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


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..


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.

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