Metric Space Verification Proofs

What is a metric space?#

In the study of mathematical (real) analysis, metric space is a commonly covered topic in typical university curricula.

A metric space can be said as a notion that produces a special kind of collection such that we can describe the distance or length between two elements in the collection using a certain function. A formal name of such function is metric.

Formally, a metric space is defined as a pair $(S,d)$, where $S$ is a set and $d : S \times S \rightarrow \mathbb{R}$ is a metric associated with $S$.

For $(S,d)$ to be a metric space, the metric $d$ should satisfy the following key properties:

• Positivity: $d(x,y)$ is nonnegative¹, i.e. $d(x,y) \geqslant 0$, for all $x, y \in S$.
• Definiteness: $d(x,y) = 0$ if and only if $x=y$, for all $x, y \in S$.
• Symmetry: $d(x,y)=d(y,x)$, for all $x, y \in S$.
• Triangle Inequality: $d(x,y) \leqslant d(x,z)+d(z,y)$, for all $x,y,z \in S$.

Verification proofs of common metric spaces#

There are some metric spaces which are commonly seen in most real analysis textbooks.

In this section, we verify that they indeed are metric spaces. To show that a given pair of set and metric $(S,d)$ is a metric space, we can show that it satisfies all four key properties as outlined above.

Usual metric space#

First, we have the set of real numbers associated with the usual metric $d : \mathbb{R} \times \mathbb{R} \rightarrow \mathbb{R}$, i.e. the pair $(\mathbb{R},d)$ where $d$ is given by $d(x,y)=\vert{x-y}\vert$ for all $x,y \in \mathbb{R}$.

Here, we show that $(\mathbb{R},d)$ is a metric space.

• Positivity: This follows from the properties of absolute values.
• Definiteness: This follows from the fact that the absolute value of a number is zero if and only if the number itself is zero.
• Symmetry: This follows from the symmetry of the absolute value of two substracted terms, i.e. $|a-b|=|b-a|$.
• Triangle Inequality: This follows from the fact that $|a-c|=|(a-b)+(b-c)| \leqslant |a-b|+|b-c|$.

Euclidean metric space#

Next, we have the set of the Cartesian product of $n$ copies of the set of real numbers, $\mathbb{R}^n$, endowed with the Euclidean metric $d_E : \mathbb{R}^n \times \mathbb{R}^n \rightarrow \mathbb{R}$, which is given by $d_E(x,y)=\sqrt{\sum_{i=1}^{n}{(x_i-y_i)^2}}$ for all $x = (x_1,x_2,\ldots,x_n), y=(y_1,y_2,\ldots,y_n) \in \mathbb{R}^n$.

We show that $(\mathbb{R}^n,d_E)$ is a metric space.

• Positivity: This follows from the fact that the sum of squares are nonnegative and its square root is also nonnegative.
• Definiteness: This follows from the fact that a square is zero if and only if the squared term itself is zero.
• Symmetry: This follows from the symmetry of the square of subtraction, i.e. $(a-b)^2=(b-a)^2$ for any $a,b \in \mathbb{R}$.
• Triangle Inequality: This follows from the Cauchy-Schwarz inequality.

‘Manhattan metric space’#

I put quotation marks around the name because I don’t think it is a universally accepted name.

Consider the pair $(\mathbb{R}^n,d_M)$, where $d_M : \mathbb{R}^n \times \mathbb{R}^n \rightarrow \mathbb{R}$ is given by $d_M(x,y)=\sum_{i=1}^{n}{\vert{x_i-y_i}\vert}$. Such metric is commonly known as the Manhattan distance function.

We verify that $(\mathbb{R}^n,d_M)$ is a metric space.

• Positivity: This follows from the fact that absolute values are nonnegative.
• Definiteness: This follows from the fact that the absolute value of a term is zero if and only if the term itself is zero.
• Symmetry: This follows from the fact that for any $a, b \in \mathbb{R}$, $|a-b|=|b-a|$.
• Triangle Inequality: This follows from the Triangle Inequality involving absolute values.

Discrete metric space#

There is also a metric space that is not limited by the choice of sets.

Let $S$ be an arbitrary non-empty set. For all $x,y \in S$, we define a discrete metric $d : S \times S \rightarrow \mathbb{R}$ by

We verify that $(S,d)$ is a metric space.

• Positivity: For any two real numbers, either they are equal to each other or they are not. Since both cases are mapped to a nonnegative value by the metric, positivity is satisfied.
• Definiteness: This follows immediately from the definition of the metric.
• Symmetry: This follows immediately from the definition of the metric.
• Triangle Inequality: Fix any two numbers $a,b \in \mathbb{R}$. If $a=b$, then $d(a,b)=0 \leqslant d(a,c)+d(c,b)$ since both terms of the right-hand side of the inequality are nonnegative, so the triangle inequality is satisfied for this case. If $a \neq b$, then $d(a,b)=1$. Take another number $c$. Since $a \neq b$, we can only have at most one equality for between $a$ and $c$ and between $b$ and $c$, so $d(a,c)+d(c,b) \geqslant 1 = d(a,b)$, and thus the triangle inequality is satisfied for this case too. Since two numbers are either equal or not equal, this proves the triangle inequality for the general case.