 So let us continue with our discussion see we were discussing meromorphic functions okay and you know we have to as I told you we have to worry about meromorphic functions because eventually our aim is to prove the Picard theorems and to prove the Picard theorems we need to actually do some topology on the space on a space of suitable space of meromorphic functions okay. So you see doing topology on a space of functions is kind of the background theme okay so you must understand what this is all about okay and for that I need to first of all tell you about spaces of holomorphic functions okay or analytic functions and of course you know analytic functions or holomorphic functions are also considered as meromorphic functions okay. So meromorphic also includes analytic okay and of course last time we saw that if D is a domain in the extended plane okay then the set of meromorphic functions on D namely those functions which are analytic on D except for poles okay that set is actually a field it is a field extension of the complex numbers and I told you that the behaviour the algebraic properties of this field extension have got a lot to do with the geometry of the domain okay and this philosophy is in fact very useful and it is exploited when you replace a domain by in fact Riemann surface okay. Now you see but whenever you are looking at a domain and of course let me also remind you when I say domain I am taking a domain in the extended plane which means that the domain can also include the point at infinity okay which means that I am also allowing study at infinity so I am allowing neighbourhoods of infinity also to be domains okay and this is very very important because the following reason you see you take a meromorphic function okay then on a domain in the extended plane okay. Now take a of course you know it is a meromorphic function so it can only have the only singularities it can have are poles so in particular it can have only isolated singularities and they are poles okay and if you go to as you as you approach a pole the function values approach infinity okay so that the function values approach infinity is can be understood in two ways one way is in a that is the way in which you would have learnt when you did a first course in complex analysis that you see saying that f of z tends to infinity is the same as saying mod f z tends to infinity okay that is it becomes orbit the modulus of the function becomes larger and larger that is one way of saying it but then there you do not really think of infinity as a point but now you know we are used to thinking of infinity as a point namely a point in the extended plane and you physically see it as the north pole in the Riemann sphere when there is a when you look at this identification of the Riemann sphere with the extended plane okay so you know you picture a you picture a domain in the extended plane as by thinking of its image on the Riemann sphere so you imagine the Riemann sphere okay that is a that is a north pole which corresponds to the point at infinity in the extended plane okay mind you the Riemann sphere is homomorphic to the extended complex plane okay it is homomorphic to the extended complex plane with the point with the north pole corresponding to the point at infinity so you can make you can clearly see that you know a small disk like neighbourhood of the north pole that will correspond to the exterior of a circle of sufficiently large radius on the plane by the stereographic projection and when you think of a domain in the complex plane or in the extended complex plane you can think of its image on the Riemann sphere okay so you can think of a domain in the extended plane as some open set on the Riemann sphere okay that is open set can include the point in can include the north pole or it need not include the north pole that depends on whether the original domain you are looking in the complex plane is actually in the complex plane or it is in the extended plane including the point at infinity okay so you must imagine whenever you are thinking of a function defined on a domain in the extended plane you must always imagine an open set on the Riemann sphere and that open set can include the north pole in which case it corresponds to a domain on the extended plane which contains a point at infinity okay and therefore you know we so the point is that if you are looking at a function which is meromorphic on a domain on such a domain in the extended plane then you know it will have only poles okay but if you now approach the you take the points corresponding to the poles on the Riemann sphere okay you take the points corresponding to the poles on the Riemann sphere okay outside those points the function will take values only in complex numbers okay whereas at the poles what happens is at the poles the function tends to infinity okay so the moral of the story is what you can do is you can define the function value at each of the poles to be infinity okay and thereby the resulting function becomes continuous at those points okay now you have to be now there is a subtlety here okay I am saying that the function is made continuous at a pole by defining its value at a pole to be infinity okay and when I am saying this I am not saying that the function is continuous in the usual sense okay a function continuous in the usual sense is if it is continuous at a point it is supposed to have a finite value at that point okay but I want so I am really thinking of infinity as an extra value okay there is a subtlety there is some subtlety involved there is a little bit of confusion involved here because when you say a function is continuous at a point okay it means you know Riemann's removable singularity theorem says if an analytic function has a isolated singularity at a point and if the function is continuous at that point then it is analytic that point is not actually a singularity but I am not saying that I am saying what you do is you take a pole of the function and you make the function continuous at that point at that pole in the following sense you declare the function value at that point to be infinity okay and regard this function as a mapping in not into C but regard it as a mapping into C union infinity the extended plane and then with respect to that now you on C union infinity which is extended plane there is a topology okay that is exactly the topology which is topological space structure which is homeomorphic to the Riemann sphere okay and so I am thinking of the function as taking values in the Riemann sphere in some sense okay and I am saying that the function is continuous at a pole by declaring its value at the pole to be the value infinity okay. Now this is a certain point of view that is very very important okay let me put this following convention let D in the extended complex plane be a domain so it is an open connected subset of the extended plane okay so what you must remember is that D corresponds under the stereographic projection to a domain on the Riemann sphere so it is an open so the image of D under the stereographic projection is an open connected set on the Riemann sphere okay that is what you must remember okay so let D be a domain let f of z be meromorphic on D okay that means f is analytic except for an isolated set of points of D at each of which f has a pole that is what it means okay that is the definition of water meromorphic functions okay so we define the value of f at pole of f to be infinity see you make this definition okay f at every pole f of a pole is equal to infinity you make this definition okay so then the beautiful thing is that f it becomes a function from D to the extended plane okay now f is a function from D to the extended plane and the point is that as a function into the extended plane this is continuous okay if you are normally taking a first course in complex analysis and you are working with a function which has a singular points then what you do is that you consider the function to be defined on the complement of the singular points okay and it is supposed to take only complex values alright now what we are doing is we are allowing meromorphic functions so that means you are restricting the singular points to be isolated and they must be poles okay so in principle you should be worried only about the function outside the poles outside the poles the function should be analytic and it should have complex values okay but what we are doing is we are also including the poles we are defining the function value at each of the poles to be infinity and therefore the function becomes a function into the extended plane okay allow it also takes the value infinity and the point is after you do this continuity is not lost and see the reason why continuity is not lost at a pole is that as if z not is a point of D where f has a pole then limit z tends to z not f of z is infinity okay but infinity is also f of z not by definition because z not is a pole we have defined the value of f at a pole to be infinity okay so f of z not is infinity and that is also equal to limit z tends to z not f z therefore you know that is the definition of continuity okay at z not so the function becomes continuous at z not okay and again let me warn you should not confuse this with continuity in the usual sense I am not saying that as an analytic function it is continuous at a pole that is not correct okay it certainly is discontinuous because when I talk about analytic functions I am only worried about functions is taking complex finite complex values not the value at infinity okay so when I say continuous here you must take it to the pinch of salt you have to be careful okay so let me write that down if z not belonging to D is a pole of f then limit z tends to z not of f of z is equal to f of z not is equal to infinity so this is the extra definition okay that we make so the actually you know sometimes in the language of Riemann surfaces we say that a meromorphic function on a Riemann surface is this same as a holomorphic function into the Riemann sphere okay we say this often and it really makes sense in this sense okay so the point is that you know by including the value at infinity you are able to give a function value at a pole namely the value infinity okay and that is and that makes the function continuous on the domain okay but the point is that the target domain the target domain becomes the extended plane and the topology on that is the you know the topology of the one point compactification which is homeomorphic to the Riemann sphere okay so you know for all purposes you know the picture that you imagine is the following so let me draw this picture it is very important so whenever you know so for all practical purposes this is the best thing to imagine on one hand you have the Riemann sphere okay so on the other hand also you have the Riemann sphere and you know well this point is a north pole okay this corresponds to the point at infinity and this is again the north pole which corresponds to the point at infinity okay and whenever I put this double sided arrow I actually refer to the stereographic projection which compares the Riemann sphere as topologically isomorphic with the extended plane with the point at infinity corresponding to the north pole n okay and what we are doing is see basically you are looking at a you are actually looking at a domain in the extended plane so you know it is going to be some domain like this okay and in fact you know I should not put you know I should not put the boundary so it is something like this so this is my domain so the way I have drawn it now it contains the north pole so that means that when you take its image under the stereographic projection it actually corresponds to a neighbourhood of infinity okay and this is an open connected set on the north pole okay and I have this function f I have the function f which is defined on that and it is taking values again in you must think of it as taking values again on the Riemann sphere okay and whenever you think of the Riemann sphere see you picture the Riemann sphere but always think of c union infinity okay the Riemann sphere minus the north pole is the complex plane by the stereographic pressure and the north pole corresponds to the point at infinity so what will happen is that your function f will is defined you must picture it as being defined on a subset of the Riemann sphere and it is the image it take the image of the function is also a subset of the Riemann sphere okay and the point is that if you take a point which is a pole of f okay then the function has a value there that is what we have done if you take a pole of f the function value is infinity so you know if I draw a pole so suppose I have z0 z0 is a pole okay then f of z0 is equal to infinity okay and so what will happen is that this f of z0 you must picture it as being mapped on to the north pole okay mind you in this the way I have drawn it is not entirely accurate because there is an identification on the left and on the right with c union infinity so what is happening is that there is this the thing on the left is c union infinity okay this thing on the right is also the c union infinity c union the point at infinity so both of them are the Riemann sphere and then I have a domain D inside c union infinity and I have this function f this is a picture so D is a domain in c union infinity f is taking values in c union infinity you allow the value infinity and that is a value you are you assign to function at a point which is a pole okay and in principle this f that I have written in red is actually the is gotten from the f that I have written in purple after identification with the stereographic projection so in principle for this f here I should use some other symbol I should use f tilde if you want and for this point z0 that I have written in purple is not exactly z0 it is actually stereographic projection of z0 the point z0 is actually as a point in D okay and after I identified the c union infinity with Riemann stereographic projection then the point z0 in D corresponds to this point z0 on the Riemann sphere okay. So you know so you know forgetting the notational complication that is involved in all the time carrying the stereographic projection okay you always imagine that things are happening on the Riemann sphere okay and the reason why this is very very useful is because you see if you are looking at only the extended plane the point infinity is something that you cannot see okay you cannot see it as a point and more importantly you cannot for example define a distance between the point infinity and some other point in the plane okay there is no way of doing it in the usual sense but if you think of the extended complex plane as a Riemann sphere then the point infinity is just the north pole and now you know if you now you can define a distance between any two points in the extended plane by simply taking the distance on the Riemann sphere of the images of these points under the stereographic projection. So you know for example therefore if you are imagining all the time the Riemann sphere you can even talk about a neighbourhood of infinity which has certain radius you can talk about a circle centred at infinity with some finite radius okay and that is something that you can picture as an interior of a circle surrounding the point n on the stereographic projection with a certain radius finite radius okay and that radius can be made smaller and smaller okay and you can imagine that as I make that radius smaller and smaller under the stereographic projection I get a larger and larger circle on the complex plane okay. So see this is a certain point of view that you should always remember you always think about a domain in the extended plane as actually a domain on the Riemann sphere and open connected set of the Riemann sphere okay and open connected set on the Riemann sphere the topology is mind you is just the induced topology from R3 the Riemann sphere is just the sphere of the sphere centred at the origin radius 1 unit in real 3 space and real 3 space is Euclidean space you know it has a standard topology it is a metric space it is a complete metric space. So you take the induced topology on the subset so there is a nice topology and it is in fact even a metric space okay therefore you think of it like that okay. So this is something that you must so you know if you want if you want I should do the following thing you know I should actually relabel this as f tilde you know and I should actually relabel this point as z not tilde where this tilde means you translate everything to the Riemann sphere and you will have to therefore bring in the stereographic projection but most of the time you do not worry about it okay see what we are going to do is as I told you we are going to do we have to worry about space of you have to look at the space of meromorphic functions on a domain okay and you have to do topology on it and more generally the simpler case is trying to do topology on a space of analytic functions on a domain. So see what is this topology all about see the most important thing that we will be worried about in the topology of a space of functions is compactness okay so you see compactness is the somehow the most important property that we have to study. So essentially what we have to study is compactness of spaces of functions what kind of functions functions essentially we have to study meromorphic functions but then before that you will have to also worry about holomorphic functions or analytic functions because they are special cases of meromorphic functions meromorphic functions which have no singularities are essentially holomorphic or analytic functions. So now let me do the following thing so let D you take it take D to be a domain in the extended complex plane okay then what you have is so domain, domain means it is non empty it is an open non empty connected subset okay and then you have this so you have this you have the complex numbers sitting inside as constant holomorphic functions in H of D so let me put this notation and this is M of D okay. So what is this M of D this is something that I told you last time this is this is the set of meromorphic functions on D okay this is so let me write this here set of meromorphic functions on D okay this is the field as we have seen last time okay and at this point you only you do not worry about the value at infinity okay at a pole okay this is completely algebraic so you are really not worried about looking at the function which looking at infinity as one of the values of the function okay. So this is the field we have seen this and you have and therefore this is a field extension of the complex numbers see this is this map C sitting inside this is you are thinking of lambda as a constant function lambda okay. If lambda is a complex number you you identify it with the constant function which is equal to lambda and what is this H of D this is actually a set of holomorphic functions on D holomorphic is same as analytic okay this is a set of holomorphic functions on D alright and the fact is that you know the set of holomorphic functions will only be a it will be a ring in fact it will be a C algebra it will not be a field because you know I cannot invert a holomorphic function to get a holomorphic function unless I know that it is non-zero is that that is not ever going to be 0 because wherever it is going to be 0 I cannot invert it okay because for the inverted function the 0 will the 0 of the original function will be a pole of the inverted function okay. So and the moment a function has a pole it is not holomorphic okay it is not analytic. So the point is that you see this is so this is a C algebra and in fact it is commutative it is a commutative algebra mind you whenever we talk about algebras the algebra and algebra is a ring which is also a vector space over a field and in general it could be the algebra the multiplication in the algebra need not be commutative for example the simplest examples are of course matrix algebras you know matrix multiplication is not commutative so but of course algebras of functions are commutative okay because they are taking values in C and C is a commutative field okay multiplication in C is commutative alright. So the fact is that if you take a holomorphic function in D it is certainly a meromorphic function on D okay if you take a function that is holomorphic on D I have to only worry about to invert it I have to only worry about points where the function has zeros but then you know zeros of a holomorphic function are isolated therefore at each of those zeros the inverted function will only have a pole therefore that should be no problem so if you take any holomorphic function on D you may not be able to invert it to get a holomorphic function on D but if you invert it you will certainly get a meromorphic function on D. So in particular you see the meromorphic functions on D will contain inverses for all holomorphic functions okay now fine so M of D contains also inverses of non-zero elements in H of D okay and H of D is mind you it is a subring it is a subalgebra okay this is this is a subalgebra and H of D is not a field but M of D is a field this is what it is algebraically okay and I told you that there is a lot of geometry in studying this field extension M of D over C it is a field extension studying the algebraic properties where extension has got to do with a lot with the geometry of D okay. Now well keep these things aside I want to go away from the algebra and I want to go into topology so you see let us look at just H of D just look at H of D and try to worry about topology on this on the on the set H of D okay so how do you do topology you think of H of D as a set and you would like to give some topology on it okay now the and you know it is not that you do not give an arbitrary topology you give a topology which makes sense with the functions with the domain of the functions okay so what is the idea the idea is you are trying to worry about topology of a space of functions okay now what I want to do is that you know I want to tell you that this topology has got to do with it has got to do with point wise convergence of functions okay on the point wise convergence with respect to points of the domain on which the functions are defined which is D in this case okay. So I want to say that the topology on H of D or if you want to even worry about more generally topology on M of D that has to do with convergence with point wise convergence with respect to points of D okay so this is what I am trying to emphasize when I say I want to do topology on the space of functions I want to worry about topology on the set of holomorphic functions or I want to worry about topology on the space of meromorphic functions set of meromorphic functions okay then the topology has to do with point wise convergence now so why is that true you know you need some motivation for that and in fact you know let me be even more accurate we are not really worried only about point wise convergence because point wise convergence is a very weak of something very very weak you know the reason why it is weak is you suppose you have a suppose you have family of functions or a sequence of functions which is converging point wise to a limit function then the limit function need not be continuous even if the original members of the family of functions each of the members is continuous okay you know that you need a uniform limit for the limit function to be continuous so it is not only point wise convergence we are worried about we need something stronger that is uniform but what happens in complex analysis you do not get uniform always on your domain what you get is uniform on compact subsets of the domain okay for example you will get uniform on closed disks inside the domain and that is a special kind of uniform convergence which is uniform convergence on compact sets that is called normal convergence and see it is so what I want to impress upon you is that the topology on the space of holomorphic functions or on the space of meromorphic functions on a domain has to be studied with the viewpoint of normal convergence okay it is point wise convergence and it is not totally uniform convergence but it is uniform convergence on compact sets okay and you know in some sense why is all this importance why is all this important because you see let me recall little bit of basic topology to help you to see the following thing let X be any set so this is some motivation X is any set take any set not even a topological space of course assume it is not empty okay then what you do is you take you know L to be well the set of all functions from X to R okay you take the set of all real valued functions on X okay take the set of all real valued functions on X so these are just functions on a set okay the only thing is that these functions have real values right now the point is that this L becomes immediately this becomes a real this becomes a real vector space L becomes a real vector space alright and you know how because point wise addition and point wise scalar multiplication makes sense so if f is a function on X with real values and g is another function on X with real values f plus g can be defined as f of X f plus g of X is just f X plus g X for each small X in capital X and similarly if lambda is a real number lambda f can be defined as lambda f value at X is lambda times f X okay so this becomes a this becomes an R vector space alright and well what you can do is among these functions you can look at those functions which are whose images are bounded okay you can look at bounded functions so in this L I can have this subset b okay which is the set of all f belonging to L such that f of X is bounded the image is bounded so these are bounded real valued functions okay the set of all real valued functions L is a vector space it is a real vector space and now you take this subset which is which causes of bounded real valued functions okay then it is also a subspace because if f is bounded and g is bounded then f plus g is bounded if f is bounded then lambda times f is also bounded okay so the point is that this is a this is a this is a subspace this is a real subspace of this vector space alright now the beautiful thing about the subspace is that because your images of functions are bounded you can make this into a Banach space you can make it into a complete normed linear space you can define a norm on it the supremum norm that makes it into a topological space in fact it makes it into a metric space the metric being induced by the norm and with respect to that metric it becomes complete and this is essentially because of the completeness of the real line okay so this object here is actually and in fact the beautiful thing is that I can in all these spaces I can even multiply two real valued functions to get a real valued function and the multiplication is commutative so these are not just spaces they are in fact commutative algebras so L is actually in fact a commutative R algebra okay B is also commutative R algebra but B is in fact a commutative Banach R algebra okay it is a Banach space it is a Banach algebra alright so let me write this so this fellow here is actually commutative R algebra and this fellow here is actually commutative Banach algebra it is commutative Banach algebra and the point is that you define norm f to be equal to supremum x small x in capital X of mod fx and the supremum will be a finite quantity because I am looking at f in B f is the image is bounded and you know that one of the properties of the real line is that you take a bounded set its supremum exists and it is a finite real number okay in fact you know that is equivalent to completeness of the real line the real line that every Cauchy sequence converges on the Cauchy sequence of real numbers converges is as strong as requiring that any subset of the real line which is bounded above has a supremum okay and that is equivalent to also saying that any subset of the real line that is bounded below has an infimum okay so if you take this norm this is a sup norm it makes B into a normed vector space and the norm induces a metric in the following way d of f, g is just norm of f minus g so with this d function distance function this B becomes a metric space okay it is the it is said to be the metric induced by the norm and with respect to this metric B is complete so it is a Banach space and in fact it is an algebra so it is a Banach algebra okay now what you do is all this is correct if you are taking X to be just a set okay now you put the extra condition that X is a topological space okay if you put the condition that X is a topological space then instead of simply looking at functions from X to R I can look at continuous functions from X to R because now the source X is a topological space it has a topology okay so I look at functions which are continuous also and when I do that then I get this familiar object that you should have seen in course in analysis namely the space of all continuous bounded real valued functions on the topological space X so this is if X is topological if X is a topological space okay and the fact is that this is a closed subspace so mind you B is already a topological space and this is a closed subspace you can show that this is a closed subspace namely you can show that if you have a set I mean if you have a if you take a point which is a limit point of continuous bounded real valued functions then the limit function is also continuous bounded real valued okay therefore this becomes a so in particular this space C X R that also becomes a commutative Banach R algebra okay and the beautiful thing is the following the beautiful thing is that for a function for a for a sequence of so you know now in C X R I can mind you C X R is also a metric space okay because there is a metric induced by the norm so I can make sense of a sequence of functions converging to a function what is that convergence if you check it out that convergence is nothing but uniform convergence okay therefore the moral of the story is that uniform convergence of real valued functions sequence of real valued functions on a topological space is the same as the convergence in the norm the so with respect to the sup norm on the Banach algebra of bounded real valued functions. So this is to give you the motivation that you know whenever you try to get hold of a topology on a space of functions okay the convergence should have to do with point wise convergence in fact it has to do with normal convergence okay or at least to begin with it has to go to do with uniform convergence but normally what happens is you cannot get uniform convergence on a whole space in complex analysis for example in the case of holomorphic functions you can get only uniform convergence with respect to compact subsets so the topology of a space of meromorphic functions or holomorphic functions needs to be studied with respect to normal convergence okay that is the motivation okay so we will continue with the next lecture so let me write this down this is also commutative Banach algebra with convergence same as uniform convergence on x okay.