are satisfied

- d(x, y) ≥ 0 and d(x, y) = 0 if and only if x=y.
- d(x, y) = d(y, x)
- d(x, y) ≤ d(x, z) + d(z, y). This inequality is known as the
**triangle inequality**.

A non-empty set X is called a metric space if there is a metric d defined on it. This metric space is denoted by (X, d).

## Examples of Metrics and Metric Spaces

Now we will provide a few examples of metric spaces.

### Discrete Metric

Let X be a non-empty set. For x, y ∈ X, define

d(x, y) = 0 if x=y

$\quad \quad \,\,$ = 1 if x≠y.

It is easy to check that d is a metric on X. This metric is called the discrete metric.

### Metric on Linear Spaces

Let K be a linear space over K. Consider the space X=K^{n}. For x = (x_{1}, x_{2}, …, x_{n}) and y = (y_{1}, y_{2}, …, y_{n}) ∈ X=K^{n}, define

- d
_{1}(x, y) = $\sum_{i=1}^n$ |x_{i}– y_{i}| and - d
_{∞}(x, y) = sup_{1≤i≤n}|x_{i}– y_{i}|

As |x_{i} – y_{i}| ≤ |x_{i} – z_{i}|+|z_{i} – y_{i}|, it can be verified that X is a metric space with respect to the metrics d_{1} and d_{∞}.

Before we talk about d_{p} metric on K^{n}, let us first note down the following two useful inequalities.

## Hölder Inequality and Minkowski’s Inequality

For 1<p<∞ and q with 1/p+ 1/q =1, we have:

**Hölder Inequality:**If 1<p<∞, then by Hölder inequality we have $\sum_{i=1}^n$ |a_{i}b_{i}| ≤ ($\sum_{i=1}^n$ |a_{i}|^{p})^{1/p}($\sum_{i=1}^n$ |b_{i}|^{q})^{1/q}**Minkowski’s Inequality:**If 1≤p<∞, then Minkowski’s inequality we have ($\sum_{i=1}^n$ |a_{i}+b_{i}|^{p})^{1/p}≤ ($\sum_{i=1}^n$ |a_{i}|^{p})^{1/p}+ ($\sum_{i=1}^n$ |b_{i}|^{p})^{1/p}

## d_{p} metric on K^{n}

For 1<p<∞ and x = (x_{1}, x_{2}, …, x_{n}) and y = (y_{1}, y_{2}, …, y_{n}) ∈ K^{n}, define the metric d_{p} on the space K^{n} by

d_{p}(x, y) = ($\sum_{i=1}^n$ |x_{i} – y_{i}|^{p})^{1/p}.

**Theorem:** d_{p} is a metric on K^{n}

*Proof: *

The first two conditions of the definition of a metric are easily satisfied. Let us check the triangle inequality.

Let x, y, z ∈ K^{n}.

Take a_{i }= x_{i} – z_{i }and b_{i }= z_{i} – y_{i} in the above Minkowski’s inequality. Thus, we have

d_{p}(x, y) = ($\sum_{i=1}^n$ |a_{i}+b_{i}|^{p})^{1/p} ≤ ($\sum_{i=1}^n$ |a_{i}|^{p})^{1/p} + ($\sum_{i=1}^n$ |b_{i}|^{p})^{1/p} ≤ d_{p}(x, z) + d_{p}(z, y).

Thus, d_{p} satisfies the triangle inequality. Hence, we have proved that d_{p} is a metric on K^{n} and this makes (K^{n}, d_{p}) is a metric space..

## FAQs

**Q1: What is metric space? Give an example.**

Answer: A pair (X, d) is called a metric space if X is non-empty and d is a distance function (called metric) defined on X. ℝ is a metric space with the metric d(x,y) = |x-y| for all real numbers x, y.