 Alright, so let me continue with this discussion about Marty's theorem, okay which is which is basically an analog of Montel's theorem except that you are working with not with analytic functions but with meromorphic functions, okay. So let us look at let us again look at the statement, so you have this domain D in the complex plane and you have this family script F of meromorphic functions on D, so the script M D is the you know it is the set of meromorphic functions on D and mind you we are considering these as continuous functions from D into the extended complex plane, okay. So you are able to do that because a meromorphic function normally at a point which is a pole it is it goes to infinity, okay but then you allow the value infinity and you declare the value at the pole to be infinity so it becomes continuous. So this set of meromorphic functions is a subset of this set of continuous functions from D to the extended complex plane C union infinity which as a metric space given the spherical metric we think of as the just the Riemann sphere, okay with infinity corresponding to the north pole, right. And so you have this family script F of meromorphic functions you can either use the word collection or family or subset whichever you prefer but the point is when is this family compact, so in this case you know normally or I should since the word normal is used technically I should say usually compactness is equivalent to sequential compactness and then that it means that you know saying something is compact is the same as saying that every sequence has a convergent subsequence. So if you want to say that the family script F is sequentially I mean is compact you would like to say it is sequentially compact if you want to say it is compact and then you would like to which means that you know given any sequence of functions in this family you are able to extract a subsequence which converges. Now what kind of convergence if we are usually the convergence that we worry about is uniform convergence but of course in the case of analytic functions and meromorphic functions you will not get uniform convergence on the whole domain you will get only uniform convergence on compact you will get only uniform convergence on compact subsets and that is called normal convergence, okay. So in other words the compactness of the family F is thought of as normal sequential compactness which means that every sequence in F admits a subsequence that converges uniformly on compact subsets of the domain, okay. So this is what compactness for us means and Marty's theorem says that this is the same as the family of spherical derivatives of F namely you take for each small f in script F you take its spherical derivative F hash of F hash small f hash and you get this family script F hash you this is the family of spherical derivatives and that should be normally uniformly bounded which means it is uniformly bounded on compact subsets. So in some sense boundedness of derivatives is giving you is equivalent to compactness I mean if you want to say it in a nutshell boundedness of derivatives is equivalent to compactness, okay and so there are a couple of aspects that I want to stress between this and the original Montel theorem see the original Montel theorem was for analytic functions, okay. So you took instead of taking a family of meromorphic functions as we have done now if you have taken analytic functions, okay then we would have put the condition that the family is uniformly bounded, the family itself is uniformly bounded, okay and the and there the uniform boundedness of the family on compact subsets that would be equivalent to the family being normally sequentially compact that is the usual Montel theorem, okay and the way we work there is you have the uniform if you restrict to a compact set you have uniform boundedness of the family, okay then from the uniform boundedness of the family you derive equic continuity because from the uniform boundedness of the family you get uniform boundedness of the derivatives and that is because of the fact that the derivatives are expressed in terms of the original functions using the Cauchy integral formula and you can make an estimation there are the Cauchy estimates. So the uniform boundedness of the family on a compact subset will give rise to uniform boundedness of the derivatives on a compact subset and uniform boundedness of the derivatives always gives rise to equic continuity. So you get along with uniform boundedness on a compact subset you get equic continuity for free if you are looking at analytic functions, okay but you see Marty's theorem is slightly different what is happening is whereas in Montel's theorem uniform boundedness on compact subsets of the family is equivalent to the family being normally sequentially compact in Marty's theorem it is not uniform boundedness on compact normal uniform boundedness of the family but it is actually normal uniform boundedness of the spherical derivatives, okay. So you move from the family in some sense you move from the boundedness of the family to the boundedness of the derivatives that is the switch, okay and the point is that in a sense this is stronger than the original Montel theorem because in the original Montel theorem if you are looking at a family of analytic functions and suppose you know that their derivatives are normally uniformly bounded suppose I am not given that the family itself is uniformly bounded but suppose I am given just the derivatives are normally uniformly bounded, okay then what happens is if the usual derivatives are normally uniformly bounded then it also happens that the spherical derivatives are normally uniformly bounded because of this reason, okay because you see the if you the spherical derivatives are bounded by 2 times the bound for the normal derivatives the usual derivatives. So if the usual derivatives are bounded, okay then the spherical derivatives are bounded so if you take a family of analytic functions on a domain such that the usual derivatives are all uniformly bounded on compact subsets that is normally uniformly bounded then also you will get you know you will get a normal sequential compactness, okay because of Montel's theorem but the only thing is that now you could have you know the because you are considering these as meromorphic functions you could have the extreme case that all these analytic functions go to infinity, okay and by that I mean they go to the function which is infinity on all points of the domain which is also considered as a continuous function, okay and mind you for such for that function the spherical derivative is 0 because it is a constant function, okay. So now what I want to say is so this is one aspect that when you move from Montel's theorem to Marti's theorem you are actually moving from uniform boundedness on compact subsets of the family of functions to the uniform boundedness of the derivatives, okay and because you are worrying about meromorphic functions usual derivatives will not work for example at poles so you will have to look at the spherical derivatives, okay now that is one aspect now that here is another important aspect. See if you know that these the Montel's theorem for example is actually a deeper version or is an application of the Arzela-Ascola theorem, okay and what is the philosophy of the original Arzela-Ascola theorem the philosophy is that if you want to say your family of functions is compact which is the same as saying that is sequentially compact namely you want to extract a convergence of sequence from any given sequence see you will have to put the conditions of the family being equicontinuous and uniformly bounded that is why the Arzela-Ascola theorem is often referred to as uniform boundedness principle, okay. So you need uniform boundedness plus you need equicontinuity together to give you sequential compactness, alright. If you are working with analytic functions uniform boundedness is enough, okay because equicontinuity will come out as immediately it will come out for free because you have the Cauchy integral formula, okay. Now in the case of Marti's theorem there is a slight advantage, the advantage is that if you see I have if I look at it in one direction that is why is it that the uniform boundedness of derivatives should give me a normal sequential compactness, okay. What you can guess immediately is that always boundedness of the derivatives gives rise to equicontinuity, okay it always gives rise to equicontinuity. So even on a compact set if you want to extract a convergence of sequence from a given sequence, okay you would like to apply Arzela-Ascola theorem. So what is missing, what is missing is uniform boundedness, okay because if you want to apply Arzela-Ascola theorem you need uniform boundedness together with equicontinuity so that you can extract from any given sequence as convergence of sequence. So if I restrict to a compact set what if I assume that the derivative spherical derivative is a bounded okay I can expect only equicontinuity, okay I will not get the I will get equicontinuity of the given family of functions but I cannot get I do not seem to be getting the uniform boundedness of the family, okay but here is where the beautiful thing is you do not need any uniform boundedness, okay. The reason is because the values are being taken in a compact matrix space, okay see the values are being taken as far as meromorphic functions are concerned where are the values being taken the values are being taken in the extended complex plane. Extended complex plane mind you is identified with the Riemann sphere and it is a compact matrix space, okay and you know a compact matrix space is of course bounded it is totally bounded, it is bounded, okay it is complete, okay so there is no unboundedness phenomena that is going to occur in a compact matrix space, okay. So this uniform boundedness condition is not necessary that is the whole point, okay. So what I want to say is that your Arzila-Asculi theorem in the Arzila-Asculi theorem, okay we were looking at functions continuous functions either real or complex valued on a compact matrix space, okay. Now I am saying and there for sequential compactness of a family of functions you needed both uniform boundedness and equicontinuity but if I instead of looking at real or complex valued functions suppose I was looking at functions with values in a compact matrix space, okay that is the change I am making you try to look at functions defined on a compact matrix space and taking values in another compact matrix space the target is not real numbers or complex numbers but the target is another compact matrix space then because the target is already compact this uniform boundedness is not needed just equicontinuity is enough and it is equivalent to sequential compactness that is the whole point, okay. So what I want to tell you is that when you go to Marty's theorem, okay you switch to the uniform boundedness of the derivatives and you do not care about the boundedness of the original family of functions locally that is because now the functions are already taking values in a compact matrix space and you do not have to worry about it, okay. So let me you know explain the proof so whatever I have circled here is to tell you that what this tells you is that if the if the family of derivatives of a collection of a family of analytic functions functions is uniformly bounded then so is the family of spherical derivatives so the boundedness of the ordinary derivatives implies boundedness of spherical derivatives, okay. So that is something that I am writing here I think I have cramped it a little bit so let me get rid of this lemma and rewrite it later, fine. So what I will do is I will try to give you the proof of this so let us go in one direction so let me again reiterate the Asiel Ascoli theorem is valid, okay in the sense that sequential compactness is same as equicontinuity without worrying about uniform boundedness if you are looking at functions which are taking values in continuous function values in a compact matrix space, okay that is if you replace real and complex numbers by a compact matrix space, right that is the whole point so just equicontinuity is enough, right and I will try to instead of trying to prove a theorem in that generality I will even explain to you how you can get the sequential compactness. So what you do is so let us start this way suppose so maybe I will use okay so suppose f is suppose f hash is normally uniformly bounded, okay suppose it is normally uniformly bounded what do I have to show I have to show that it is normally sequentially compact that means you will have to pick up given any sequence in the family script f you have to show that there is a convergence of sequence, right convergence in the sense of normal convergence that is convergence on compact subsets so that is what I have to do we need to so let me write that down we need to show that any sequence f1, f2 and so on admits this sequence in of course in I should not say well when I put subset this I am not writing this sequence as a set okay because that could be repetitions in a sequence okay so this is why this notation let me put belongs to okay so I mean that f1, f2 etc is a sequence in f you have to show that any sequence admits convergence of sequence and of course it should be a normally convergence of sequence that is something that converges on compact subsets okay uniformly on compact and of course on compact subsets the convergence is uniform, right so uniform convergence so now so how do I go about this so as usual the moment usually if you have boundedness of the derivatives the first thing that you do is you get equicontinuity of the family that is always you should remember as a philosophy boundedness of the derivatives is a strong condition to imply that will imply equicontinuity of the original family so what you do is that so that is what I am going to demonstrate we will demonstrate that this family script f is equicontinuous okay so we will show script f is equicontinuous how do I do that you check equicontinuity at every point so what you do is fix z0 in D and a disk a close disk centered at z0 at z0 in D of sufficiently small radius okay so now you know so the situation is like this you have this you have the complex plane and you have this you have some you have this domain D okay and so this is D I am always trying to draw a bounded domain but it need not be a bounded domain okay because if it is an unbounded domain with I cannot show it on a picture so here is so here is the domain D it is the boundary is this dotted line and what I am having is a point z0 in D and I am choosing a sufficiently small disk such say of radius rho okay rho sufficiently small so that the whole close disk is inside D okay that the open disk with z with center z0 radius rho along with the boundary circle that is also in D okay and what do I do I just so I remember that you know my if you take a function f small f in script f mind you the function is being now thought of as going into the Riemann sphere okay see it is going into C union infinity and the C union infinity is identified so I put a triple line okay this is identified with the Riemann sphere so what is it it is just so this is just S2 the real two sphere in three space real three space radius 1 center at the origin so it is this you know it is this thing so this is Riemann sphere and this point corresponds to the north pole which corresponds to so this infinity corresponds to the north pole okay so your function is taking values on the Riemann sphere that is how you should think about it right and now what is it what is it that I am given I am given that the I am given that the family I am given that the family of spherical derivatives is normally uniformly bounded so that means it is uniformly bounded and compact subsets of D and this close disc center at z0 radius rho is a compact subset of D so it is uniformly bounded on that okay by hypothesis of normal uniform boundedness of the family script f there exists an m such that the spherical derivatives of all the spherical derivatives in the family are bounded by m so let me just put in mod z minus z0 this is not equal to so I have this okay this is just the uniform boundedness of the spherical derivatives restricted to this compact subset given by this disc right. Now what do you do mind you that in this situation since the functions are taking values in the extended complex plane okay on the target the target metric space is extended complex plane and the target metric is a spherical metric that is what you have to remember okay the target metric space is the extended complex plane and on the extended complex plane the metric is the spherical metric it is actually the spherical distance on the Riemann sphere transported by the homeomorphism of the Riemann sphere with the extended plane okay So it is the so you should remember this is the big point remember you have to remember that whenever you are working with values in the external complex plane okay in the target space the external complex plane the metric involved is the spherical metric okay. So if you keep that in mind this is what is going to happen you see if I take 2 points I take so let me use a different colour suppose I take 2 points say Z1 and Z2 okay inside this closed disk and I take the straight line segment from Z1 to Z2 okay then and suppose I call this segment as L okay then under if I take the image of the segment the straight line segment under this map f where f is any function any meromorphic function in the collection of in the collection script f okay what I am going to get is I am going to get something on the I am on the Riemann sphere I am going to get something okay. So it is going to be again it is going to be a contour with starting point f of Z1 and ending point f of Z2 mind you now f of Z1 and f of Z are being thought of as points on the extent in the extended plane okay and the image contour the image contour is going to be just f of L okay and what is the if you now you know you can you know that from f Z1 to f Z2 on the Riemann sphere that is in the external complex plane the spherical distance is actually the shortest distance on the on the sphere it is just it is the minor arc of the greater circle passing through f of Z1 and f of Z2 on the on the sphere okay and so what you can write is the distance the spherical distance between f Z1 and f Z2 this is certainly it is the shortest distance because it is a geodesic okay curves of shortest length on a surface are called geodesics okay in general if you have space with the metric then the if you give me 2 points in the space it is not necessary that the straight line distance if it makes sense is the shortest okay that could be some other curves depending on the metric especially you could have you could find the distance along a curve to be smaller than the straight line distance in some cases for example for spaces with negative curvature okay but in any case if you take a space where a metric is defined and if you take 2 points in the space then the shortest distance the curve shortest distance from this point to that point on the space is called a geodesic and that is the geodesic distance on this on the sphere the geodesics are all given by the minor arcs of the arcs of the major circles okay so that is the spherical distance and this is certainly this is the smallest distance and so this is certainly less than the length this spherical length so let me not abbreviate it spherical length of f of L okay certainly and well what is the spherical length of f of L you know how to get the formula for the spherical length the formula for the if you give me a curve on the plane that is a contour on the plane then the length of the contour is just given by integrating mod dz okay where z is a variable you integrate mod dz from the initial point of the contour to the final point of the contour you get the length of the arc or contour on the plane but if you want to get the length of the image of the arc what you will have to do is you have to multiply by the factor which is given by the spherical derivative okay if you multiply the ordinary derivative and if it is an analytic function you will get the length of the image arc in the complex plane itself okay that is if you use modulus of the derivative of the analytic function as a scaling factor but if you use the spherical derivative as scaling factor and you will take the spherical derivative corresponding to meromorphic function then you will get the spherical length of the image of this arc on the Rayman sphere okay so what is this is going to be just integral from z1 to z2 of f hash of z mod dz this is the spherical length alright and what will happen is that you see now since you know now the point is that this integration is being carried out from z1 to z2 and of course this integration is over let me put L here because this integration is along the straight line path from z1 to z2 okay and that path lies inside this closed disk okay and on this closed disk all the spherical derivatives are all bounded by m so you know mind you the spherical length is always a non-negative quantity okay it is a positive it is a non-negative real number okay so what I will get is that this is certainly less than or equal to m times mod z1 minus z2 this is what I will get because I can replace this f hash of z by m because m is an upper bound and the integral from z1 to z2 mod dz is just the it is just along the straight line segment it is just the length of that segment that is the mod just mod z1 minus z2 okay so I get this but now what is the advantage of this now it tells me I have got equicontinuity see so for epsilon greater than 0 okay if we if we choose for epsilon greater than 0 if we choose delta to be you know epsilon by m okay then mod z1 minus z2 less than delta will imply that the spherical distance between fz1 and fz2 is going to be less than epsilon I will get this inequality given epsilon greater than 0 whenever the distance between z1 z2 is less than delta I can find a delta say that whenever distance between z1 z2 is less than delta this is a spherical distance between fz1 and fz2 is less than epsilon and this works for all f in the family script f so long as z1 z2 lying that closed disk so what have I got I got equicontinuity I have got a kind of uniform equicontinuity you can think of this as either equicontinuity at z1 or thinking of z2 as a variable or you can think of equicontinuity at z2 thinking of z1 as a variable in any case it is a uniform equicontinuity okay so what I have got is that f from this disk mod z minus z0 less than or equal to rho to the extended complex plane is equicontinuous and this but then of course I can cover the source domain D I can cover every point by such a closed disk lying in that domain therefore I have got equicontinuity at every point so this implies that so and this is equicontinuity for all f in script f so basically what I am saying is that f is this family script f is equicontinuous on D so I get equicontinuity okay which is what which so basically what I have done is I have just shown that is boundedness of the spherical derivatives gives me equicontinuity and that is a very general philosophy whenever you have boundedness of the derivative you integrate and you get equicontinuity that is a very general thing alright. Now what I have to show what do I have to show I have this I started with this I have this sequence here in script f okay and I will have to extract a subsequence which converges uniformly on compact subsets that is what I have to do what I want to indicate is that you can now do it exactly in the way you proved equicontinuity and uniform boundedness implies sequential compactness in one way of the proof of the Arzela-Askoli theorem okay so what you do is you do so these are the steps okay so what you do is we retrace so oops we retrace the steps in one way of the proof of the Arzela-Askoli theorem to extract a normally convergent so I am using CGT for convergent as an abbreviation subsequence from the given sequence okay so what you do so I will put it as a star list so the first thing is find the accountable dense subset X1, X2, etc so you start with a compact subset K of D given a compact subset K of D first find accountable dense subset okay here it is just the general statement that a compact metric space is separable okay then what you do is now you have now go back and think about the proof of the Arzela-Askoli theorem what you would do is that you would take the original sequence you will apply it to X1 okay and you apply it to X1 you get all these real or complex numbers okay and now you will use the fact that the original sequence is uniformly bounded to say that you have a sequence of bounded sequence and you will extract a subsequence okay any bounded sequence of real numbers or complex numbers admits a subsequence a convergent subsequence that is all you are using but now you see look at the present situation if I apply F1, F2 if I apply this sequence to X1 mind you let me change just change the notation to from X1, X2 if you want to Z1, Z2 because all my points are actually my compact subset K is actually a point is a subset of D and all my points are complex numbers so I will let me change it to Z1, Z2 and so on okay now what I will do is I will apply to Z1 I will apply the sequence okay and I will get a convergent subsequence I will get a convergent subsequence why is that that is because if I apply F if I apply these functions I am going to get a sequence of points on the Riemann sphere which is compact therefore it sequentially compact therefore every sequence gives me a convergent subsequence you see so it works that is the whole point. So apply the sequence F1, F2, 2, Z1 what will you get you get F1 of Z1, F2 of Z1, F3 of Z1 on the Riemann sphere okay but this is compact it is a compact matrix so it is sequentially compact and because of that what I will get I will get a subsequence F1, F1, F2, F13 such that F1J of Z1 converges okay so this is the key step okay this is the key point of difference when we were looking at real numbers a complex number when we are looking at real or complex valued functions okay when you apply the sequence to a point you got a sequence real sequence or a complex sequence but then you extracted a subsequence because you knew it is bounded and where from did the boundedness come here it came from the uniform boundedness of the original family okay but now you do not need any uniform boundedness in this case to extract a subsequence because the values are already being taken in the external complex plane which is compact and it is already sequentially compact I do not need anything more to extract a convergent subsequence that is the big difference okay. Now what you do is now you iterate what you do is apply to Z2 okay and you get a subsequence a further subsequence which is F21, F22, F23 and so on such that if you take F2J Z2 this converges okay and you do this ad infinitum what you will end up with is that you will end up with this matrix as usual so you will get this matrix of functions 1, 3 and so on F21, F22, F23 and so on F31, F32, F33 and so on so it goes on like this and you know it is the diagonalization trick that we use what we do is that we extract this diagonal subsequence okay then what is the advantage of this diagonal subsequence this diagonal subsequence will give you a sequence which will converge at all points of this dense subset this countable dense subset of K okay so F11, so G1 is equal to F11, G2 equal to F22, G3 equal to F33 and so on is a subsequence that converges on the countable dense subset Z1, Z2 of K okay alright and now what you do is I want to repeat those steps now you use we have just now proved that all the functions in this family are equicontinuous okay we have just now proved that so just use equicontinuity and on this sequence of functions to cook up to show that this sequence is actually Cauchy on the whole space okay and therefore it is convergent okay so the moral of the story is that at this point you use the equicontinuity of the family and mind you that equicontinuity came from the boundedness of the spherical derivatives that is what you have to remember okay so use the equicontinuity of use the equicontinuity of this as a family script F to get to show that G1, G2 etc is convergent on all of K okay so this is exactly as we did in the proof of the Azela Ascoli theorem I am not going to repeat it okay. Now so what have we succeeded using what we have succeeded is given any compact subset of D I have given a sequence given any compact subset I am able to extract uniformly convergent subsequence okay but then what do I need I need one if I change the compact subset okay my subsequence could change but I want one global subsequence which works on every compact subset and how do you get that you again get by another diagonalization argument what you do is you fill up D by a sequence of increasing compact sets okay with the property that any compact subset is contained in one of one set of the sequence okay and then use again a diagonalization argument as we use in the proof of Montel's theorem to extract from this a global subsequence which is going to be convergent uniformly on every compact subset and that finishes the proof in one way of proof of Montel's theorem that boundedness of spherical derivatives implies the family is normally sequentially compact okay boundedness of derivatives on compact subsets okay so normal boundedness of derivatives implies normal sequential compactness. So let me write that very quickly as in the proof of Montel's theorem fill out D by an increasing sequence sequence of compact subsets and use a diagonalization argument extract a global subsequence that converges uniformly on every compact subset of D. So this proves one way what is the other way you have to show that if you have a normal family you will have to show that it is the spherical derivatives are bounded and the other way is proved by contradiction if the spherical derivatives are not bounded okay then I can extract a sequence I can find a compact set and a sequence of functions and a sequence of points at which the spherical derivatives are becoming bigger and bigger and bigger okay. Now from this sequence of functions because I have assumed normal sequential compactness I can also get a subsequence which converges okay if the functions you know if a family of functions converge meromorphic functions converges to a limit function then the family of spherical derivatives also will also converge okay but mind you the spherical derivative of any function is always a finite quantity the spherical derivative of any function meromorphic function is only a finite quantity even if you take the function which is uniformly infinity the spherical derivative is 0 okay you will only get a finite quantity. So if the sequence of functions converges to a function then the sequence of spherical derivatives converges to the spherical derivative of the limit function and that is a finite quantity but on the other hand the original sequence had points where the values were becoming larger and larger so that is a contradiction. So that contradiction will prove that you know if you assume that the family is normally sequentially compact the spherical derivatives have to be normally uniformly bounded that is the other way of the proof of Marty's theorem okay and with that we are through with the proof of Marty's theorem, right.