Metric Spaces - UiO

1 Chapter 1 Metric SpacesMany of the arguments you have seen in several variable calculus are almostidentical to the corresponding arguments in one variable calculus, especiallyarguments concerning convergence and continuity. The reason is that thenotions of convergence and continuity can be formulated in terms of distance ,and that the notion of distance between numbers that you need in the onevariable theory, is very similar to the notion of distance between points orvectors that you need in the theory of functions of severable variables. Inmore advanced mathematics, we need to find the distance between morecomplicated objects than numbers and vectors, between sequences, setsand functions. These new notions of distance leads to new notions of con-vergence and continuity, and these again lead to new arguments suprisinglysimilar to those we have already seen in one and several variable a while it becomes quite boring to perform almost the same argu-ments over and over again in new settings, and one begins to wonder if thereis general theory that cover all these examples is it possible to developa general theory of distance where we can prove the results we need onceand for all?

2 The answer is yes, and the theory is called the theory of Metric space is just a setXequipped with a functiondof two variableswhich measures the distance between points:d(x, y) is the distance betweentwo pointsxandyinX. It turns out that if we put mild and naturalconditions on the functiond, we can develop a general notion of distancethat covers distances between number, vectors, sequences, functions, setsand much more. Within this theory we can formulate and prove resultsabout convergence and continuity once and for all. The purpose of thischapter is to develop the basic theory of Metric Spaces . In later chapters weshall meet some of the applications of the 1. Metric Definitions and examplesAs already mentioned, a Metric space is just a setXequipped with a functiond:X X Rwhich measures the distanced(x, y) beween pointsx, y the theory to work, we need the functiondto have properties similarto the distance functions we are familiar with.

3 So what properties do weexpect from a measure of distance ?First of all, the distanced(x, y) should be a nonnegative number, andit should only be equal to zero ifx=y. Second, the distanced(x, y) fromxtoyshould equal the distanced(y, x) fromytox. Note that this is notalways a reasonable assumption if we , , measure the distance fromxtoyby the time it takes to walk fromxtoy,d(x, y) andd(y, x) may bedifferent but we shall restrict ourselves to situations where the conditionis satisfied. The third condition we shall need, says the distance obtainedby going directly fromxtoy, should always be less than or equal to thedistance we get when we stop at a third pontzalong the way, (x, y) d(x, z) +d(z, x)It turns out that these conditions are the only ones we need, and we sumthem up in a formal space (X, d)consists of a non-empty setXanda functiond:X X [0, )such that:(i) (Positivity) For allx, y X,d(x, y) 0with equality if and only ifx=y.]

4 (ii) (Symmetry) For allx, y X,d(x, y) =d(y, x).(iii) (Triangle Inequality) For allx, y, z Xd(x, y) d(x, z) +d(z, y).A functiondsatisfying conditions (i)-(iii), is called :When it is clear or irrelevant which metricdwe have in mind,we shall often refer to the Metric spaceX rather than the Metric space(X, d) .Let us take a look at some examples of Metric 1:If we letd(x, y) =|x y|, (R, d) is a Metric space. The first twoconditions are obviously satisfied, and the third follows from the ordinarytriangle inequality for real numbers:d(x, y) =|x y|=|(x z) + (z y)| |x z|+|z y|=d(x, z) +d(z, y) DEFINITIONS AND EXAMPLES3 Example 2:If we letd(x,y) =|x y|, (Rn, d) is a Metric space. Thefirst two conditions are obviously satisfied, and the third follows from thetriangle inequality for vectors the same way as above :d(x,y) =|x y|=|(x z) + (z y)| |x z|+|z y|=d(x,z) +d(z,y)Example 3:Assume that we want to move from one pointx= (x1, x2)in the plane to anothery= (y1, y2), but that we are only allowed to movehorizontally and vertically.

5 If we first move horizontally from (x1, x2) to(y1, x2) and then vertically from (y1, x2) to (y1, y2), the total distance isd(x,y) =|y1 x1|+|y2 x2|This gives us a Metric onR2which is different from the usual Metric inExample 2. It is ofte referred to as theManhattan metricor thetaxi in this cas the first two conditions of a Metric space are obviouslysatisfied. To prove the triangle inequality, observe that for any third pointz= (z1, z2), we haved(x,y) =|y1 x1|+|y2 x1|==|(y1 z1) + (z1 x1)|+|(y2 z2) + (z2 x2)| |y1 z1|+|z1 x1|+|y2 z2|+|z2 x2|==|z1 x1|+|z2 x2|+|y1 z1|+|y2 z2|==d(x, z) +d(z, y)where we have used the ordinary triangle inequality for real numbers to getfrom the second to the third 4:We shall now take a look at an example of a different that we want to send messages in a language of N symbols (letters,numbers, punctuation marks, space, etc.)

6 We assume that all messages havethe same lengthK(if they are too short or too long, we either fill them outor break them into pieces). We letXbe the set of all messages, allsequences of symbols from the language of lengthK. Ifx= (x1, x2, .. , xK)andy= (y1, y2, .. , yK) are two messages, we defined(x,y) = the number of indicesnsuch thatxn6=ynIt is not hard to check thatdis a Metric . It is usually referred to as theHamming- Metric , and is much used in coding theory where it serves as ameasure of how much a message gets distorted during 1. Metric SPACESE xample 5:There are many ways to measure the distance between func-tions, and in this example we shall look at some. LetXbe the set of allcontinuous functionsf: [a, b] R. Thend1(f, g) = sup{|f(x) g(x)|:x [a, b]}is a Metric onX. This Metric determines the distance beween two functionsby measuring the distance at thex-value where the graphs are most means that the distance between two functions may be large even ifthe functions in average are quite close.

7 The metricd2(f, g) = ba|f(x) g(x)|dxinstead sums up the distance betweenf(x) ogg(x) at all points. A thirdpopular Metric isd3(f, g) =( ba|f(x) g(x)|2dx)12 This Metric is a generalization of the usual (euclidean) Metric inRn:d(x,y) = n i=1(xi yi)2=(n i=1(xi yi)2)12(think of the integral as a generalized sum). That we have more thanone Metric onX, doesn t mean that one of them is right and the oth-ers wrong , but that they are useful for different 6:The metrics in this example may seem rather strange. Al-though they are not very useful in applications, they are handy to knowabout as they are totally different from the metrics we are used to fromRnand may help sharpen our intuition of how a Metric can be. LetXbe anynon-empty set, and define:d(x, y) = 0 ifx=y1 ifx6=yIt is not hard to check thatdis a Metric onX, usually referred to as 7:There are many ways to make new Metric Spaces from simplest is the subspace Metric : If (X, d) is a Metric space andAis a non-empty subset ofX, we can make a metricdAonAby DEFINITIONS AND EXAMPLES5dA(x, y) =d(x, y) for allx, y A we simply restrict the Metric toA.

8 Itis trivial to check thatdAis a Metric onA. In practice, we rarely bother tochange the name of the Metric and refer todAsimply asd, but rememberin the back of our head thatdis now restricted are many more types of Metric Spaces than we have seen so far, butthe hope is that the examples above will give you a certain impression of thevariety of the concept. In the next section we shall see how we can defineconvergence and continuity for sequences and functions in Metric we prove theorems about these concepts, they automatically hold inall Metric Spaces , saving us the labor of having to prove them over and overagain each time we introduce a new class of for Section Show that (X, d) in Example 4 is a Metric Show that (X, d1) in Example 5 is a Metric Show that (X, d2) in Example 5 is a Metric Show that (X, d) in Example 6 is a Metric A sequence{xn}n Nof real numbers is calledboundedif there is a numberM Rsuch that|xn| Mfor alln N.

9 LetXbe the set of all boundedsequences. Show thatd({xn},{yn}) = sup{[xn yn|:n N}is a Metric IfVis a (real) vector space, a function| |:V Ris called anormif thefollowing conditions are satisfied:(i) For allx V,|x| 0 with equality if and only ifx= 0.(ii)| x|=| ||x|for all Rand allx V.(iii)|x+y| |x|+|y[ for allx, y that if| |is a norm, thend(x, y) =|x y|defines a Metric onV7. Show that if (X, d) is a Metric space.|d(x, y) d(x, z)| d(z, y)for allx, y, z Assume thatd1ogd2are two metrics onX. Show thatd(x, y) =d1(x, y) +d2(x, y)is a Metric 1. Metric SPACES9. Assume that (X, dX) and (Y, dY) are two Metric Spaces . Define a functiond: (X Y) (X Y) Rbyd((x1, y1),(x2, y2)) =dX(x1, x2) +dY(y1, y2)Show thatdis a Metric onX LetXbe a non-empty set, and let :X X Rbe a function satisfying:(i) (x, y) 0 with equality if and only ifx=y.(ii) (x, y) (x, z) + (z, y) for allx, y, z :X X Rbyd(x, y) = max{ (x, y), (y, x)}Show thatdis a Metric Convergence and continuityWe begin our study of Metric Spaces by defining convergence of sequences.]]

10 Asequence{xn}in a Metric spaceXis just a collection{x1, x2, x3, .. , xn, ..}of elements inXenumerated by the natural (X, d)be a Metric space. A sequencee{xn}inXconvergesto a pointa Xif there for every >0exists anN Nsuchthatd(xn, a)< for alln N. We writelimn xn=aorxn that this definition exactly mimics the definition of convergence inRogRn. Here is an alternative sequence{xn}in a Metric space(X, d)converges toaifand only iflimn d(xn, a) = :The distances{d(xn, a)}form a sequence of nonnegative sequence converges to 0 if and only if there for every >0 exists anN Nsuch thatd(xn, a)< whenn N. But this is exactly what thedefinition above a sequence converge to more than one point? We know that itcannot inRn, but some of these new Metric Spaces are so strange that wecan not be certain without a sequence in a Metric point can not converge to morethan one CONVERGENCE AND CONTINUITY7 Proof:Assume that limn xn=aand limn xn=b.