 Okay so our context is the proof of the Riemann mapping theorem okay and what we have done is in the last few lectures we have looked at hyperbolic geometry okay now there is one more technical detour that we have to take to be able to complete the proof of the Riemann mapping theorem and that has got to do with the so called complex version of the Arzila Ascoli theorem and the so called Montel theorem okay. So that is what I am going to discuss now right so basically the Arzila Ascoli theorem and the Montel theorem they are all theorems which you know guarantee that you know given a family of functions okay on a compact domain okay that any sequence in that family has a uniformly convergent subsequence okay so let me tell you the general idea the general idea is you see I have some a family of functions okay a family of functions defined on a domain alright and let us assume that the family is defined on I mean for example if you are thinking of the simplest case of real valued functions okay then you assume that if the real valued functions are all defined on a closed and bounded interval okay which is a compact subset compact connected subset of the real line right or more generally if you are thinking of functions on the plane okay then you think of functions which are defined on a domain on the plane okay and in fact you assume at least to begin with you assume that you they are all they also extend continuously to the boundary of the domain that the domain is bounded and it is you know if the domain is bounded and then you add its boundary it becomes compact okay so you have a family of complex functions complex valued functions defined on the on the domain of course all functions you we are interested in are all certainly continuous okay then the question is if you take a sequence of functions from this family okay to expect that the sequence will converge any sequence of functions will converges too much okay to expect that any given sequence of functions will converges too much okay but what you can always expect is that there is at least a subsequence which converges okay so the idea of the Arzela Ascoli theorem and you know Montel theorem is that under good conditions okay you can always ensure that you give me any sequence of functions in a family satisfying of course certain conditions on a compact set okay it will always have a subsequence which converges uniformly okay so the so the general point is that you want you have a family of functions okay and the domain effectively where you are studying it is compact alright or in the or if you are looking at for example analytic functions you are looking at analytic functions the property that I mean you look at the analytic functions on close discs in your domain which are compact closed and bounded discs okay and the result that you want is that you want to you want that given any sequence from this family there is a subsequence which converges uniformly okay. Now why is this so important it is important because you see the moment you say there is a subsequence which converges it tells you that at least there is a limit function for a subsequence though the whole the given sequence of functions need not converge but at least a subsequence converges but the fact that the converges uniform tells you that all the properties of the functions will also carry over to the limit for example if you are looking at a uniform convergent a uniform limit of analytic functions it continues to be analytic a uniform limit of continuous functions continues to be continuous. So you just do not want you know just convergence of functions that does not help because if you just take ordinary convergence of functions it can go wrong in the sense that the limit function may not have the good properties of original functions you can have a sequence of continuous functions which converges to a limit function which is not continuous you might have the limit function may lose continuity okay but you do not want such things to happen. So there is at least when you are studying convergence of functions it is natural that you at least demand uniform convergence so that all good properties pass on to the limiting function okay. So this are the both the Arzela-Ascoli theorem and the Montel theorem they basically situations which guarantee that you can always find a subsequence of uniformly convergent functions okay a subsequence that converges uniformly a subsequence of functions that converges uniformly okay. So that is the general idea now let us get to the technicalities okay so you know so here is a so here is the so here is the situation let script F be a collection a family or collection so let me do the following thing let me put the title as Arzela-Ascoli and Montel theorem. So these are very technical theorems but they are reasonably easy to prove and they are very powerful okay. So let F be a family of family or collection of functions defined on defined and continuous and complex value on a compact subset in complex plane okay. So I have a compact subset of the complex plane namely it is both closed and bounded and I have a collection of functions defined on points of this capital E taking values in complex numbers the condition is that to begin with I am assuming they are all continuous okay and they are complex value right. Now we all know what the definition of continuity at a point is the definition of continuity at a point is that you know given an Epsilon okay the function values at the point the function values in a neighbourhood of the point can be brought to within an Epsilon value within an Epsilon distance for the function value at that point if you choose points in a delta neighbourhood of the given point okay that is continuity at a given point and so let me just write that down recall that F a function in this family is continuous at a point the set if given Epsilon greater than 0 there exist delta greater than 0 such that whenever the distance between z and z0 is less than delta and of course z is point of E then the distance between Fz and Fz0 can be made less than Epsilon okay. So this is just ordinary Epsilon delta definition of continuity of the functions small f in this family at the point z0 okay now you see what you should notice is that you know delta this delta depends on of course delta depends on Epsilon okay and this delta also depends on the point z0 and it also depends on the point f it also depends on the function f okay so delta delta is actually delta of f, f, z0, Epsilon okay certainly if you change for you know if you keep the function f the same if you keep the Epsilon the same but if you change z0 the delta will change okay that is the dependence of delta on z0 okay and of course if you change Epsilon or f also the delta will change okay so this delta depends on these three things alright now you see suppose that you know you are able to find a delta that is independent of this f okay suppose you are able to find delta that does not depend on f okay that means the same delta for the given Epsilon the same delta will work for every f okay for a given Epsilon and given z0 the same delta will work for every f small f in the family script f if that happens we say that the family is equicontinuous at the point z0 okay so if delta is independent of f if a delta independent of f and depending only on Ez0 and Epsilon can be found for every Epsilon greater than 0 we say that the family f is equicontinuous at z0 so this is the motion of equicontinuity okay. So for all the functions in the family you know you are saying that the function values can be made the function values near the function the function values at points near to the point z0 can be made to within an Epsilon distance of the function value at z0 if you choose a sufficiently small neighborhood of the point z0 but the same neighborhood works for all functions okay it works uniformly for all functions right. So you know so the point is that you are able to find there is no dependence on the particular member of the small f of the family script f that is the whole point of course it is clear that if a family is equicontinuous at a point then it should be continuous at that point okay because equicontinuity is stronger than ordinary continuity alright and the point about this equicontinuity is that you know this is one of the ingredients one of the hypothesis that in the context of the Arzela Ascoli theorem or the Montel theorems it will ensure that you know you can always extract a subsequence of functions which converges uniformly okay so this is equicontinuity and so that is one of the technical ingredients so you know if you take any two points if you take any two points in this disk okay then the distance between them is certainly less than 2 delta alright and the distance between the function values by the triangle inequality is less than 2 Epsilon alright so what this also tells you is that it tells you that it tells you that each f will be kind of uniformly continuous on each of these disks okay so but anyway see the other thing that one wants to worry about is so called uniform boundedness so we say script f is uniformly bounded on E if of course you know in all these in this argument I have not I have still not used the compactness of the subset E alright but I could have defined it for any subset E of the complex plane right these definitions make sense for any subset E of the complex plane but the point is that the compactness is one of the ingredients for the theorem okay so of course in all these things I need not have assumed E is compact but I am keeping E compact in view of these theorems right so we say f is uniformly bounded on E if mod f of z is less than or equal to m for all z in E and for all small f in script f so this is uniform boundedness okay so of course boundedness of a function complex value function means that it is modulus is bounded okay so for all values of the function you take the modulus all these moduli they are bounded above by some positive real number okay you are able to find some positive real number m such that mod f z is always less than or equal to m so this m is a bound for f for mod f okay and you want the same bound to work for every small f in script f if that happens you say the family is uniformly bounded on E okay so you have so you have these two facts and now comes the now I can state the Arzela Ascoli theorem of course I am stating only one version of the theorem which is the version that we need but there are versions of the theorem for defined on compact housed of spaces with functions taking values in matrix spaces and so on and so forth there are very general versions okay but this is the version that we need okay that is the version that I am going to that I am defining that is the version that I am going to state and prove so here is the theorem let script f be a family of continuous complex valued functions on a compact subset E of the complex plane okay suppose f is uniformly bounded on E okay then the following are equivalent number 1 f is equicontinuous at each point of E number 2 every sequence f sub n has a uniformly convergence of sequence so this is the version of the Arzela Ascoli theorem that we need okay so you have so again let me explain you have this compact subset E of the complex plane so this compactness is very very important alright and you have script f a family of continuous functions defined on this compact set E and taking complex values alright and you put the condition that this family is uniformly bounded on E okay so there is a positive m which is an upper bound for the modulus of fz for all ez in E and for all f in script f okay then the Arzela Ascoli theorem actually tells you that the condition for being able to extract a uniformly convergent subsequence from any sequence in the family is equivalent to just demanding that the family is equicontinuous at every point of E okay so equic for a family of when you have uniformly when you have uniform boundedness okay then equicontinuity is equivalent to being able to extract a uniformly convergent subsequence okay this is how you can state it elegantly if you are having functions defined on a compact set okay which are uniformly bounded if you have a family of functions which are defined on a compact set and suppose the family is uniformly bounded then what is the condition that is equivalent to being able to extract a uniformly convergent subsequence from any given sequence of functions the condition is simply the equicontinuity of the family at each point of the compact set okay so this is the Arzela Ascoli theorem right and what I am going to do is I am going to next go on to the Montel theorem which I need to which I need to use in the proof of the Riemann mapping theorem but the Montel theorem is but to prove the Montel theorem I need only the implication that 1 implies 2 I do not need the other part of the equivalence which is 2 implies 1 so what I will do is I will just indicate how to prove 1 implies 2 and 2 implies 1 is a reasonably easy exercise okay and in fact even the proof that 1 implies 2 just parallels the proof that you would have seen in the real case in a first course in real analysis okay so let me do that so proof of 1 implies 2 so you see so you know this is a this is a standard technique of diagonalization that is used to prove this implication of the Arzela Ascoli theorem even in the even for real valued functions defined on you know a close bounded interval on the real line okay the same proof will work okay. So how does 1 begin so what is given is you are given a compact subset E in the complex plane you are given a family script F of continuous complex valued functions on E and you are given that this family is uniformly bounded so there is this constant M which bounds the modulus of the function values at each point of E and for every function in the family okay uniformly. Uniformly means the same constant works regardless of the point and regardless of the function okay and what is given to me is that it is equicontinuity is given to me okay so what you do is you make use of the fact you make use of the fact that if you if you you know on the real line if you take the rational numbers okay take the points that are rational that is countable and that is dense okay. So similarly if you take the plane which is just R2 okay if you take all the points with the rational coordinates then that is countable okay and it is also dense okay so this existence of a countable dense subset is what is used okay. So what you do is so you we do the following thing let E sub Q be the set of points of E with rational coordinates okay so of course you know here so in other words I am looking at points in the complex plane as points on points of R2 and when I say with rational coordinates I mean both the real and imaginary parts are rational okay so that is E sub Q is actually set of all X plus IY which are in E such that X and Y are rational okay then of course you know that then you know that E sub Q is countable okay and it is dense in E in Q in E because its closure will be its closure will be equal to E okay then E sub Q is countable and E sub Q closure will be E because E sub Q is just I mean E sub Q is just you know E intersection with Q cross Q, Q cross Q if you think of CS R cross R then the points in the complex plane namely complex numbers with rational coordinates will be Q cross Q and how do you get E Q you just get E Q by intersecting E with Q cross Q and you know subset of a countable set is countable you know Q is countable therefore Q cross Q is also countable alright and therefore a subset of a countable set is also countable therefore E sub Q is also countable okay and the rational numbers are dense and therefore E sub Q will be dense in E namely if you take the closure of E sub Q in the complex plane you will get back E right. So we need these facts right now the whole you see this is something of a mystery for example you know I mean these are all facts that you keep using all the time but if you really ponder over them a little deeply in a certain way then you are only perplexed for example you know rational numbers are countable okay which means that you know all the rational numbers can be put in a single sequence okay. So I can write rational numbers as a sequence X n okay and that is very that is something that you cannot imagine okay because given any rational number you are enumerating rational numbers in some order alright but then the usual order that you know of in the usual order that you know of on the real line you cannot tell what is the immediate next rational number to given national number simply because however close you go you can always find another rational number close to it okay. So you cannot say what is the next rational number but here is where you are using some you know very abstract set theory to say that the countability allows you to index and you know enumerate all the rational numbers okay. So this is a highly so it is a very abstract thing that you use alright but something that in practice you really cannot do it you cannot expect to do it right. So well so in some sense that will be connected to the axiom of choice okay which is which as you know is equivalent to the you know well ordering principle and zon's lemma and these are all and zon's lemma is not a lemma it is actually a result that is actually an axiom which you accept okay and you do not you cannot you cannot prove it alright only you can prove it only if you assume it is other equivalent forms namely zon's I mean nearly axiom of choice are the well ordering principle right. So these I mean this is the depth of abstraction that is involved but anyway we use it it is bread and butter when you do analysis okay. So you so therefore you know I write this eq of e sub q as z i as z n so I can do this okay n greater than I can do it okay so this is the really this is the really perplexing thing you write all the points in e you enumerate them alright and you do not I mean you write this completely existential you really do not know what z1 is or z2 it is all you know is that you can write all these points like this and that is used we will use that as you will see. So you write like this so this corresponds to what this corresponds to the fact that this is countable okay or it also corresponds to the fact that this is I mean it is countable and therefore you can order it using real numbers I mean you are using natural numbers therefore if you order it if you choose some ordering with respect to the natural numbers you will get a sequence and that is the sequence I am writing here okay. This is something very abstract right but what you will do is now what you do is you do the following thing you take and of course you know what I am supposed to do is I am supposed to take a sequence in the family and I am supposed to produce a subsequence which is which converges uniformly okay not just converges but I want it to converge uniformly right So start with a sequence fn in this family okay so consider so take the first point okay and apply fn to it okay consider this what will happen is since modulus of all these fellows is less than or equal to m these by the Bolsana Weierstrass theorem we have a convergent subsequence which we write as fn1 of z1 okay. So this is see fn of z1 is a sequence of complex numbers which is bounded so it has a convergent subsequence and call that convergent subsequence is fn1 of z1 okay so this fn1 is a subsequence of fn okay mind you fn1 is a subsequence from fn right. Now what you do is you repeat this process you repeat the process with z2 and with the functions in the subsequence okay now consider what you do is you take this fn1 take the subsequence apply it to z2 and let n1 vary from 1 to infinity. So I get a sequence of function values at z2 okay again the same argument works again modulus of fn1 of z2 is less than or equal to m will tell you and the Bolsana Weierstrass theorem will tell you that there exists a subsequence fn2 of z2 of fn1 of z2 which converges. So you see in the first step I am applying all the members of the sequence to the point z1 and from that I get a subsequence of functions in the second step what I do is I forget z the point z1 but I only look at the subsequence fn1 and apply z2 to it and again apply the Bolsana Weierstrass theorem and use uniform boundedness to show that there is a further subsequence of the subsequence okay which converges when at z2 okay and notice see fn1 already converges at z1 and fn2 is a subsequence of fn1 therefore fn2 will not only converges at z2 it will also converges at z1 okay so note that fn2 of z1 also converges okay so now you go by induction okay by induction we get for every n greater than or equal to 1 a subsequence so let me use m or k a subsequence fnk of fnk-1 such that fnk of zj converges for j less than or equal to k okay so you so you know so this is this is what is happening right so you know if you if you write it you know pictorially so you have fn1 which is with the property that fn1 converges at z1 alright therefore you know if you write it I will get f I can write as fi1 of z1 fi2 of z2 and so let me write fi3 of z3 I mean of z1 again and so on so this is this is convergence at z1 alright then I get so this is sequence fn1 then for this I get a subsequence fn2 okay this fn2 is subsequence of fn1 that means all the indices that occur here they are among the indices among these indices but still I write it only in this order so now I will write it as fj1 of z2 fj2 of z2 fj3 of z2 and so on and this will converge at both z1 and z2 okay and mind you the j1, j2, j3 is a subsequence of i1, i2, i3 and i1, i2, i3 is subsequence of the natural numbers okay then if I again repeat the process once more I get fn3 I get this subsequence which is further subsequence of this and if I write the indices as fk1 of z3 fk2 of z3 fk3 of z3 and so on then I get the subsequence of this subsequence which converges at z1, z2 and z3 okay I get this situation like this right and now what you do is that you know you take the diagonal sequence of functions namely you take this, you take this, you take this, you take this, you take the diagonal, this is called diagonalization you take the diagonal subsequence okay so consider the subsequence so you know I will give it a special symbol I will call it capital F, fl is actually lth member of fnl this is how it is defined so f1 is first member of fn1, f2 is second member of fn2, f3 is third member of fn3 that is how you are defining it okay so this guy here is f1, this guy here is f2, this guy here is f3 and that is how it goes so this diagonal sequence you are looking at okay. Now the beautiful thing about this diagonal sequence is that it will converge at every point of eq okay so that is the I mean that is the power of the diagonalization process you are able to extract this see after all you want a sequence which converges on all of e okay but then you know because everything is continuous okay if you can get uniform continuity on a dense open subset of e okay then you will get everything alright and what is a dense open subset of e it is this open subset and what helps in the diagonalization process is the fact that this is countable okay that is what allows you to enumerate and then extract this diagonal alright. So what is the point so the fact that is nice is that fn converges or f I think I used fl tl greater than equal to 1 converges on eq so this is the beautiful fact why because you see why is that true because you see f what is fl fc fl is actually lth member of fnl where if you take this sequence fn l converges at z1 etc of tzl okay see if see this is the lth member of fnl alright and the but if you look at fnl that sequence converges at all points up to zl alright and therefore you know if you give me any point of eq that point will be because of this enumeration okay which is very abstract as I told you any point of eq is some zl any point of eq is some zl and if but you know fnl the sequence fnl will converge at zl alright therefore all ft is for t greater than l which after all come from subsequence of this will also converge at zl therefore this itself will converge at zl okay so let me write that if z belongs to eq then z is equal to zl and since fnl of tzl converges we have that the subsequence ft of zl t greater than or equal to l converges okay and you know for a sequence of functions to converge at a point it is enough it converges beyond certain stage so what I am saying is that this sequence fl will you know converge at the point zl at least after the stage l alright and this is so this will prove this fact alright so the moral of the story is you are able to extract a subsequence which converges point wise on this set of rational points okay now from this and the eq continuity we can show that you have that this sequence this diagonal sequence that you have extracted the convergence is actually uniform and that will give the proof of the theorem okay so I will continue with that in the next talk.