 So what we are supposed to worry about is you know see we are worried about compactness of families of meromorphic functions, okay. So basically you are trying to do topology on collection of meromorphic functions and see this is the technical background that is required to prove the Picard theorems and many other theorems in fact because the root is through so-called Montel's theorem, okay. So you know what I want to do is I want you I want to go back to some topology and tell you about compactness, okay so that you realize how whatever we are going to do is connected with all this. We will have to bring into the discussion Arzela Ascoli theorem and then Montel's theorem, okay and then we will you see so let me say the following thing. You know what we have done so far is the following. We have defined a spherical derivative, alright. So first of all so let me sum up what we have done so far. We have first we have tried to think of a meromorphic function as a continuous function even at a pole, okay that is because we allowed the value infinity and so we are not only looking at complex valued functions we are looking at functions with values in the extended complex plane. So you allow the value infinity the advantage of allowing the value infinity is that a meromorphic function at a pole can be given the value infinity and it becomes a continuous map. It becomes a continuous map when you consider it as a map into the extended complex plane which is identified with the Riemann sphere, okay and you know it is a complete compact metric space, right. Now so first we had to deal with the point at infinity, okay so we try to think of infinity as a isolated singularity, when is infinity an essential singularity, when is infinity a removable singularity, when is infinity a pole, okay all these things we discussed behavior at infinity, okay and then value of the function at infinity that also we have we worried about, okay. So you allow in principle you allow functions not only to take the value infinity but you also want to study functions at infinity, okay so you see these are two different concepts in the code domain of the function usually we are interested only with complex functions but now you allow the value infinity, the advantage is that you can make a meromorphic function in a continuous map even at a pole, okay. Then not only that in the domain normally the domain of the function is usually a domain in the complex plane but then you also want to study the function at infinity itself so you want to put infinity also in the domain, okay so you have to define you have to understand the behavior of the function at infinity, okay. So a function may have a pole at infinity it may go to infinity at infinity which is the case for example if you take polynomials on constant polynomials they all have poles at infinity. So you want to be able to work in this kind of generality that is the reason why we have to study the function behavior at infinity think of infinity as a isolated singular isolated singularity and classify that kind of singularity and we also want infinity to be a value taken by the function for example the value of a meromorphic function at a pole, okay. So we have to deal with infinity that was the first thing then the second thing is we were worried about this defining the spherical derivative, okay we were concerned about defining the spherical derivative and see the important thing about the spherical derivative is that the spherical derivative will not change you can first of all you can define it for a meromorphic function, okay. So it is a derivative that will work even at a pole. If you take a meromorphic function by which by definition is a function which is which has only pole singularities, okay of course it may be completely holomorphic it may be completely analytic but we are interested in the situations we are going to really encounter those in which there are actually poles, okay. So if you look at meromorphic functions honest meromorphic functions you take a pole at the pole it is not differentiable because after all you know at the pole the function goes to infinity and it is not differentiable because it is a singular part it is not a removable singularity it is a pole. The function is not differentiable in the usual sense of the term, okay and the function value is also not defined in the usual sense of the term but what we do is we define the function value at the pole to be infinity that is an extra definition we make and then since you cannot differentiate the function at a pole. So what you do is you do this clever trick of differentiating not with respect to the usual metric which is the Euclidean metric but you try to differentiate with respect to the spherical metric. So you introduce what is called the spherical derivative, okay so that gives you a derivative of a function which will work even at a pole you see that is the advantage, okay. If I take a meromorphic function at a pole I cannot differentiate it but if I take the spherical derivative the spherical derivative will exist and I have told you the spherical derivative we calculated it last time I think it was 2 divided by the modulus of the residue at the simple pole if it is a simple pole and it is 0 if it is not a simple pole if it is a pole of higher order. So even the spherical derivative makes sense and on top of all this one more beautiful thing about the spherical derivative is that the spherical derivative will not change if you change the function by its reciprocal that is if you take a meromorphic function f and calculate the spherical derivative you will get the same thing if you took 1 by f, okay mind you which is also meromorphic with only the only thing is that the poles and zeros will get interchanged when you move from f to 1 by f but for 1 by f also if you calculate the spherical derivative you will again get the same thing as a spherical derivative of f. So what it tells you is that if you are studying the spherical derivative you can actually apply the theory of analytic functions and stop worrying about even poles because at a pole of f I can simply if I am working with the spherical derivative in the neighbourhood of a pole of f it is the same as a spherical derivative in a neighbourhood of that point for 1 by f but for 1 by f that point is a 0, okay and therefore it is analytic 1 by f becomes analytic at that point that is the advantage. So working with a spherical derivative allows you to reduce to analytic functions, okay you do not have to even worry about poles that is an advantage and the other thing is it gives you a derivative that works even at poles, okay. Now so this is the this is what we have done so far now why did we do all this is we did do the idea is there are 2 concepts on the one hand we are worried about compactness of a family of meromorphic functions that is our main aim we want to do topology on a collection of meromorphic functions on a space of meromorphic functions. We want to what kind of topology of course topology means there are many things right there is connectedness compactness and so on and so forth but we are interested in compactness, okay and so that is on that is on the one end. On the other end what we have is this spherical derivative that is what we have which is close to a derivative in the case of a in the case of a meromorphic function, okay. So now I need to I need to tell you people how I need to tell you people how to connect these 2 things, okay. So we need to do some topology so I will give you some topological background. So topological background so this is very very important because only then you will understand what is going on, okay in the in the broad sense what are we trying to do, okay. So if you want to get an idea of that this is very very important. So what we will do is we start with let us say let us say you are working with a metric space suppose you are working with a metric space, okay. Mind you the topology I am worried about the topological property that I am worried about is compactness, okay. So we will try to do try to understand everything connected with compactness, right. So start with a metric space which is the simplest kind of topological space that you can think of which naturally occurs, okay. Then what do you have? The following are equivalent is compactness number 2 is sequential compactness and number 3 is the so-called bulls and a way of stress property, okay. So we have these 3, these 3 properties are equivalent, okay. So I am just trying to recall what is equivalent to compactness, okay. Just if it helps to translate a property in different ways to find out equivalent properties so that you can work with them, okay. So compactness is a so this is something that you should have done in the first course in topology. What is compactness? Compactness is that every open cover admits a finite sub cover, okay that is when you are given a collection of open sets whose union is the full space then it is enough to pick only finitely many among those collections in that collection whose union is also the whole space. You can extract a finite sub cover from every open cover that is compactness, okay. It is a very, you see it is defined only in terms of open sets and it is a very general thing. So it works for any topological space, compactness can be defined for any topological space because for any topological space open sets make sense, okay. Defining a collection of open sets is exactly what giving a topology is, okay. So compactness makes sense for any topological space but it is a very abstract notion. At least for metric spaces where the topology is induced by a metric, okay that means that you know your open sets are defined to be unions of open balls and open balls are they are the analog of open balls in Euclidean space, you take points of the space and then you take all points which whose distance from the given fixed point is less than some positive number which you call the radius of the open ball, okay and of course you say strictly less than because if you put less than or equal to then you will also include the boundary and it will not remain an open set it will become a closed set, okay. So you put strictly less than, the distance should be strictly less than some positive radius, okay and if a set is called open if it is union of such open balls and this is how you and you know this involves a notion of distance that is where the metric in this space is used. So the metric space the metric induces a topology. So when we say metric space and you think of it as a topological space you always mean the topology induced by the metric, okay. The open sets are precisely those which are given by union of open balls and open balls are defined by the metric, alright. Now for such a metric space compactness which is a very abstract thing is connected with what is is equivalent to sequential compactness and what is sequential compactness it has to do with sequences what it says is that you give me any sequence of points in the space I can always find a convergence of sequence that is what sequential compactness is, okay. If you give me a sequence in the space the sequence itself may not converge but at the worst you can pick out a subsequence which converges, okay that is sequential compactness and that is equivalent to compactness that is what this basic result says. And then there is a third property which is called the Bolzano-Weierstrass property what is this Bolzano-Weierstrass property it is just a property which is satisfied by the Euclidean spaces which you namely which you would have come across namely the fact that you take any infinite subset that it has an accumulation point or a limit point, okay. Given any infinite subset all that is a cluster point there is a point where there is a point of the space such that if you take any open neighbourhood of the point and delete that point there is a point of your infinite subset there, okay. So points of your infinite subset come closer and closer and closer to at least one point of the space and that point is a limit point of that set, okay. Now that every infinite subset has a limit point is the Bolzano-Weierstrass property and that is also equivalent the space having this property is also compact, okay. So all these three are three different avatars of compactness, okay, alright sequential compactness and then Bolzano-Weierstrass property, okay. And well if you are looking at Euclidean spaces, okay that is Rn n dimensional real spaces finite dimensional real spaces then what happens is that this is also equivalent to if you look at a subset of Euclidean space compactness is equivalent to closeness and boundedness put together, okay and that is what we most of the time when you are working in Rn n dimensional real space we keep using that all the time whenever you want to say something is compact you say it is you just verify that it is closed and bounded. For example if you take a closed disk in the complex plane that is closed disk in the complex plane is compact because it is disk of finite radius so it is bounded and it is closed so it is both closed and bounded so it is compact. So we keep using this all the time, okay. So let me write that down for Euclidean spaces R to the n we also have the equivalence of the above with 4 so this is for in fact I should say for subsets of Euclidean spaces so the subsets should be closed and boundedness so if something is closed and bounded is compact and conversely, okay. So mind you you know you know my background what is the background of our trying to understand all this the background of our trying to understand all this is you want to do this for functions for a space of functions you want to do this for a space of functions for a space of functions if you take a space of functions it will be a subset of all functions of the given type. So for example if you take a space of continuous functions real valued functions it will be a subset of the space of all continuous if you want continuous bounded real valued functions, okay and or you might be looking at a space of analytic functions or you might be looking at a space of meromorphic functions that is the background in which that is the generality in which you want to do all this and you want to make sense of compactness for such a set of functions. So usually we use various terms sometimes we say family of functions if you want to specify an index set or sometimes we say sequence of functions if you want to think of a sequence of elements which are each of which is a function or you take a subset of the space of all functions, okay. So you refer to it in different ways but then basically you are looking at a subset of functions and you want to study compactness for that, okay. Now you see the question is of course that you know how do you go from this to something else. So there is a very important property and that is called total boundedness, okay. There is something called total boundedness, okay. Now what is this total boundedness? It is a very very strong form of boundedness, it is a very very strong form of boundedness. So what is this total boundedness? So I will try to explain to you. So basically what happens is that you know you have some space X, okay and let us assume that X is a say metric space. Suppose X is a metric space, there is something so the idea of total boundedness is like is to you know fill out the whole space by finitely many disks open disks of a fixed radius, okay no matter how small that radius may be that is the idea total boundedness. So here is my space X, it is a metric space, okay and then for every epsilon positive no matter how small it is there exists a subset A epsilon subset of X A sub epsilon and this is the point is a finite set so it is only finite collection of points A epsilon finite, okay such that you see the union if you take the union of all the if I take the union of all the open balls centered at points Xi of A epsilon and take radius epsilon and I do this for I equal to I so you know in fact let me not put a subscript let me get rid of the subscript and just put X belongs to A epsilon so when I say X belongs to A epsilon there are only finitely many such X because A epsilon is finite and for each such X I so you know so here is one X here and then I have this ball centered at X this is open ball centered at X and radius epsilon, okay and I do this for all the points of A epsilon I take the open ball centered at each of the points of A epsilon with radius epsilon, okay and if I take the unit that should be equal to X that is the requirement. So I can cover X by finitely many such open balls and the beautiful thing is that all these balls have the same radius epsilon, okay and there are only finitely many of them they cover all of X, okay and this must happen for every positive epsilon this should happen for every epsilon greater than 0 if it happens for a particular epsilon such a collection of points finitely many points A epsilon is called an epsilon net, okay so this is called an epsilon net so this is called an epsilon net and this is the net condition, okay. Now this is this see you are saying that no matter how small an epsilon you take I can make sure I can find I can make sure to find only finitely many points in X such that every other point of X is at a distance less than epsilon from at least one of these points that is what you are saying, right. So let me repeat it what is this epsilon net condition given an epsilon no matter how small, okay you are able to find finitely many points that they will constitute the elements of the set A epsilon such that given any point of X its distance from at least one of these points is less than epsilon that way you cover every point of X, okay it is a very very strong condition and you know the point is that this is a very strong form of boundedness because this implies boundedness because you see why does this imply boundedness if you see you know so let me say it in words so let me put this here this implies or rather let me write it above so let me put it here this implies boundedness and why is that true see in fact what it will tell you is that you know it will tell you that the diameter of X is comparable to the diameter of any of these A epsilon that is what it will tell you see what is the diameter of a space it is a metric space the diameter is a supremum of the lens between two of its points and you allow those two points to just vary so it is like trying to draw the longest line segment through that space if you want to think of it and measure the length of that of course this longest may not exist so it may become infinite so your space may have infinite diameter so that is the reason instead of taking maximum we take supremum so basically what you do is you take supremum of the distances between two points of your space and you allow the points to vary, okay if that has a finite value that is called the diameter of your space and the point is if your space is totally bounded then its diameter can be compared to any epsilon net so for example you know if you take an epsilon net such as A epsilon, okay and you measure the distance between two points of the space what you can do is that each of these points is within an epsilon from one of the points in the net and the distance between two points in the net cannot exceed the diameter of A epsilon mind you A epsilon is only a finite set so it has a finite diameter a finite subset always has a finite diameter because you are just going to take supremum of the finitely many distances between pairs of points in that set and that is only finitely many pairs, okay so the diameter of any finite subset is of course finite alright and you know if you look at the diameter of A epsilon, okay that will be an upper bound for the distance between any two points of A epsilon, okay now if you take any two points of the space for each point I can find a point of A epsilon which is to within an epsilon so what this comparison will tell you by triangle inequality is that the diameter of the space cannot exceed the diameter of A epsilon plus two times epsilon if you write it out alright if you use a triangle inequality the diameter of the space cannot exceed the diameter of A epsilon plus two epsilon for every epsilon greater than 0 that will tell you that the space has finite diameter, okay so you can see it is a very very very strong condition and of course if the diameter is finite it means the space is bounded if the diameter of the space is finite that means if the diameter of the space is say lambda positive number lambda then this space is of course bounded because you take any point in that space and take a disc of take an open ball of radius greater than lambda the whole space will be contained in that so it becomes bounded. So totally bounded is very strong it implies boundedness alright and in fact actually for Euclidean spaces you see boundedness is the same as total boundedness, okay and in fact more generally if you take a Banach space if you take a complete normed linear space if you take a Banach space even for a Banach space you see the fact that every subset that is bounded is also totally bounded is a very strong condition it will happen if and only if the Banach space is finite dimension, okay it cannot happen in infinite dimensional Banach space, okay so if you go to infinite dimensional spaces okay which is like non-Euclidean kind of spaces then you are in trouble, okay there is a difference between total boundedness and boundedness okay but total boundedness a priori is a very very strong condition, right. So for example you know if you take r infinity infinite sequences of you know infinite sequences of real numbers then what will happen is that if you take the unit ball there that is of course you know bounded but it is not totally bounded because if you take the diagonal sequence which consists of 0 everywhere except 1 in the ith place for i equal to 1, 2, 3, 4 it is called the diagonal sequence, okay then that sequence will never have a convergent subsequence because distance between any 2 points of that sequence is finite quantity so it is a finite positive quantity it is a constant, okay and therefore you cannot have a convergent subsequence because if there is a convergent subsequence then distance between points should come closer and closer but this does not happen all distance between any 2 points in that sequence is equal to some fixed positive quantity, okay. So if you take r infinity the unit ball is bounded but this is certainly not totally bounded, okay and what I am trying to say here is basically a theorem in fact what I am trying to say is that you know if you have compactness which I have written on the left side in its various avatars in its various avatars I have written compactness, sequential compactness see all these things they all imply total boundedness, okay compactness or sequential compactness or Bolzano-Weierstrass property they all imply total boundedness of course what I wrote below is that you know they all imply for Euclidean spaces they all imply closeness and boundedness, okay but it is not just compactness in general gives you a very strong thing it gives you total boundedness. Now the question is how do you come back from total boundedness how do you come back to compactness, okay and the answer to that is a theorem if you want to come back this side what you need to do is you will have to put the condition that your space is complete, okay. So with completeness so let me try to use a different color so that you understand the implication that is involved with completeness so if I go like this plus completeness if I take a metric space that is totally bounded and I add completeness to it, okay completeness is the condition that every Cauchy sequence converges, okay that is you put this completeness condition then you will get compactness, okay this is a so you know in so what I am trying to tell you is see we are trying to move from compactness which is a very abstract thing to something that is related to boundedness, okay and why we are doing this is because when you are studying functions or spaces of functions it is easier to verify something is bounded if you want to say a function is bounded that is easy to verify, okay whereas if I want to say that a collection of functions is compact it is very very abstract, okay. So boundedness is something that for functions is easy to verify under many situations so that is why we are trying to move from compactness to boundedness and this is the root compactness implies boundedness for example in Euclidean space, okay and in fact it is equivalent to closeness and boundedness but if you forget Euclidean spaces compactness gives you total boundedness which is a very strong form of boundedness but from total boundedness if you want to come back to get compactness you need completeness. So the translation so far is we have so basic topology teaches us that you can translate from compactness to completeness plus total boundedness, okay. Now what I will need to do is that I will have to now translate all this to functions, spaces of functions, okay and that is where what we will come across is the so called Arzela Ascoli theorem and then there so what we will do is there we will try to see how to decide a certain collection of functions is compact, okay. So you will you can expect that you know you will say the condition that will be needed is total boundedness and completeness but completeness you will get if the collection is already a close subset because a close subset of a complete space is complete. So if you are working for example with the Banach space of real valued functions or complex valued bounded continuous functions, okay then any subset any close subset that any close subset will automatically be complete so the only thing that is required for it to be compact by what I just said is that it should be totally bounded, okay but then from total boundedness you want to even remove the totalness and come down to boundedness that is where you have to bring in the Arzela Ascoli theorem, okay so I will explain that in the next lecture.