 Welcome to module 3 of point sector biology course. Last time we had taken a, you know, cursory look on what are called as normed linear spaces. They were seen to be an immediate generalization of the modulus function on the field of real numbers or complex numbers. Today we are going to take one more step towards the generalization. Once again, we go back to this modulus function on k. This k denotes remember is set of all real numbers or set of all complex numbers, any one of them. So, take the distance function dxy where x and y are elements of k to the modulus of x minus 1. Inside the real numbers or inside r cross r complex numbers is actually coincides with the standard notion of distance. So, it is what you can call the distance function. What are the fundamental properties of distance function? Just like yesterday what we observed about the modulus function itself, those three properties will be picked up today also. What are they? The first thing is the distance function is always a non-negative real number and it is 0 if and only if x is equal to y. The second thing is distance function is symmetric. dxy is same thing as dyx. Distance that is why it is called distance between x and y. Distance from x to y is same thing as distance from y to x. So, that is the meaning of this. The third one is more clearly this time is triangle inequality. Distance between xy is less than or equal to distance between x and z plus distance between z and y. So, if x, y, z are three points formally triangle. So, this just says that the length of any side the x to y is less than or equal to some of the lengths of the other two sides namely x to z and z to y. So, this is much direct as compared to the norm of x plus y was less than or equal to norm of x plus norm of y. To interpret that one as a triangle inequality you have to use the vector method represent x represent y as vectors you know lines join from 0 to 1 point x to take the arrow from 0 to x 0 to y and then take the sum that will be another third point. So, now the end points become a triangle that is the meaning of the norm of x plus y is less than or equal to norm of x plus norm of y. So, we take exactly those three, but formulated now in terms of d. So, dxy equal to 0 what does it mean modulus of x minus y equal to 0 ok with the norm. So, modulus of x minus y is equal to 0 x minus y is 0 that means x is equal to y. So, this is identical with the condition 1 which we had called norm 1. The beauty of this one is now the condition is independent of the additivity on the vector space ok. So, that is what has happened. Similarly, the third condition is independent of the additivity you know you do not have to take norm of x plus y there is no x plus y distance to the x and y it less than or equal to x plus this plus c is by the way on the right hand side this is just real numbers addition ok. So, the left hand sides here or the starting things here the condition they are independent of the vector space structure on v and that allows us to make a sweeping generalization that is what we are going to do. So, I am repeating it here once again if you have missed it. The condition 1 and 3 here are parallel to n 1 and n 3 these n 1 and n 3 were the norm for the norm conditions same condition this was whatever you call positive definiteness this was a triangle inequality. But this n 2 seems to be it is just an additivity no something weaker right whereas the n 2 there was something about alpha times x if you take the norm of that modulus of alpha came out ok norm of alpha x was equal to modulus of alpha into norm of x. So, that was quite a strong condition, but this is just there does not seem to be place here at all. So, it may be true but why this is so fundamental. So, that is something which you have to think about ok, but how to how to derive this one that is not all that difficult ok we will check that into what. So, like right now let us make the sweeping generalization namely start with any set x now. Remember for a normed space we have to start with a vector space v. So, there is no extra structure is just a set here x. Now, if take a function d from x cross x to 0 1 the norm was a directly a function on the vector space. The distance is between two points that is why I have to take x cross x to 0 infinity a function like this is called a metric. So, that is the name we are going to give ok. So, you can you could have called a distance metric. So, classical it has been called as metric. So, we are going to call it as metric otherwise it would distance function also. So, what is the condition? There are three conditions which you can call it as axioms of metric space. Positive definiteness which corresponds to d x y equal to 0 if filled only if x equal to y. Symmetry d x y equal to d y x. Trander inequalities d x y is less than equal to d x that plus d y z for every x and y and z. So, take these three conditions here which which are true for modulus function on k generalize it to those exactly three functions. It is what we have. Such a thing will be called a distance function or a metric ok. So, what we have is take any set x and a metric on it the ordered pair x comma d is going to be called as metric space ok. But, quite often we will just say x is a metric space without mentioning what is our d. Quite often the distance function the metric is understood by the context ok or it is mentioned just a few minutes back. So, you do not have to again and again mention it. That is the only reason why you have to do that one just to save some time. Otherwise logically every time you have to say x comma d is a metric or metric space then only it makes sense ok. Is that clear? Now, one sitting generalization definition is over that is all. So, slowly you have to build up the theory ok. Just like what we have done for subspace of a normed linear space. What is it? It is a vector subspace and then the norm is restricted to the subspace. Similarly, take any any metric space ok x comma d take a subset x prime restrict the metric d to x prime cross x prime then you will get a metric subspace. So, then what you can say x prime comma d prime is a metric subspace of x d ok. So, this is definition x prime is subspace subset of x d prime is a restriction of d on the x cross x prime x prime cross x prime which is subset of x cross x. d is defined on x cross x take the restriction. All those properties will be automatically true for x prime d prime ok. So, we can just say instead of my mentioning d prime etcetera we can quite often write it as same d also quite often restricted function we use the same notation ok just to save time and you know instead of cumbersome too many notations that is all. We can just say x prime is a sub metric space of x ok. So, that is the just the second definition after making a metric space we have made a metric subspace. So, slowly we have to develop a number of terminologies and theories. So, all based on similar experience with modulus function what it is doing for real numbers or complex numbers. So, remember that ok. So, here are a few more terminology. So, start with a metric space take any point in the space take any real number between 0 and infinity any positive real number. Now, by an open ball of radius r and centre x at x this is the definition now ok. What is the meaning of open ball? An open ball always has a centre and a radius. A radius must be some positive number no unless you mention it it is not clear what it is open ball is an open ball ok. What is it? It is set of all points y inside x which are at a distance less than r from x dy x is less than r that is an open ball just put in equality that dx dy x less than or equal to r. Hello equality also here then you get a closed ball. So, we will use br over open ball and drxt for closed ball this x denotes the centre r denotes the radius and d denotes the metric. In the same space x same set x if you change the d to some other metric these balls will be different obviously ok. So, that is why you have put that d also there. But if you understand what this d is just like I can call x a metric space here also I do not need to mention d then I will have a shorter notation namely br of x is br of x d that d is not mentioned ok that d is understood similarly dr of x is dr of x d ok this provided you know what metric we are speaking with respect to all right. So, now you know a few more definitions. So, let me again go back to emphasize that our standard example with which we begin is just the field decay itself together with dx y equal to x minus y. This is the starting example after all. So, this first example this is called the usual metric on r or c whatever coming out of the modulus function ok. It is also called Euclidean metric because it is so so ancient goes back all the way to Euclidean ok more than 2000 years old ok. So, in that sense if you take whatever we have defined namely take a metric on c restricted to r then you get a metric subspace. But this function is the same ok when you take x and y as real number when you take the modulus of x minus y it is the same thing as when you take them as complex numbers ok. So, this is the first example of a metric space and a sub metric space r is a sub metric space of c ok with the usual metric. So, the second example here is take any non-linear space that we have studied last time a vector space together with a norm. Then you define d of u e just like we defined modulus of x minus y you define norm of u minus v d of u is norm of u minus v. Then d 1 is first one follows directly from n 1 right. So, what is d 1? d f d u v is 0 if and only if u is equal to v. d u v is 0 means norm of u minus v is 0. Then u minus v is 0 that means u is equal to v. So, that is n 1. Similarly, n 3 which is triangle inequality stated in terms of vectors is equivalent to triangle inequality stated in terms of the distance function d 3. So, they are fine, but what we wonder about is number 2 how do you get u v equal to u minus v that is very easy here d of v u is by definition. I want to show that it is equal to d of u v right d of v u norm of v minus u. So, if this has a symmetric that is a point it is minus 1 times norm of u minus v v minus u can be written like this. But this now minus 1 comes out with a modulus that is now just 1. So, that is the quality of. So, the symmetry property of the norm function which is much weaker than modulus of alpha times something coming out alpha coming out and so on that is just forgotten we do not need that one. But we have retained something else namely the symmetry. So, that is the beauty. So, this is definitely going to be something more general. If I just try d of alpha u and alpha v here there is no way this alpha will come out mod alpha will not come out in general that is not a part of the axiom. Whereas if your metric is defined using the norm then that is true. So, that is what I am going to repeat it here. First of all whenever we have a metric coming out of a norm then we call it a metric associated with the norm. In short we can call it as linear metric. The linearity is coming from modulus of norm of alpha times x is equal to modulus of alpha times norm of x. And of course triangle inequality, triangle inequality is always there. So, number 2 makes it a linear metric here. But in general this is not a part of the definition of a metric. You have to be careful about that. And whenever such a d happens we may indicate it by writing d suffix norm whatever norm we have taken. Suppose there are two different norms on the same vector space v. They will give rise to two different metrics. So, which one? So, you better tell that. So, in that case suppose the norms are written as norm 1 and norm 2, two different norms then I guess it correspondingly d1 and d2 are the metrics. So, this is the way we will treat how these metrics are related to the corresponding norms. If d is a linear metric it has two additional properties. So, I repeat it I have already told you. One is this d of rx comma ry is mod r times dxy. xy belong to v, r belong to k. The second property which is also hidden there is distance between x plus z, y plus z. This x plus z, y plus z it makes sense because I am working inside a vector space v now. This is same thing as what? It is same thing as what? Norm of x plus y, x plus z minus y plus z which is z and z cancels away is equal to norm of x, x minus y is equal to d of xy. So, this property is addiquity you can directly state on a vector space if you start with a metric suppose you can put this as a condition. So, one can have this as condition it is satisfied with the linear metric. But in addition I can put this as a condition for an arbitrary metric on a vector space. Then it will be called translation invariant metric. A linear metric is automatically translation invariant. This condition itself is called translation invariance of d. Similarly, its multiplication invariance you can say this is scalar multiplicative namely r does not count out directly r comes out as modulus of r. So, something is not a linear metric if even this condition or this condition is not satisfied. That is easy way of checking whether a given metric comes from a norm or not. If it is then this will satisfy but that does not guarantee that it is coming from then there is a likelihood that it may be coming from. On the other hand if this any of this condition is not satisfied then you are happy. This is not norm. So, let us have something now which are actually not coming from norm at all or may be coming from norm but we have not bothered. We can directly define them. So, there are lots and lots of such metrics which arise without reference to any norm. Let us see say a few of them. Okay. Start with any set. Define this delta delta of x, y equal to 0 if x is equal to y and equal to 1 otherwise. As soon as x and y are different it is 1. This delta is actually the opposite of the delta function direct delta function in analysis. So, that is the contrast here. Straight forward check that this is a metric. What you have to check? D of x, y is 0 implies x is equal to y. That is built in in the definition here. Okay. D x, y equal to dy x. That is how it is built in the definition. Triangle inequality takes two more seconds. You just think about it. Why dissatisfaction triangle inequality? It does not take more than two seconds. That is all you have to think about. Okay. So, this is a metric and it has a name. It is called discrete metric not direct metric. Okay. So, this is a discrete metric. Next I will give you something which is quite non-trivial out of the blue. I will not explain how it came today, but I will just give you. Today you have to only understand the computational aspect here. Part of it I will do. Part of it I will leave it as an exercise. So, what is this? The called metric, it is defined on the set of complex numbers. It is not going to be coming from any norm. Okay. At least not on C as a vector space. So, this dc, c for called tc from c cross to 0 infinity is defined as dc of z1, z2 is twice the modulus of z1 minus z2 which is quite near and you are looking at normal activity. It is multiplied by 2. That is okay. But this part, the numerator is quite close to a norm now. It is twice the norm of modulus of z1 minus z2. But there is a denominator here which will kill all these properties. Okay. What is it? 1 plus mod z1 square into 1 plus mod z2 square whole thing raised to half. The square root of the product of these 2. 1 plus mod z1 square and 1 plus mod z2 square. Okay. So, that is the denominator. So, this is the definition. See, I have divided, but this is never 0. So, I can divide. So, this makes sense. So, this is now a real number. Okay. And positive non-negative real number. If this is 0, the numerator must be 0. That means z1 is equal to z2. Mod z1 is equal to mod z2. Z1 minus z2 is 0. So, z1 is actually equal to z2. Okay. So, this satisfies condition d1. Look at d z2 z1. It is perfectly symmetric here because modulus of z1 minus z2 is modulus of z2 minus z1. And these things are all, you can change this term here. That is all. So, this is symmetric. So, what is difficulty here? The difficulty is it is not very easy either. It is not very difficult, but it is not quite easy either. Is it triangle inequality? By the way, in all these cases, triangle inequality verification is the most difficult part. Okay. Other things, if they are there, they come easily. If they are not there, you cannot help it. They are not there. Okay. So, let me help you to see how triangle inequality comes here. I will not do the entire computation. The first thing is there is some inequality of complex numbers. It takes z1 and z2 are complex numbers. Okay. 1 plus mod z1 z2, 1 plus z1 z2 modulus square is less than or equal to 1 plus mod z1 square, 1 plus mod z2 square. Where are these coming from? Look at here. So, it is a square of that. If you take the square root, so modulus of 1 plus z1 z2 is less than or equal to this one. So, this is the first inequality that you have to see. So, this denominator here is bigger than 1 plus mod z1 z2 square or a square is not there now. If you denote half half, you have taken already. So, 1 plus mod z1 z2, that is what you have to see. This itself is not difficult. You have to just expand this one. 1 plus mod z1 z2 square is nothing but 1 plus mod z1 z2 into 1 plus mod z1 z2 bar, which is same thing as 1 plus z1 bar z2 bar. Now, expand it, collect it. One of the terms, so central term namely twice real part of z1 z2, you can replace it by what? Mod z1 square mod z2 plus mod z2 square and then you will get this. Okay. So, that is the explanation for this one. There is another though looking somewhat threatening here, but this is much simpler. This is an identity. See, this is a product of two terms here. It is a product of two terms plus product of two terms. So, there are four plus four eight terms here. There are four terms here, but you can immediately see that four of the terms cancel out 2 by 2 minus z3 here plus z3 here minus z3 z2 z3 bar minus z2 minus plus z3 z2 z2 bar. So, something will cancel out. What you will get is same term here. So, this is an equality. Okay. Once you do this one, combine this one, triangle inequality will come very easily. Okay. So, that part I will leave to you. So, I leave the further calculations to you as an exercise. All right. Now, we do one step ahead here. Look at what is called as extended complex plane. The extended complex plane. In this, you take the complex numbers along with infinity, one single infinity. Okay. Do not do much of algebra there, but you can do lot of topology and analysis. That is why it is there. This extended complex plane. So, one extra point, set theoretically that is all. But now I want to extend the metric, the called metric to this set C hat. What you have to do? Whenever the whole, both of them are in C, this is the formula. So, if one of them is infinity, I have to define what is the meaning of this. Namely, I have to define take any z in the second chord in second term here infinity. What is dc of z infinity? This is what I have to do. So, what I do? Take z1, z2 here and let z2 go to infinity. In this, in this, in this rh says take z2 go to infinity, take the limit. Okay. What you get is this formula. Of course, instead of z1, I am writing z here, that is all. So, 2 divided by 1 plus modulo square raised to half. To work it out, it is not very difficult. When you, how do you take the limit as z2 infinity? So, this is infinity by infinity method. Oh, do not worry, differentiation and so on. Yet, pull out z2, that is all. Pull out mod z2. Mod z2, mod z2 here. Okay, mod z2 square will come, then raise to half. So, mod z2, that will cancel out. So, then take the limit, which is easy, it will be just this much. Okay. So, that will define whenever one of the point is infinity. When both the points are infinity, what should I do? I am forced to define that equal to 0. There is no other chance. Okay. Take this as definition. Now, we have to verify that for this extended thing also, the triangle inequality is true. Okay. So, z infinity, you take some other point here. Okay, z1, z2, some other z prime, and then you have to show that this inequality is not very difficult. So, that also I will leave it to you as an exercise. So, extended complex plane has a metric here. So, because of this denominator here, we were able to do that. If you just take the linear matrix z1 minus z2, you could not take this limit as z tends to infinity. Now, you see why the, you know, the geometry or the analysis, whatever you want to say is reflected here. So, more about this one, I will tell you when the time comes. All right. The chord metric is very important in the geometric function theory of one-variable complex numbers. Okay. So, the geometry node here will be explained to you a little later. All right. So, let us take a break here. Today, we have done some good work. So, next time we will see more. Thank you.