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, …, x_n) and y = (y₁, y₂, …, y_n) ∈ X=Kⁿ, define

• d₁(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 d₁ and d_∞.

Before we talk about d_p 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 $\sum_{i=1}^n$ |a_ib_i| ≤ ($\sum_{i=1}^n$ |a_i|^p)^1/p ($\sum_{i=1}^n$ |b_i|^q)^1/q
2. 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 Kⁿ

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

d_p(x, y) = ($\sum_{i=1}^n$ |x_i – y_i|^p)^1/p.

Theorem: d_p 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 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 Kⁿ and this makes (Kⁿ, d_p) is a metric space..

FAQs