Introduction to metric spaces

Introduction

In may courses, concepts such as continuity and convergence are invoked without much discussion of their formal definitions, instead relying on the reader's intuitive understanding of these matters. However, for purposes of proofs, including some in the main body of material for this course, a greater rigor is required. This module will discuss metric spaces, a mathematical construct that provide a framework for the study continuity, convergence, and other related ideas in their most concrete but still formal senses. This is accomplished by formalizing a notion of the distance between two elements in a set. The intent in this and subsequent modules in this chapter is not to give a complete overview of the basic topics of analysis but to give a short introduction to those most important to discussion of signal processing in this course.

Metric spaces

A notion of distance

In many situations in signal processing it is often useful to have a concept of distance between the points in a set. This notion is mathematically formalized through the idea of a metric space. A metric space $(M,d)$ is a set $M$ together with a function $d:M\times M\to R$ that assigns distances between pairs of elements in $M$ while satisfying three conditions. First, for every $x,y\in M$ , $d(x,y)\ge 0$ with $d(x,y)=0$ if and only if $x=y$ . Second, for every $x,y\in M$ , $d(x,y)=d(y,x)$ symmetrically. Third, for every $x,y,z\in M$ , $d(x,y)+d(x,z)\ge d(y,z)$ , which is known as the triangle inequality.

There are, of course, several different possible choices of definitions for distances in a given set. Our typical intuitive understanding of distance in ${\mathbb{R}}^n$ fits within this framework as the standard Euclidean metric

as does the taxicab or Manhatten metric

that sums individual components of vectors, representing, for example, distances traveled walking around city blocks. Another simple yet more exotic example is provided by the discrete metric on any set defined by

in which all pairs of distinct points are equidistant from eachother but every point is distance zero from itself. One can check that these satisfy the conditions for metric spaces.

Relationship with norms

It is not surprising that norms, which provide a notion of size, and metrics, which provide a notion of distance, would have a close relationship. Intuitively, one way of defining the distance between two points in a metric space could be the size of their difference. In other words given a vector space $V$ over the field $F$ with norm $\left|\right|·\left|\right|$ , we might ask if the function

for every $x,y\in V$ satisfies the conditions for $(V,d)$ to be a metric space.

Let $V$ be a vector space over the field $F$ with norm $\left|\right|·\left|\right|$ , and let $d(x,y)=\left|\right|x-y\left|\right|$ . Recall that since $\left|\right|·\left|\right|$ is a norm, $\left|\right|x\left|\right|=0$ if and only if $$x=0$$ and $\left|\right|ax\left|\right|=\left|a\right|\left|\right|x\left|\right|$ for all $a\in F$ and $x\in V$ . Hence $\left|\right|x-y\left|\right|\ge 0$ for all $x,y\in V$ and $\left|\right|x-y\left|\right|=0$ if and only if $x=y$ . Since $y-x=-(x-y)$ and $\left|\right|-(x-y)\left|\right|=\left|\right|x-y\left|\right|$ it follows that $\left|\right|x-y\left|\right|=\left|\right|y-x\left|\right|$ for all $x,y\in V$ . Finally, $\left|\right|x\left|\right|+\left|\right|y\left|\right|\ge \left|\right|x+y\left|\right|$ by the properties of norms, so $\left|\right|x-y\left|\right|+\left|\right|x-z\left|\right|\ge \left|\right|y-z\left|\right|$ for all $x,y,z\in V$ . Thus, $(V,d)$ does indeed satisfy the conditions to be a metric space and is discussed as the metric space induced by the norm $\left|\right|·\left|\right|$ .

Metric spaces summary

Metric spaces provide a notion of distance and a framework with which to formally study mathematical concepts such as continuity and convergence, and other related ideas. Many metrics can be chosen for a given set, and our most common notions of distance satisfy the conditions to be a metric. Any norm on a vector space induces a metric on that vector space and it is in these types of metric spaces that we are often most interested for study of signals and systems.