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- 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.
- 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:

- (Positivity) For all
*x*,*y*∈*X*,*d*(*x*,*y*) ≥ 0 and*d*(*x*,*y*) = 0 if and only if*x*=*y*, - (Symmetry) For all
*x*,*y*∈*X*,*d*(*x*,*y*) =*d*(*y*,*x*), - (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. - (Positivity) For all
**Examples**- 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**.

- 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.

- The
*plane***R**^{2}with the usual metric*d*_{2}obtained from Pythagoras's theorem.

*d*_{2}((*x*_{1},*y*_{1}), (*x*_{2},*y*_{2})) = √((*x*_{1}-*x*_{2})^{2}+ (*y*_{1}-*y*_{2})^{2}).

The picture looks similar to the complex numbers case.

- The
*plane***R**^{2}with the*taxicab*metric*d*_{1}.

*d*_{1}((*x*_{1},*y*_{1}), (*x*_{2},*y*_{2})) = |*x*_{1}-*x*_{2}| + |*y*_{1}-*y*_{2}|.

- The
*plane***R**^{2}with the*supremum*metric*d*_{∞}.

*d*_{∞}((*x*_{1},*y*_{1}), (*x*_{2},*y*_{2})) = max{*x*_{1}-*x*_{2}|, |*y*_{1}-*y*_{2}|}.

In the above three examples the first two properties of the metric are easy to check. The triangle inequality is a bit harder.**Remarks**- To visualise the last three examples, it helps to look at the unit circles. That is the sets {
*P*∈**R**^{2}|*d*(**0**,*P*) = 1 }

- 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}((*x*_{1},*y*_{1}), (*x*_{2},*y*_{2})) = [|*x*_{1}-*x*_{2}|^{p}+ |*y*_{1}-*y*_{2}|^{p}]^{1/p}for any*p*≥ 1.

If you let*p*→ ∞ you get the example*d*_{∞}. - Examples 3. to 5. above can be defined for higher dimensional spaces
**R**^{n}also. The next two examples show that one can even use them in some infinite dimensional spaces.

- To visualise the last three examples, it helps to look at the unit circles. That is the sets {
- Let
*X*be the set of all bounded real sequences (*x*_{1},*x*_{2},*x*_{3}, ... ). Define a metric*d*_{∞}on*X*by

*d*_{∞}((*x*_{1},*x*_{2},*x*_{3}, ... ), (*y*_{1},*y*_{2},*y*_{3}, ... )) = lub{*x*_{1}-*y*_{1}|, |*x*_{2}-*y*_{2}|, |*x*_{3}-*y*_{3}|, ... }.(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".

- 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".

- Let
*C*[0, 1] be the set of all*continuous*functions on the interval [0, 1].

Then define a metric*d*_{1}by*d*_{1}(*f*,*g*) = |*f*(*x*) -*g*(*x*)|*dx*.

Here is a picture:

**Remarks**- 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.
- 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. - This last example can be generalised to metrics
*d*_{p}with formulae like:

*d*_{p}(*f*,*g*) = [|*f*(*x*) -*g*(*x*)|^{p}*dx*]^{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).

- 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.

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(Convergence in a metric space)

- The prototype: The set of

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**