 Welcome to this NPTEL course namely Real Analysis. This is the course which we shall be giving. And maybe I should begin by saying that this is a very basic course. You should take any MSc program anywhere in the world. This real analysis will be one of the core courses in this. Real analysis and linear algebra these two form a very pair of very basic courses in every MSc program all over the world. And whatever course you do after this that is say differential equations or functional analysis or topology or complex analysis all those courses require concept from the real analysis. And what do you do basically in real analysis? Whatever you have learned in calculus concepts like function, continuity, differentiability, integrability we look at those concepts more closely more rigorously in this course. And there is one more thing it is that this is a course which gives more importance to proofs. Unlike for example something like a differential equation course where you are where usually both places given to solve the equations and find the solutions etc. Here there are very few things where you solve anything. Main thing to do in real analysis is to learn some new concepts and prove things. And while talking about the proofs let me also mention here that something that is said by Siemens that when you look at the proof of a theorem of theorem will consist of two parts. There will be some hypothesis and some conclusions. And the idea of a theorem is to start from the conclusion and go to sorry start from the hypothesis and go to the conclusion. And this may be done in a number of steps. So suppose you understand each step let us say how you know how to go from say jth step to j plus one step for each j that does not really mean that you have understood the proof completely. The main idea of the proof you may still not have understood. So when do you really know that you understood the proof of the theorem. It means that you have to understand the proof as one single idea or what is the main idea in the proof. Then you have to ask questions like suppose you drop some hypothesis in the theorem will the conclusion still remain valid. And now how does one say such a thing. Arguments like this particular proof does not work will not be a complete answer to that because even if that particular proof does not work some other proof may work. So questions like this can be decided only by giving a counter example. What is meant by a counter example again here that it is an example in which except that one particular hypothesis all other hypothesis should be true but the conclusion is false. So that only will prove that particular hypothesis is essential for proving that theorem. So these are the kind of things which are very important in real analysis. Let me also mention a couple of books that we shall be following. The one book is by Walter Rudin principles of mathematical analysis. We will also occasionally see the book by Siemens. The title is introduction to topology and modern analysis. One of the basic difference between the calculus and analysis is that in calculus one is usually concerned with one function and properties of that function and what happens to that functions. In analysis we usually deal with the classes of functions various spaces of functions and in particular one of the objects that we shall be studying is what are called metric spaces. Now before proceeding with those things let us fix some notations which we shall be using throughout this course. Of course you may have already used that notations but it is better that to fix them once for all. This is n the set of all natural numbers set of all natural numbers that is natural numbers that is numbers like 1, 2, 3, etc. Then this is z the set of all integers that is the natural numbers then 0 and then minus 1, minus 2, etc. Then Q the set of all complex not complex set of all rational numbers. Let me mention here that this symbol is for example this is n with just one vertical bar here or similarly z with another bar here, etc. These are basically device so that we can use the usual letters n, z, etc. in as other symbols. This is like freezing of that particular symbol. This is red as blackboard n similarly blackboard z, etc. That means n with this another extra bar here similarly this r blackboard r that is the set of all real numbers. Also we may not use this set very often but this is z the set of all complex numbers. These are notations for these sets which we shall be using throughout. Then we shall now begin with because as every such course in real analysis and for example every elementary course in mathematics requires various operations with sets. So we shall begin with what is called review of set theory which means that we shall just take a quick review of elementary set theory because that is something which we shall require fairly often. You are already familiar with what is known as say union or intersection of sets. Suppose you are taken given set two sets a and b then you know what is spent by things like a union b or a intersection b or a minus b. This is perhaps may not be familiar to you. Let us let this means the set of all x in a with the property that x is not in b. x is in a and x is not in b that is the set of all x which are not in b that is called a difference b or a minus b. Now it is okay to write things like that when there are only two sets. For example if there are three sets you may write that as a union b union c or if there are n sets you may write that as a 1 union a 2 etc. But suppose there are large number of sets say possibly infinite family of sets then how do you write this and so for that what is done is that let us take the case of suppose I start writing this a and b suppose I take this as a 1 and a 2. Then this 1 and 2 these are called indices and so I will take the set i to be the set of indices. So this set consists of just 1 and 2. So instead of calling now a 1 union a 2 we write this as union a i where small i belongs to big i small i belongs to big i. Now advantage here is that this i can be now any set it need not consists of just two symbols here. So such a set so now we can take i as any arbitrary set this i is called indexing set and so indexing set means what it will consist of some indices. So a particular index will be loaded by small i in i and what we are dealing with now is what is called a family of sets that means for each index in this indexing set i you have one particular set and that set will denote by a i and the whole family we may denote by this suppose is script f that is family of the sets a i it is a suffix i for each small i in big i. So this is what is called family of sets family of sets family of sets index by the indexing set big i. Now once we have such a family then we can talk about various operations on that family of sets. So for example one can talk of what is meant by union a i so we can write now union a i union taken over every small i in big i or similarly if we want to talk of the intersection of for example what will this mean it will mean that the set of all those x such that x belongs to this a i for some i for at least one small i in i similarly we can talk of what is meant by intersection a i small i intersection taken over all those small i belonging to big i. So what will this mean instead of saying for some i it will mean for all for all i that is this will be the set of all x such that x belongs to a i for all a i in i. By the way let me again since this is the beginning lecture let me say that this is symbol as it stands for for all and similarly this stands for there exists standard symbols use in mathematics. Then once we understand these unions and intersections over an arbitrary family then we can also imitate the various laws that we know about the intersections and unions about this finite family for example for example what we know here is that suppose you take say two sets a union b and take the union a union b and suppose we take the intersection of this with c then what we know is that this is same as a union c intersection b union c. And so what we can say is that in a similar way whatever is true for just union of two sets similar thing is true for such a union and a similar law for if you interchange the intersection union let us just mention one of them. So for example suppose I want this union a i union of a i i belonging to i and suppose I want to take the intersection of this whole thing with some other set c then this will be same as union over i a i intersection c and this can be proved by usual elementary methods basically use this definition what is meant by a i intersection c what is meant just use the definition of left hand side and right hand side. And similarly you will get another law by interchanging the unions and intersections. In a similar way suppose all these sets are a part of some big set suppose x is some set which contains every set here sometimes this is called universal set. Then we can talk of what is meant by complement of that that is we can just as here we have dealt with a minus b. So similarly one can talk of x minus a x minus a. So this is sometimes called complement and where we take the complement also the so called de Morgan's laws are also true in this case. Again you would be familiar with de Morgan's laws when it comes to the union or intersection of two sets a similar thing is true for example what you know is this a union b. Complement of that is the thing but a complement intersection b complement and similarly suppose you take a arbitrary union like this that is union a i i belonging to i then its complement its complement means x minus this x minus this whole union a i i that will be same as intersection i belonging to i a i complement. Now so that is about unions, intersections, complements etc. Another concept of the set theory that we use very often is what is called Cartesian product or cross product and what this means is suppose you take two non-empty sets a and b then this is something is familiar with you this a cross b it is nothing but the set of all ordered pairs of the form a comma b where this small a belongs to big a and small b belongs to big b. Now that we are here with this notation suppose we take b and a are same suppose we take a cross a then we denote this by a square or a superscript 2. So, quite the rotations quite frequently used are the following for example r 2 this is nothing but r cross r. So, this is the set of ordered pairs of real numbers. So, this will be the set of all pairs of the form a comma b where a and b both are real numbers and similarly one can define what is meant by r 3 or r 4 r 3 will be r cross r 2 or r 2 cross r 3 that does not matter and in general we can define what is meant by r n it will be r cross r cross r Cartesian product taken n times. Similarly one can define a cross b cross suppose there are n sets sets a 1 cross a 2 cross a 3 cross a n we can define what is meant by Cartesian product of those n sets the real problem occurs if we want to define the Cartesian product of an arbitrary family that is suppose now you are given a family like this a i for i belonging to i suppose this is the family and suppose I want to talk of what is meant by the Cartesian products of sets in this family. So, that is pi a i this is pi a i pi for the product pi a i i belonging to i what is the meaning of this and how this is defined now to do that that is a very non trivial questions and for that we need we need to go to another concept from the set theory namely that of functions we will come back to this we will first discuss something about functions because this is defined in terms of functions there is no simple way of defining this in terms of ordered pairs or ordered triples or ordered quadruple etc because this is an infinite family in general. Now, let us just recall quickly that when you talk of a function there are basically three things or three objects or three concepts in mind function what are those three things first is there are two sets a and b both are non empty sets that is a and b both are non empty sets and f is a function which goes from a to b f is a function which goes from a to b. So, functions means these three things two non empty sets and what is f f is a rule f is a rule which assigns to every element in a some unique element in b that is a function let us let us take it like that of course it is possible to differentiate some other way by as a some kind of subset of a cross b etc let us not go into that right now. So, this is a function this set a is called domain of f and this set b is called a co domain of f of course the set a and b did not be different those can be same also. So, the function can go from a to a and quite frequently in our real analysis course the functions will go from either r to r or some subset of r to some subset of r that is the now given such a function for example let us just take some example that is when you take x in a then f x suppose I call this f x as y y is equal to f x in b I think I will make a slight change in notation here that is very convenient instead of taking the sets as a and b let me take the sets as big x and big y so that I can take this a and b as subsets of x and y so x belongs to x y is equal to f x that belongs to y. So, this big x is a domain and big y is a co domain now for any element x in x small x in big x this y is equal to f x this is called image of x image of x so x is called the object and y is called the image of x and so other way if if an element y small y in y is given then this x is called a pre-image x is called a pre-image of y x is called pre-image of y now it can happen that of course once from the definition of a function it follows that given an object small x in x there will be a unique image but given an element small y in y there can be several pre-images that depends on what that function is and that determines certain properties of functions or certain definitions involving functions. Now, to make the idea somewhat clear let us let us just take some couple of functions let us and as I said our most examples will come from functions which go from r to r or some subsets of r some subsets of r let us take it like that. So, let us say I will take one function f from r to r suppose here this function is something like this f x is equal to let us say 3 x plus 5 this is what function and I will take one more function g from r to r and where g x is let us say x square x in r here also x in r then we come to the general definition once again these are just examples which will illustrate the concepts. As I said right in the beginning in order to understand any concept or any definitions as soon as you learn a particular definition you should also look at several examples which satisfy that definition and also some examples which do not satisfy that definition then only one will understand what is involved in that particular definition. Now, what are the definitions here that I am going to look at first thing is we will say what is meant by saying that f is 1 1 or also sometimes it is called some books prefer to call it injective this is means the same thing this is this means what I was saying just now that if we if you are given an element small y in y then that particular element may have one or more pre images if it has only one pre image then it means that that function is called 1 1 that means given any y in y there can exist at the most 1 x for which f x is equal to f or which is same as saying that suppose you take 2 elements x 1 and x 2 and if f x 1 and f x 2 coincide then x 1 and x 2 must coincide. So, we will say we will take it that way f is said to be 1 1 1 1 injective if for all x 1 x 2 in x f x 1 is equal to f x 2 this implies x 1 is equal to x 2. In fact, this the dot this if x 1 x 2 if you take 2 elements x 1 and x 2 in x and if it so happens that their images coincide then those elements themselves must coincide. So, this must happen for whatever x 1 and x 2 is given for example, this function here if you take say suppose f x 1 is equal to f x 2 it will be in that 3 x 1 plus 5 is equal to 3 x 2 plus 5 and that will immediately lead to x 1 is equal to x 2. Whereas, this function is not 1 1 because if you take say x is equal to plus 1 or x equal to minus 1 still g x will be same as 1. So, this is not a 1 1 function then another concept what is said f is said to be on to what is meant by on to or also some whichever books call this as injective this is also called surject. In simple language it means that every element here has a pre image every element here has a pre image is said to be on to if for every y in y there exists x in x such that f x is equal to y looking at these two functions here this function is on to. Suppose, you are given any y here then it will be in that f x is equal to 3 x this will f x is equal to y this will be in that 3 x plus 5 equal to y and you can easily solve this and write x as y minus 5 by 3. So, that shows that f is on to no this is not on to this is not on to because for example, suppose you take y equal to minus 1 suppose you take y is equal to minus 1 then there is no x in R for which g x is equal to g x is equal to y if a function is both 1 1 and on to it is also sometimes called 1 1 correspondence 1 1 correspondence 1 1 correspondence or it is also there is another name for that is bijection bijection or bijective function question is called bijective function or all just called bijection if it is both 1 1 and on which means for every element here there is precisely one element here and vice versa for every object there is exactly one image and for every image there is exactly one pre image this is an example of bijection this is both 1 1 and on to this is neither 1 1 nor on to one more concept which is related to this we have talked of what is been by domain of f and co domain of f we should also say what is spent by range of f range of f. So, range of f is nothing but it is the set of all images it may or it may happen that every y every small y in y is image of something but there may be some elements which are not images of this. So, you exclude those elements that is just collect set of all images then that is called range of f. So, what is the range of f there is we can say it is a set of all elements of this form f x or x in x or which is the same thing we can say that you can set of those y those small y in y such that there exists x in x such that f x is equal to y. So, this range of f this is always a subset of the co domain range of f is always a subset of the co domain and these two sets coincide if and only if f is on to that is range. So, we can say that f is on to if and only if range of f is equal to y range of f is equal to y in fact that is the same thing said in the said in different languages. Now, for example if you look at this example here what is the range of g here range of g is because it takes only non negative values. So, we can say the range of g is this 0 to infinity which is not same as r is not same as r. So, this is not an on to function where the range of f is r because range of f is r given any such set x or y you can always define some functions. For example, I can define a function which is which is called i suffix x i suffix x from x to x this is called identity function on x. That means x goes to itself what is meant by identity function on x we can say that i suffix x apply to x is equal to x for x in x. Similarly, one can talk of what is meant by i suffix y that will be a function which will go from y to y and it will take every element of y to itself. Similarly, one can think of what is meant by constant function what is a constant function it is suppose you take some element y not here send all elements in x to that fix element y not that is a that is a constant function that will obviously not on to unless y contains just one point of this. Now, when the function is both verb on on to or when the function is a bisection then you can define a function which goes from y to x which in a certain sense what is called inverse function that is so let us come back to this. So, suppose f from x to y is a bisection that means it is one one and on to what does it mean it means that given any y any small y in y there exist a unique small x such that f x is equal to. So, suppose I call that x had some g of y then so we can say that define f inverse from y to x as follows from y to x as follows let take y take small y in y then since first of all since f is on to take y since f is on to there exists x in x such that f x is equal to y since f is one one this x is unique there cannot be more than one x. So, since f is one one x is unique such x is unique this is unique. So, define that x as inverse of y define that x as inverse of y define f inverse of y equal to x f inverse of y equal to x. So, this function f inverse from y to x that is called inverse function inverse is called this f inverse from y to x this is called inverse function we shall come to the properties of inverse function just in a moments. But before that let us also come to one more concept which is very frequently used in analysis and what is meant by composition of functions this is here idea is as follows suppose you have three sets x y and z let us say all these three are non-empty sets and suppose f is a function from x to y and g is a function from y to z. So, these two are functions then we can compose these two functions and form a function which goes from x to z that is we suppose we call that function h which is denoted as g composed with f this will be a function which goes from x to y sorry x to z and that is defined as follows that is suppose you take some small x here then look at f of that look at f of that then that will be an element in y and then apply g to that apply g to that then that will be an element in z because you define that element as h of x. So, what is h of x is g of f of x that is called a composition that is called a composition and remember here that composition is not a commutative operation that is you can talk of g composed with f, but f composed with g may not be defined at all if x y z all of all three are different sets then f composed with g may not have any meaning. It is also possible that g composed with f and f composed with g both are meaningful both can be defined, but still they can be different. So, now coming back to this inverse function what is the properties of this inverse function it is the following that is suppose we compose f inverse with f that is f goes from x to y and f inverse goes from y to x. So, suppose we look at say f inverse composed with f f inverse composed with f is it clear that will go from x to x because f f takes an element x from x to y then f inverse brings it back to x. Now, suppose you take x it will be it will be f x and f inverse of f x what will be f inverse of f x by this definition it will be same as x. So, in other words f inverse composed with f is nothing, but what we had called identity function on x f inverse composed with f will be nothing, but the identity on x similarly if you look at f now in this case we can also compose this way f composed with f inverse this is also possible. Now, what happens f inverse takes an element from y to x and f brings this back to y. So, this will be identity on y this will be identity on y. So, again you see if x and y are different sets f composed with f inverse and f inverse composed with f those will be two different functions of course both of those are identity functions, but identity functions on different sets. Now, related to this there are a few more things we also want to know till now we have talked about what is meant by image of a point. We can similarly talk of what is meant by image of a set similarly we can talk of what is meant by pre image or what is also known as more frequent is inverse image of a set. So, to do that let us take some subset a of x then we talk of what is meant our f from x to y is a function. Now, given a subset a of x we want to talk of what is meant by f of a. Now, this is clear how it is defined you take all elements in a and take their images and collect. So, that will be a set that will be subset of y that is called f of a. So, this is nothing but we will say set of all f of x such that x belongs to a or another way of saying that is it is set of all those small y in y such that there exists x in a satisfying satisfying f x is equal to y. This is called image of a under f image of a under f similarly one can look at just as instead of taking a subset a of x I can start from some subset b of y. So, I can start some subset b of y by the way here a can be an empty set also in which case f of a also will be empty because there will be no nothing here and before going to this let me say there are a few obvious things about for example, suppose you take two sets a 1 and a 2 then it something like this is clear f of a 1 union a 2 that is same as f of a 1 union f of a 2 etcetera. And again this is not necessary for any two sets this two for any in general we can say this f of union a I belonging to some indexing set I this is same as union of f a I small I in big I small I in big I whereas if it comes to intersection we can say only this f of intersection a I I in I this is contained in intersection f a I belonging to I but the two sets may not be equal two sets may not be equal and one can give examples where the where the this is properly contained in this it is not very difficult to give those examples. Now, let us come to the other concept just as here we have started with taking a subset a of x now this time let us start from taking some subset b of y. So, suppose b is a subset of y then we define what is meant by the set of all pre images of b or also sometimes called inverse images of b and that set is loaded by f inverse b of course, one should be one should understand here is that we can talk of f inverse of b even though f inverse is not defined remember f inverse is as a function is defined when f is 1 1 and on 2, but we can define f inverse of b even when f is neither 1 1 not on 2 because this is a set what is f inverse of b it is the set of all pre image that is it is a set of all x in x such that f x belongs to b that is take all elements in b for which there is some pre image that of all x in x such that f x belongs to b this is called inverse image of course, if f inverse as a function exists that will coincide with whatever we would mean by because f inverse will be a function from y to x and f inverse b will be have the same meanings f inverse y for y in b, but this is defined even when f inverse does not inverse image of course, set is defined even when f inverse as a function may or may not be defined and inverse images behave in a much better manner when we take unions and intersections what is let me just state this property here. So, suppose we look at properties something similar for example, we would suppose we take a family of this a family like this family b i where i in i and each b i is a subset of y each b i is a subset of y then we would like to say how this inverse images behave with respect to intersections and unions. So, what we can say is the following f inverse of union b i small i in i this is same as union f inverse b i small i belonging to b i and similar thing is true about the intersection that is one property second property is the following f inverse of intersection b i small i belonging to i this is same as intersection of f inverse b i small i belonging to this are simple exercises elementary set theory just we have to look at the definition of both sides see the properties of intersection unions and their definition and show that if you take any element here that belongs to here and similarly if you take any element here that belongs to the left hand side all right. So, that is about the elementary properties related to functions and subsets now let me again remind you why did we come to the discussion of functions because I said that if you want to define the product of infinite family of sets you have to take the make use of the concept of functions. So, where does that come into picture. So, let us again come back to that question. So, suppose this is a family suppose script f is a family a i i belonging to i and we want to define what is by and by product of a i product of all these sets small i in i now let us look at the analogous thing about the finite sets and what we shall do is that we shall rewrite or reinterpret our definition of the Cartesian product of two sets in a slightly different manner and we shall observe that that different manner is capable of generalization here also. So, suppose we had only two sets a 1 and a 2 then we would call a 1 cross a 2 we will call a 1 cross a 2. So, what will be a 1 cross a 2 that will be the set of all elements of this form x 1 x 2 where x 1 belongs to a 1 and x 2 belongs to a 2. Now, suppose I want to imitate this then here I cannot write pair or triple or because I do not know how many of them are there right it will be something like this will depend on what this set i is there, but what I can do is reinterpret this in the following manner see here there are two sets by indexing set here is i that contains just two symbols 1 and 2 and suppose I consider of I interpret this x 1 x 2 as the values of a function as the values of a function what function it is a function which says 1 to x 1 and 2 to x 2. So, that is so we can say that is a function I will call that function f it is a function it is goes from i to a 1 union a 2 a 1 union a 2 with what is the property? Property is the following that is f 1 belongs to a 1 and f 2 belongs to a 2. So, this f 1 belongs to a 1 and f 2 belongs to a f 1 is my x 1 and x 2 x 2 is nothing but f 2 right. So, for each such pair x 1 x 2 I can say that there is there is a function f says that first coordinate is f of 1 second coordinate is f of 2. So, suppose I collect all such functions which have this property suppose I collect all such functions which have this property then I get all such coordinates here. So, that is nothing but the cross product. But now this is a concept which is capable of. So, let me just rewrite here what is this we can say that this is nothing but the set of all functions f from I to a 1 union a 2, but not all functions what should happen is that f of 1 should be in a 1 and f of 2 should be in a 2 f of 1 should be in a 1. So, satisfying this f of 1 belongs to a 1 and f of 2 belongs to a 2. Now, this is something which I can generalize here I cannot talk of pairs or triples or anything, but I can say that I will consider all those functions which go from this indexing set I to union of the family you know the family which are already defined and what should be the what should be the property for each index I f of i should belong to that corresponding set a i. So, we will say this is this is the set of all f from I to union a i I to union a i satisfying satisfying f of i belongs to a suffix i and this for every i in i. Remember that is what is happening here every this i contains only two symbols 1 and 2 f 1 belongs to a 1 f 2 belongs to a 2. So, here we do not know how many i is there it can be 2, 3, n countable infinite or uncountable anything. So, this should happen for every i any function which satisfies this such a thing is called a choice function by choice function because it chooses one element for each i it chooses picks up an element on the corresponding set a i. Now, the whole question is how do we know that such functions exist how do you know that such functions exist and that is where we come to the basic foundations of set theory as you know in mathematics you have to start from some basic concepts. In each subject there are some basic concepts which are called undefined terms like for example, if you want to define what is meant by set and then you try to define that is a collection of objects then somebody may ask what is a collection then you go to define what is meant by collection again you will have some other word which will have a similar meaning. So, that way one end up in circles. So, circularity of the argument. So, in each discipline of mathematics there are some terms which are left undefined those are basic terms and similarly there are some axioms that is some things which we are starting hypothesis or starting axioms. So, in set theory we do not define what is meant by a set or what is meant by an element or what is meant by being a member of a set we assume that those things are understood. So, there is so the set theory similarly starts from some undefined terms and some axioms and that the choice functions exist is one of those axioms it is a very famous axiom it is called axiom of choice I will just take this axiom of choice and then maybe with that we will stop for today. What is axiom of choice it is the following if the axiom of choice means the following given a family basically in simple words it simply means that choice function exists if each of this a i is non-empty given a family a i i belonging to i of non-empty sets there exists a function we will call choice function then that is why this axiom is also called axiom of choice function f going from i to union a i small i in i such that f of i belongs to a i for every small i in i another way of saying the same thing is that what this means that if each of this a i is non-empty at least one such function exists that means if each of this a i is non-empty this product is also non-empty in that that is another way of stating axiom of choice and that is also given very frequently in some books. So, equivalently we can say that equivalently if each a i is non-empty each a i is non-empty then the product is also non-empty product a i i belonging to i is also non-empty or in more simpler words Cartesian product of a family of non-empty sets is non-empty that is one version of axiom of choice we will stop with that.