Metric spaces: definition and examples

The ideas of convergence and continuity introduced in the last sections are useful in a more general context. In particular we will be able to apply them to sequences of functions.
The basic idea that we need to talk about convergence is to find a way of saying when two things are close. A metric space is something in which this makes sense.

Definitions

Let X be a set. A metric on X is an assignment of a distance d(x, y) ∈ R to every pair of "points" x, y in X
(that is d: X × X→ R) satisfying the following:

1. (Positivity) For all x, y ∈ X, d(x, y) ≥ 0 and d(x, y) = 0 if and only if x = y,

2. (Symmetry) For all x, y ∈ X, d(x, y) = d(y, x),

3. (The triangle inequality) For all x, y, z ∈ X, d(x, y) + d(y, z) ≥ d(x, z).
   

A metric space is a set X together with such a metric.

Examples

1. The prototype: The set of real numbers R with the metric d(x, y) = |x - y|.
   This is what is called the usual metric on R.

   
   

2. The complex numbers C with the metric d(z, w) = |z - w|.
   Although the formula looks similar to the real case, the | | represent the modulus of the complex number. The picture looks different too.
   
   
   
3. The plane R² with the usual metric d₂ obtained from Pythagoras's theorem.
   d₂((x₁, y₁), (x₂, y₂)) = √((x₁ - x₂)² + (y₁ - y₂)²).
   The picture looks similar to the complex numbers case.
   
   
   
   
4. The plane R² with the taxicab metric d₁.
   d₁((x₁, y₁), (x₂, y₂)) = |x₁ - x₂| + |y₁ - y₂|.
   
   
   
   
5. The plane R² with the supremum metric d_∞.
   d_∞((x₁, y₁), (x₂, y₂)) = max{x₁ - x₂|, |y₁ - y₂|}.
   
   
   In the above three examples the first two properties of the metric are easy to check. The triangle inequality is a bit harder.

   Remarks

   
   1. To visualise the last three examples, it helps to look at the unit circles. That is the sets {P ∈ R²| d(0, P) = 1 }
      
      
      
   2. The subscripts on the d's are explained by the fact that there is a whole family of metrics :
      d_p given by
      d_p((x₁, y₁), (x₂, y₂)) = [|x₁ - x₂|^p+ |y₁ - y₂|^p]^1/pfor any p ≥ 1.
      If you let p→ ∞ you get the example d_∞.

   3. Examples 3. to 5. above can be defined for higher dimensional spaces Rⁿ also. The next two examples show that one can even use them in some infinite dimensional spaces.
      
   
   
6. Let X be the set of all bounded real sequences (x₁ , x₂ , x₃ , ... ). Define a metric d_∞ on X by
   d_∞((x₁ , x₂ , x₃ , ... ), (y₁ , y₂ , y₃ , ... )) = lub{x₁ - y₁|, |x₂ - y₂|, |x₃ - y₃|, ... }.

   (You had better have the sequences bounded or the lub won't exist.)

   This is a metric space that experts call l^∞ ("Little l-infinity").

   The other metrics above can be generalised to spaces of sequences also.

   Let us look at some other "infinite dimensional spaces".
   
   

7. Let B[0, 1] be the set of all bounded functions on the interval [0, 1]. Then define a metric (again called the supremum metric) by d_∞(f, g) = {|f (x) - g(x)|}.
   

   
   Here is a picture:

   Although we have drawn the graphs of continuous functions we really only need them to be bounded.
   Note that d_∞ is "The maximum distance between the graphs of the functions".
   
   
   
   

8. Let C[0, 1] be the set of all continuous functions on the interval [0, 1].
   Then define a metric d₁ by d₁(f, g) = |f (x) - g(x)| dx.
   

   
   Here is a picture:
   
   

   Remarks

   1. Deciding whether or not an integral of a function exists is in general a bit tricky. In this case, however, it is OK since continuous functions are always integrable.

   2. The hard bit about proving that this is a metric is showing that if d(f, g) = 0 then f = g. For this you need to use the fact that f and g are continuous.

   3. This last example can be generalised to metrics d_p with formulae like:
      d_p(f, g) = [|f (x) - g(x)|^pdx]^1/p.
      The case p = 2 is particularly important to theoretical physicists and leads to something called Hilbert Space named after the mathematician David Hilbert (1862 to 1943).
      

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JOC September 2001