If any general observation could be made concerning the use of the word "space" in mathematics, it is perhaps this: A mathematical space is a set, together with some geometrical structure defined on top of it.

  Here we discuss a very useful class of mathematical spaces called metric spaces. Very briefly, a metric space consists of two parts: a set, and a measure of distance over that set.

  The "geometrical structure" mentioned in the first paragraph, so far as a metric space is concerned, is distance. It is possible to ascertain the distance between any two points in a metric space. It is as simple as that.

  We will see, elsewhere, that the mathematical machinery used in engineering leans heavily on the use of metrics. We like to embed signals of interest in metric spaces. Then we can measure the similarity or dissimilarity of signals, fidelity and distortion, as distance. Similar applications abound in physics and mathematics. We will have much more to say about this.
 

A Metric is a Function

  Let   be a function from the product set   to the set of real numbers such that the following four axioms (or rules), A1 through A4, hold true:

  The symbol ">=" is used here for "greater than or equal to," since the usual symbol for this relation is not available in HTML. For those to whom the notation and terminology used here is unfamiliar, I'd like to say that we will go back and explain it sometime soon. Please do write us your comments and suggestions. They will determine the evolution of this site.

  The pair (A,d) of objects (A, a set, and d, a function) is said to define a metric space. We say also that d defines a metric on A.
 

  We want a distance to never ever be a negative number. Therefore we have axiom A1.

  We want a distance from point a to point b to vanish if, and only if, a and b are one and the same point. Hence A2.

  We want the distance from a to b to be the same as the distance from b to a. Hence A3.

  Last of all, if we travel directly from a point a to a point b we want the distance so travelled to be less than or equal to the distance we would have to travel if went through a third point c in between. Hence A4.

  To see why the axiom A4 is called the axiom of triangle inequality, recall from elementary geometry the fact that the sum of any two edges of a triangle must be greater than or equal to the the length of the third edge.

  We have seen that axioms A1 through A4 are natural rules to impose on a metric.
 

Example: A Space That Is Not A Metric Space

  Let A = {a,b,c} be a set of three elements. Define

d(a,b) = d(b,a) = 100,
d(b,c) = d(c,b) = 10, and
d(c,a) = d(a,c) = 80.

  Then (A,d) is not a metric space because d(b,c) + d(c,a) < d(b,a). Axiom A4 fails to hold.
 
 

A Metric Space Example

  Given two points in the Euclidean plane , say u = (x₁, y₁) and v = (x₂, y₂), let the distance between them be defined as .

This is the ordinary distance between two points, as in Euclid's geometry, and it satisfies the axioms A1 through A4. Therefore  is a metric space.
 

Another Metric Space Example

  Let us work again with the set of points in a plane, just as in the last example, but define a different measure of distance, d' (spoken dee-prime) over this set. Let

.

  Then  is also a metric space, but is very different from the space M. To see the difference, we draw in Figure 1 a picture of unit circles in M and M'.

Pseudo-metric Spaces

  A pseudo-metric space is just like a metric space, except that it is possible in a pseudo-metric space for two points to be distinct and yet be zero distance apart. In other words, axiom A2 above may not hold.

  Every metric is also a pseudo-metric; but not every pseudo-metric space is a metric space. We say therefore that metricity is a stricter requirement that pseudo-metricity. It takes more axioms to enforce metricity.

  It is always an application that decides our choice of an axiom-set and the flavor of the mathematics used. There is nothing inherently right or wrong about one mathematical system as against another based on a different axiom-set. One application may require us to work in a metric space, while another may require a pseudo-metric. Our mathematics must bend to the will of Mother Nature.
 

A Word on Topological Spaces

  Another useful class of spaces are topological spaces. The geometrical structure in a topological space is even more primitive, in a sense, than that found in a metric space.

  Every metric space is a topological space; but not every topological space is a metric space.

  We will discuss topological spaces in another writeup at Kahany.
 

Metric Space Applications

  In signal processing and control theory we are very interested in three kinds of mathematical spaces: Banach Spaces, Hilbert Spaces, and Hardy Spaces. We will see that these are, all three of them, different kinds of metric spaces. We will use them extensively in the engineering section of Kahany.
 

End of the Metric Spaces page