 Now, there are a few things that are remaining in the discussion of the series and those we shall complete today. We discussed for a long time the series of non-negative terms and only in the last class, we discussed the series of the terms which may have positive and negative signs and we discussed what is known as Abel's test. A series which is convergent, but which is not absolutely convergent, it has a special name, it is called conditionally convergent. That is to give a more formal definition, suppose it is a series sigma a n, n going from 1 to infinity. This is said to be conditionally convergent, conditionally convergent or we say converges conditionally if the series converges, that is if sigma a n converges, but it is not absolutely convergent. That means and sigma mod a n diverges. We have seen an example of a series of this kind. For example, we saw this series yesterday 1 minus half plus 1 by 3 minus 1 by 4 etcetera. We have seen that this series converges, but it is not an absolutely convergent series. So, this is an example of a conditionally convergent series. Now, in case of this conditional and absolute convergence, there appears one very important question and that is what we shall also discuss today and that is what is called rearrangement. Now, to just motivate this, after all series is something like an infinite sum. Now, if we take let us say a finite sum, suppose we take say something like a 1 plus a 2 plus a 3 plus a 4 plus a 5 etcetera. Since the addition is commutative as well as associative, it does not matter in what way I take the sum whether I add a 1 to a 2 then add to a 3 or I take a 5 here or a 3 there. It does not matter whichever way you write this, the sum is going to be the same. But that may or may not happen in case of the infinite series. So, now actually what is meant by rearrangement? It is that you write these terms of the series in some different order that is called a rearrangement or to make it more precise. Suppose we take a map tau from n to n, suppose it is a bisection, it is 1, 1 and 2 as you know such maps are also called permutation. If it is a finite set, you call 1, 1 and 1 to map permutation. So, if you consider a series sigma a suffix tau n that is suppose you are given a series sigma a n instead of that you consider a sigma a suffix tau n then this is called rearrangement of sigma a n. That means basically what the term says that rearrangement that you just rearrange the terms of the series, write the terms in some different order. Now what is the obvious question here? If the original series converges does this rearrangement also converge and does it converge to the same sum? If it converges does it converge to the same sum? Now it so turns out that that is true if the series is absolutely convergent and the things are very bad if the series is conditionally convergent. We can just see an example here. Let us take this same example. We have seen that this series is 1 minus etcetera, we have seen that this is a convergent series. Suppose its sum is s. Now let us just rewrite the series in some different order. What I will do now is that I will write the series as follows. So, 1 minus half minus 1 by 4. Instead of taking 1 by 3 I take the next term as 1 by 4 then I will take plus 1 by 3 then I will take the next 2 negative term 1 minus 6 minus. So, 1 minus 4 the next negative term will be minus 1 by 8 then take the next positive term that is 1 by 5 and then minus 1 by 10 minus 1 by 12 etcetera. I suppose the original series was there was 1 positive term 1 negative term etcetera followed by that. Now what I am doing is I am taking the same series I am taking the first positive term then the next 2 negative terms then the next positive term again followed by the next 2 negative terms and do like that. So, this is a rearrangement of this same series. So, now let us see a few things here. For example, this 1 minus half is half 1 minus half is half. So, this is 1 by 2 minus 1 by 4 plus again this 1 by 3 minus 1 by 6 you can say it is same as 1 by 6. So, the next will be 1 by 6 minus 1 by 8 then similarly you can say that 1 by 5 minus 1 by 10 again 1 by 10. So, that is so the next 2 terms will be 1 by 10 minus 1 by 12 etcetera this will be the series of the simplification. You can see that I can this half is a common factor from all of them. So, suppose we take that common factor out what remains half into 1 minus this will be 1 by 2 that will be plus 1 by 3 then minus 1 by 4 etcetera plus 1 by 5 minus 1 by 6 etcetera. Do you see that it is the same series here. So, now what is it some it is half s it is 1 by 2 s. So, it is the rearrangement of the same series, but it converges to a different number it converges to different number. So, this idea of this example is to show that if the conditional convergence behaves very badly with respect to rearrangements and in fact what is known is something much worse. Not only that rearrangements will converge different number in fact given any real number we can find some rearrangement of the series such that that rearrangement converges to that given real number not only that we can also find a rearrangement. So, that that the new series with that new rearrangement that rearrangement diverges and on the other hand if the series converges absolutely then all its rearrangements converge and all those rearrangements converge to the same sum. So, let us just write this as a theorem. So, if sigma a n converges absolutely then all its rearrangements converge all its rearrangements converge and converge to the same sum on the other hand if sigma a n converges conditionally given any real number we can find a rearrangement such that that rearrangement converges to that. So, what I would say that for every s in r for every s in r there exists a rearrangement sigma I will call it rearrangement sigma a suffix tau n converging to s converging to s also we can find a rearrangement. So, that that rearrangement diverges. So, we can say that also there exists a rearrangement also there exists a rearrangement sigma a tau n that diverges. So, that is the importance of absolute convergence if you know that the series is absolutely convergent or if the series is of non negative terms remember this if the series is of non negative terms there is no difference between convergence and absolute convergence. So, in that case you can rearrange the terms of the series in any manner you like and that will not change the convergence or divergence or it will not also change the sum whereas, in case of conditional convergence things are quite bad all right. Now, there are a few elementary properties of the series which perhaps we should not discuss immediately after discussing the series and those are as follows. Suppose, we take two series sigma a n n going from 1 to infinity and let us say sigma b n n going from 1 to infinity. Suppose, both of them converge suppose both of them converge then we can say that if you take the series sigma a n plus b n that should also converge and sum should be same as the sum of sigma a n and sigma b n. So, let us just see that if sigma a n and sigma b n converge then sigma a n plus b n also converges and sigma a n plus b n n going from 1 to infinity this is same as sigma a n plus sigma b n equal to by the way you may wonder we are not going to discuss the proof of this theorem here because at proof it is somewhat lengthy those of you who are interested in the proof you can see the proof in Rudin's book it is this theorem is given in Rudin. Coming back to this so what it says is that if the two series converge there are some also converge and similarly if you multiply the series by some real number lambda then the new series will also converge that is also we can say that also sigma lambda a n this is also convergent series and its sum will be same as lambda times sigma a n going from 1 to infinity and this will follow simply by taking the partial sums. Suppose s n is a partial sum of the series suppose s n is a 1 plus a 2 plus a n and t n is say b 1 plus b 2 plus b n then saying that sigma a n converges is same as saying that s n converges s n converges to s and t n converges let us say t then use the corresponding theorem about the sequences then s n plus t n converges to s plus t and similarly lambda s n converges to lambda times s that is all in there is the proof. I said already that whatever we want to say or prove about the series everything can be done using the sequence of partial sums and using the corresponding theorem about the sequences. But what you will also notice further is that we cannot give similar characterization if we take the product. For example, if we take this series let me again recall that is suppose we take this s n as the sum sigma let us say sigma a j j going from 1 to n and say t n as sigma b j j going from 1 to n then partial sum of the series sigma a n plus b n that is same as s n plus t n that is fine. But suppose I take this series sigma a n b n n going from 1 to infinity and then the partial sum of this will be a 1 b 1 plus a 2 b 2 plus a n b n and that is not the product of s n n t n. So, we cannot say that if s n converges and t n converges this series also converges. So, in general we cannot say that if the two series converges that is if the sigma a n and sigma b n if both of them converge we cannot infer from that that sigma a n b n is also convergent series. So, to consider the products there is totally different notion what is called Cauchy products and Cauchy product is something like this. For example, the first term suppose I denote the terms of the product as sigma c n I do not know terms of what as sigma c n I think going from 1 to infinity then this first term I think for this for considering this it is convenient to start with 0 to it take both the series starting from 0 to infinity that is sigma a n also going from 0 to infinity and sigma b n also going from 0 to infinity just a minor convenience here and so sigma c n will also I will take from 0 to infinity. So, the first number here is c naught the first number here is c naught. So, that is taken as a naught b naught that is just the product of the corresponding terms then the next number c 1 that is taken as a 1 b naught plus a naught b 1 a 1 b naught plus a naught b 1 and now you can understand how we will proceed for example, next number c 2 that will be taken as a 2 b naught plus a 1 b 1 plus a naught b 2 that is what we are doing here we are taking all those indices such that the sum becomes 2 here. So, now you can understand how the general term will be. So, in general the term c n that will be sigma a k b n minus k k going from 0 to n. So, suppose you form a series like this then that series is called the Cauchy product of these two series sigma a n and sigma b n and we can say something about the Cauchy product if sigma a n and sigma b n converges then whether Cauchy product also converges or not there are some conditions for that. But, since that is not very important right now for us we shall not go into theorems of those kind for the time being all that you should remember is that convergence of these two series sigma a n and sigma b n certainly does not imply that sigma a n b n converges and of course, directly it also does not imply that sigma c n converges you need some additional conditions for that. Now, I think for the time being we shall close the discussion of the series and let us move on to next topic. But, to move on to the next topic let us start something with which depends on the series I shall define a new set let us say I will call that set l 1 l superscript 1. This is let us say this is a set of real sequences and real as you know a sequence is nothing but a real sequence is nothing but a function from n to r. So, suppose I did not any such function as x x from n to r then image of any number n here by using this notation I should denote by x of n. But, in a sequence this customary to denote it as x suffix n we can continue to use this notation. So, I will just say that x means this sequence x n x means the function from n going to r and that is nothing but same as the sequence x n. Now, this notation is sometimes more convenient when you also want to talk about sequence of sequences then for example, suppose I want to talk of sequence of this sequences then you will need some more either superscript or something. So, instead of that for that this notation is more convenient. So, what I want to do is that I shall take the set of all sequences x. So, x is a sequence such that sigma x n is absolutely convergent that is you take the series sigma x n either you write x n or this way whichever way you want like sigma x n is absolutely convergent. That means what sigma mod x n is convergent that is sigma mod x n is convergent. For example, if you look at this theorem here can I say that if sigma n and sigma b n if both are absolutely convergent will it follow that sigma a n plus b n is also absolutely convergent. Because you take mod a n plus b n that is less than or equal to mod a n plus mod b n and since sigma mod a n and sigma mod b n those are convergent. So, what you can say that sigma mod a n plus mod b n that is a convergent series and then use comparison test. So, that is trivial. So, if there are two series which are absolutely convergent then their sum is also absolutely convergent. In other words if I use this notation I can say that if x and y belong to l 1 then x plus y also belongs to l 1 and next is I will say that if x belongs to l 1 and let us say some alpha belongs to r then alpha x also belongs to l 1. Because the series alpha x n will be nothing but mod alpha times mod x n that is also convergent series. So, what it means is that if you take this set l 1 then if you take any two elements in this set their sum is also in this set and product of a real number and any element in this is also in this set. In other words this set has these two operations addition of two elements and multiplication of a scalar and a element in this scalar is real number. So, what is the obvious thing to do next? You are also learning linear algebra simultaneously. So, you all heard of vector spaces. So, vector spaces structure which has these two operations. So, what is the next obvious question that we should ask that whether this is a vector space and what is the answer? Because see after all what is the operation here? If you take two anyway it is a sum of two functions operation. So, we can say operation is coordinate wise x plus y entry of x plus y is x n plus y n. So, what is the 0 element constant sequence 0 and so it is this operation of x plus y it is associative commutative and simple also you can also verify all other actions that alpha times x plus y is alpha x plus alpha etcetera. So, I will simply say that l 1 is a vector space well that is nothing great, but there is something more about this. Now, because this I could have said even if I did not use this for absolutely convergent. Suppose I have taken the sequences only of convergent sequences still I could have done this. I could have converted that into a vector space that is also done. Now, I shall make use of this fact because of this fact what I know is that the series sigma mod x n is a convergent series. So, what I will do is that I shall call that I shall use a notation for that sum I shall call it norm of x this is used it is called norm of x norm of x norm of x it is nothing but sigma n going from 1 to infinity mod x n. We know that once x is in l 1 this is a real number that it is absolutely convergent. So, this is defined. So, norm of x is defined. So, norm now it means it is a function from l 1 to r it is a function from l 1 to r. Now, let us ask some very obvious questions what properties does this function have it is clear. So, we have a function norm which goes from l 1 to r. Of course, we are using the notation in a slightly different manner since if norm is a function from l 1 to r I should have denoted norm of x by this something like f of x. But anyway this is customary we do not bring this x here bring let x here all right. What are the properties let me just say first property for example, can you say that this norm of x is bigger than or equal to 0 for all x in l 1 norm of x is bigger than or equal to 0 for all x in l 1 and what is the norm of the 0 element 0. And suppose norm of some element is 0 then what can you say if sigma mod x n is 0 then all of this x n must be 0. So, can we say this that norm of x is equal to 0 if and only if x is equal to 0 norm of x equal to 0 if and only if x is equal to 0 second property. I want to say something about norm of x plus y norm of x suppose you take two elements x and y in l 1 I want to know how is norm of x plus y related to norm x and norm y and that something we saw just now. We can say that for example, what we are asking is this what is the relationship between norm of x plus y is nothing but sigma mod x n plus y n that is norm of x plus y and what is norm x it is sigma mod x n and what is norm y it is sigma mod y n. Now, how are these three numbers related it is clear because mod of x n plus y n is less than or equal to mod x n. So, this is less than or equal to this right. So, we can say that this is less than or equal to norm of x plus norm of y for all x and y in l 1 and lastly I want to say that norm of alpha times x norm of that is of suppose alpha is a real number norm of alpha times x should be sigma mod alpha x n is it clear that this should be same as mod alpha times norm x mod alpha times norm x this should be true for every x in l 1 and alpha in r alpha in r. Now, what can you say about these three properties have you seen something similar earlier it is these are properties very similar to the absolute value function on the real line on real line you define the function modulus of a real number and when we listed the properties of that function those properties were very similar all these for example. So, suppose x where a real number all these things are true if x and y are real number. So, this is a function which basically has a properties which very similar to the properties what is called absolute value of real number. Now, such functions can be defined on several vector spaces and when it can be done that function is called a norm and corresponding vector space is called a normed vector space or normed linear space. So, let us just make a formal definition as follows that is suppose v is a suppose v is a real vector space suppose v is a real vector space and a function let us say norm going from v to r is called a norm is called a norm on v if it satisfies these three properties if it satisfies these three properties all right if you want we will write once again if first is norm of x is bigger than or equal to 0 for every x in v and norm of x is equal to 0 if and only x is equal to 0 and the second property is norm of x plus y is less than or equal to norm of x plus norm of y for every x y in v and third property is norm of alpha times x is same as mod alpha times norm of x for every x in v and alpha in r and what is it last is that a normed linear space normed linear space normed linear space is an ordered pair is a pair v norm is a pair v norm. So, this is term that we are termed linear space or we can also called norm vector space where v is a vector space and this is a norm on v. So, is a pair where v is a vector space and norm is a norm on v and why we say that why we talk in terms of this pairs. So, it is possible that on the same vector space there may exist several functions satisfying this or the same vector space you may be able to define different we will see examples of this kind of thing little later. So, as a vector space those two objects will be same, but as normed linear spaces those two objects will be different. So, v with some let us say norm one and v with norm two as vector spaces those are the same, but as normed linear spaces those are different. So, that is why we usually talk about the pair of course, again as is the practice when it is clear from the discussion what is the norm that you are talking about then we will simply say v is a normed linear space. Also there is one obvious question here why we are taking real vector space what can we not take vector space on some other field. Of course, we can also take vector space on complex numbers you can take complex vector space then see as far as this first two are concerned there is it has no reference to the scalar only this last action that refers to the scalar and that will change this will become for every x in v and alpha in c every x in v and alpha in c that will be that will be called complex vector space. And so corresponding it will be a complex normed linear space and this is what we can call real normed linear space. Since there is not much difference as far as the definition is concerned we shall not bother too much about this. Now, this is an example of a real vector space now instead of taking the sequences from n to r suppose I have taken sequences from n to c then also you can define absolute convergence and all that in the usual way that would become an example of a complex vector space. Now, the next question why exactly we are discussing all these things and what is the idea of discussing this normed linear spaces to understand it let us again look at our definition of the convergence of a sequence. How did we define the convergence of a sequence suppose x n is a sequence in r suppose we take a sequence in r sequence in r. Then when did we say x n converges to x we say that x n converges to x x n converges to x if you remember what we this we said that this meant that for every epsilon bigger than 0 there exists n 0 in n there exists n 0 in n such that n bigger than or equal to n 0 n n 0 implies mod x n minus x less than epsilon mod x n minus x less than epsilon. In other words the concept of convergence of a sequence depends on this function mod x n minus x and we proved several theorems about the convergence sequences about real numbers using this properties of this absolute value function and of course some theorems using the order completeness of the real numbers etcetera. But you can say that sequence can be defined on any set after all what is the sequence? Sequence is a function whose domain is the set of all natural numbers co-domain can be anything. So, instead of take considering sequence of real numbers I can consider a sequence of any objects sequence of say elements in R 2 or R n sequence of vectors, sequence of matrices, sequence of functions and suppose I want to ask the question how do we define what is meant by such a sequence converges. Suppose you are given a sequence of matrices and suppose that sequence is a n each a n is a matrix of some fixed order let us say 3 by 3 and I want to say that this sequence a n converges to a. What is the meaning of that or how does one define? We can say that if we had some notations like this norm on that then I could have simply imitated this. This will simply change to norm of x n minus x. So, instead of sequences in a real line sequences in a real line I can take sequence in any normal linear space. I can take sequence in any normal linear space and define what is meant by the sequence converges in that normal linear space or more generally this is one idea that is the reason for discussing normal linear spaces. More generally see by mod x n minus x is nothing but a distance between these two numbers x and x n distance between two numbers x and x n. So, if you remember what we had said all the time saying that sequence converges means distance between x n and x n becomes small as n becomes that was the idea. So, one can say similarly that if we have a concept of distance in any set suppose we have a concept of distance then we can talk about the convergence of a sequence in that set all that we need is a concept of a distance. Similarly for example, other concept of limits continuity etcetera all those concepts depend in some sense either on this like absolute value function or on the concept of distance. So, we can also develop all those concepts in more general sets like that. What is the advantage see now we have proved let us say some theorems about convergent sequences. For example, we have proved that every convergent sequences Cauchy or every convergent sequences bounded we proved all these things for the sequences of real numbers. Let us say sometime later we talk of sequences in R 2 or R 3 or R n or sequences of matrices or sequences of functions. Then again we can define what is meant by convergent there and again we may have to again separately prove that every convergent sequences Cauchy or every convergent sequence is bounded and things like that. That means essentially we will be repeating the same proof again in various different contexts. That is the way to avoid that instead of avoid this repetition that is that is that method is what is called abstraction and it is very commonly used in mathematics. You may have heard this word that mathematics is a very abstract subject and people use it in some sort of a negative way that mathematics is an abstract subject, but abstraction is a very powerful tool it is used in all sciences. As I said because of this abstraction we can avoid these repetitions it saves lot of time and energy and it is more efficient we are doing this. So, what we will do is that what we do is that we see for example what we have done here this norm linear space is an abstraction abstraction of what real line and then that l 1 and so many spaces whatever common to all those spaces those properties we have taken and defined that as a norm linear space. So, similarly we will do about the distance similarly we will do about the distance and then follow the idea then after that we shall just develop all the theory in those particular either the norm linear spaces or those new objects. By the way let me just tell what those new objects are called those are called metric spaces those are called metric spaces and then we shall develop all these theory in metric spaces and once we develop in metric spaces it will be applied to it can be applied to any different any of these other specific examples R R 2 R n l 1 and all those things. Now, let us come to this what is what is a metric space or what is a metric this is something more general than norm linear spaces here what we have seen we shall subsequently show that every norm linear space is also metric space. But before that which is basically same as saying that metric spaces are more general concept because norm is defined only on a vector space we to starting point has to be a vector space whereas metric can be defined on any set. So, we take x as a x as any non-empty set x as any non-empty set then what is a metric it is nothing but a function which says something suppose you take two points x and y in it says what is the distance between those two points. So, that function which satisfies the properties which we normally associate with the distance between the two functions distance between the two points whatever we commonly associate some of those properties are taken and those are taken as a definition of a distance or definition of a metric. So, obviously we talk about the distance between the two points. So, it means it is a function from the pair of points it will associate some real number to a pair of points. So, we will say that a function it is a function d from x cross x to r function d from x cross x to r is called a metric if it satisfies some properties if it satisfies some properties what are those properties those properties are again very similar to this first property is that suppose you take two points and if you take distance between it is called a metric and metric is same as distance this is just a different word. So, d x y distance that is this is distance between two in fact strictly speaking I should write one more bracket here because it is d of this some element in x cross x that element is x comma y. So, strictly speaking I should use this notation, but we will understand what we mean is this. So, d this is let me just remove this just for the convenience. So, d x y this is bigger than or equal to 0 for every x y in x that is what we normally expected distance between any two points should be a non negative number and it should be 0 only when or if x and x distance between the x and the point it should be 0 and the distance between the two points are 0 those two points must coincide. So, which is same as saying this and d x y is equal to 0 if and only if x is equal to y then second property is that distance between x and y this should be same as distance between y and x it should not matter whether I call distance from x to y or from y to x that should be the same this should be true for every x y in x and lastly by the way this property has a name. In fact, it is an obvious name this is called symmetric this is called symmetric. So, you express this by saying the distance is a symmetric function this property is called symmetric then last property. Suppose, we take three points x y and z then we want to compare the distance between x and z and distance between x and y and distance between y and z. Suppose, we take three points you imagine that those three points form a triangle then distance between x and z is a length of one side and distance between x y and z are the other two sides. So, what we should expect is that this should be less than or equal to that this should be less than or equal to that. So, this is true for every x y z in x and because of the comments which I might just now this last property is called triangle inequality this last property is called triangle inequality this is called triangle inequality by the way similarly in this definition of a norm this property two is also called triangle inequality this is also called triangle inequality and we will see the reason for this little later. So, that is about a metric and so what is a metric space again in a similar way metric space is a pair x d where x is a non-empty set and d is a metric defined on it. So, let us just record it for instance. So, metric space is a pair x d where x is a non-empty set here also I should have said is called a metric on x this is called a metric on x. So, coming back to this a metric space is a pair x d where x is a non-empty set d is a metric on x and again why pair again because when one can define several metrics on the same set x. So, for example, I can define say d 1 as one metric d 2 as the other way d 3 as another way. So, the set underlying set may be the same, but the metrics may be different. So, in which case those become different metric spaces. Now, you can see that all these axioms which we have written here are the properties which you associate with the usual concept of distance and those are the ones which are taken for defining the distance. Now, you may ask there are some many other things also which we associate with a distance. For example, we also know that given two points we can talk of something like a midpoint of the two and then that has not come here in the axioms, but again which of the properties to be chosen for making definition that is a matter of convenience and also matter of history because this definition is like this arise after several years of efforts from various mathematicians by trying various axioms and which work better etc. And ultimately it is decided which exactly the things that go into the definition. So, let us not go into that kind of history right now. Let us see some examples of the metric spaces. So, in example means what it should be some non-empty set and function defined like this and usually given a function like this to check whether it forms a metric or not. Actually these two properties are very easy to check in fact quite trivial and if at all anything takes some time it is this last property it is this last property triangle inequality. This one very famous example this is a metric which you can define on any set suppose x is any non-empty set and suppose you define say d x y is equal to 0 if x equal to y and 1 if x not equal to y. It is easy to see that it satisfies all these properties only this last property will take some time to check as I said other two properties are trivial. So, this is also a well-known metric it is called a discrete metric and this space is called a discrete metric space. It is called a discrete metric space and the main use of this discrete metric space is basically for understanding it is not much of practical importance. You do not come across discrete metric in any applications, but in order to understand the various concepts in metric spaces and to check whether you have understood or not this example is very useful. Then the next obvious example is that of a real line you can take the real line and define d x y as mod x minus y distance between x that is the usual distance between the two real numbers and again it is easy to see that satisfies all these three properties and now let us come back to this. Let us come back to this we will just finish this. So, we have seen that this function norm is nothing but the generalization of the function of the absolute value. So, we can use this idea in any normed linear space. Suppose I take instead of taking x and y as two real numbers suppose I take x and y as two elements in a vector space and define distance between x and y as norm of x minus y then that should also satisfy all these properties because those are basically followed from the properties of the absolute value. Let us just quickly see how this happens and then we will stop with that. So, now suppose let us say v is a normed linear space and take any two x and y in v and define d x y as norm of x minus y. Then we will just quickly verify these properties one by one what is the first thing that we require that distance between x and y should be bigger than or equal to 0. Is it true norm of and this should be 0 if and only if x equal to y is this is also true that follows from this pattern norm of x minus y will be 0 if and only if x minus which is same as x equal to y. What about this distance between x y is equal to distance between y x. So, the distance between y x will be norm y minus x by this definition are these two things same by what it follows from what you it is nothing but minus 1 times this and that follows from this last property you take alpha is equal to minus that is basically norm of minus x is same as norm of x for every x. So, this symmetry follows from this property 3 what about the triangle inequality this is norm of x minus z this is norm of x minus y and that is norm of y minus z. Is that true that norm of x minus z is less than or equal to norm of x minus y plus norm of y minus z again you see you can first suppose you take a as x minus z b as x minus y and c as y minus z then you can say that norm of a plus b is less than or equal to norm a plus norm b and it will give this. So, this property to implies this triangle inequality here and that is why that is also called triangle inequality. So, what it what follows from it is that every norm linear space can be made into a metric space every norm can will lead to a metric on that vector space and. So, this is a big class of examples of metric spaces and that is what is most important applications most metric spaces which are important from the point of view of applications are basically norm linear spaces I think we will stop with that we shall see more examples of this in the next class.