# Metric Spaces: Definition, Examples, Properties

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.

## 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

1. d(x, y) ≥ 0 and d(x, y) = 0 if and only if x=y.
2. d(x, y) = d(y, x)
3. 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=Kn. For x = (x1, x2, …, xn) and y = (y1, y2, …, yn) ∈ X=Kn, define

• d1(x, y) = $\sum_{i=1}^n$ |xi – yi| and
• d(x, y) = sup1≤i≤n |xi – yi|

As |xi – yi| ≤ |xi – zi|+|zi – yi|, it can be verified that X is a metric space with respect to the metrics d1 and d.

Before we talk about dp metric on Kn, 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:

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

## dp metric on Kn

For 1<p<∞ and x = (x1, x2, …, xn) and y = (y1, y2, …, yn) ∈ Kn, define the metric dp on the space Kn by

dp(x, y) = ($\sum_{i=1}^n$ |xi – yi|p)1/p.

Theorem: dp is a metric on Kn

Proof:

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

Let x, y, z ∈ Kn.

Take ai = xi – zi and bi = zi – yi in the above Minkowski’s inequality. Thus, we have

dp(x, y) = ($\sum_{i=1}^n$ |ai+bi|p)1/p ≤ ($\sum_{i=1}^n$ |ai|p)1/p + ($\sum_{i=1}^n$ |bi|p)1/p ≤ dp(x, z) + dp(z, y).

Thus, dp satisfies the triangle inequality. Hence, we have proved that dp is a metric on Kn and this makes (Kn, dp) 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.

Share via: