 We will from the next class try to see a method which will be very familiar to many of you, you have already done it. I mean from donkey's ears you have been solving these equations. So initially the kind of stuff we will be covering will seem very familiar to you particularly those of you who have done a preliminary course on linear algebra you might find the first few lectures are kind of repetitive. But it is hoped that as soon as we get into a more formal description of stuff you will begin to see the application of these ideas how they are abstracted out right. So we will no longer deal with just systems of equations that look like matrices and stuff. We will see that you will define generic objects like vector spaces ok. And when we deal with vector spaces we will see that the same ideas that we have learned for solving these linear systems of equations they can be just extended to problem domains that you might not have thought of. For example you can be dealing with spaces of functions and you can talk about how close is one function to another function right. These sort of ideas will come in when we describe things like norms. Speaking of which by the way I can also say there is another kind of algebra you might have already done without maybe knowing about it formally which is something called a Banach algebra. Anyone's heard of this? At least maybe some of you have heard of something called a Banach space. Remember what a Banach space is putting you through a lot ok no problems. So you see that when we learn linear algebra the notions that you acquire from this will help you in learning and also grasping the sense of other more sophisticated domains of mathematics. So this is like a stepping stone in that sense. So what is a Banach space? I will be using some terms please allow me to do that cut me some slack here. So when we talk about a Banach space right we have the following ideas in mind it is a normed vector space if that doesn't make sense at all welcome about this course. Normed is basically the abstract idea of a distance we shall see how it can be extended to arbitrary space vector spaces not just Euclidean spaces. So that's norm for you for the time being just keep that in mind. What is a vector space? Well that is a whole new kettle of fish altogether. I will not trivialize it by just wishing it away and saying oh you know it's this or that. We will go through a formal description of what a vector space is but that is why it is important to know linear algebra because then you will be able to appreciate what these things are. But here's one more notion that again you might have been familiar with earlier that is complete anyone knows what complete completeness implies. Let's just do this as an exercise in some mathematical terms and figuring things out okay. Don't think of this like some formal syllabus kind of thing that you're doing here. This will not be even discussed in this course okay but just you know it's good idea to just exercise our muscles a bit. Do you know what is a complete space? Have you heard of it? Yes. So that again brings us to the idea of what is a Cauchy sequence yeah. So what is a Cauchy sequence? It's after the name after the famous French mathematician Cauchy sequence. I'm sure you know what a sequence is right. Sequence I'm not going to define here right. It's just an arrangement of objects. Most cases you've encountered real numbers right. So Cauchy sequence is a very special kind of a sequence. What is it? So the way to look at these things these definitions is to look at them piecemeal term by term okay. I could already see some of you like you know throwing your hands up in the air and say oh here we go again with this Cauchy sequence and completeness and no tired of these jargons. Well irony is you are in the math department so yeah live with it. So this is a Cauchy sequence. What's the Cauchy sequence then? Here goes. Suppose you have this sequence some xk and heaven knows how many terms probably infinite but countably infinite terms Cauchy sequence. So suppose this is a sequence. So suppose is a sequence okay. If again we'll get into some domain that you may again think oh again come on but just bear with me a while. If for every positive epsilon okay epsilon, epsilon delta the continuity arguments you've heard again from time immemorial right. If for every positive epsilon there exists get used to the symbols. So maybe that's why it's a good exercise today right because we'll be using this in our proofs these symbols at least. There exists a natural number right n yeah. So this is a set of natural numbers. So some natural number begin in the set of natural numbers such that for all I'll just let you absorb those symbols slowly. For all m small m, small n greater than n the distance between which we denote by this xm minus xn is less than epsilon okay. That doesn't complete it. Then x is a Cauchy sequence seems a little familiar now. That's Cauchy sequence right. It's a Cauchy sequence or how to view this? I always try to view these things like a game of cards. So it's like two players playing one player throws in an epsilon yeah. If I have to trump that card I have to find my trump card in the form of n so that I can find a point in the sequence somewhere which is this big n to the right of which it doesn't matter which two terms I pick. This is very important. It's not two consecutive terms remember. It's not xm and mxm plus 1. It's not how the consecutive terms are varying. In fact you can actually take as an exercise a term like this okay. A sequence like this where you will see that each consecutive terms the difference gets small but this is not Cauchy. You can check. So the point is that no matter which two terms I pick to the right of this I can always fit them in to be closer and closer to each other. Closer than a certain predefined card that has been played or if you like football maybe this is the goalkeeper you are setting before the goal. If you set an M.E. Martin is you are setting a very small epsilon yeah. You wouldn't want to give me a goalkeeper who's incapable right. Then I can just play any shot blindfolded and probably it will go to the goal. Maybe not blindfolded but anyway in the right direction it will go for a goal. You want to set your best goalkeeper. What's the best goalkeeper for you? Give me a smaller and smaller epsilon to challenge me because if you allow me a large epsilon I'll have my job very easy right. I can always choose some number. It is only when you choose epsilon to be smaller and smaller that my job gets tougher. So your job is to throw the most challenging epsilon possible at me and if I am still able to spit out some N so that this is true then I prove that the sequence is cauchy and won the game. If not then you win the game it's not cauchy. That's the way to view this okay. That's the way to view these definitions. Don't think of them like some terms being thrown in stewed aside just for confusing people. It's there's enough clarity there that's very important. Now the point is is every cauchy sequence convergent. Convergent in what? That is the important thing. It seems like cauchy sequence is a very nice kind of a sequence that should converge. But the point is you look at a sequence like this 1 by N over the interval the open interval 0, 1 and you see that 0 and 1 are not part of this interval but this as limit N tends to infinity tends to 0. So this 0 is not part of this. So over this interval this sequence even though it's cauchy it doesn't converge. So therefore this is an example of a set that is not complete because by definition a complete space or complete you know if anything has a complete structure like the one we were talking about a while back this Banach space every cauchy sequence must have its limit over there inside of that set must be contained in that set. If it's not contained in it yeah if there are some cauchy sequences whose yeah which do not converge to points inside that you cook up cauchy sequences from elements in that set or elements in that space it's a space because it's got a norm it's got a structure. You cook up cauchy sequences from elements in that space and if those sequences do not converge to points inside that space then it's not a complete space. So in order for something to be a Banach space it has to be a normed vector space that is also complete so that every cauchy sequence converges to a point therein but that's just a definition. So we have the definition of a Banach space so fully of course we haven't cleared what is a Banach what is a vector space what is a norm those we will learn by the time you're done but why is this Banach space of importance because some of you have done some courses maybe on control theory or things like that you have actually dealt with Banach algebras which are defined on Banach spaces. So what is a typical Banach algebra just like we've seen a sigma algebra while back a Banach algebra is basically a Banach space. So b is a Banach space and you have a multiplication operation therein yeah so when I say a multiplication operation the way we describe it is this is not such that okay this takes objects two tuples or two objects from b and after this operation it spits out another object in b that's all that this symbol means okay don't read too much into it just try and understand this I'm just trying to give you a light introduction to some of the symbols and things that we shall be using okay what this means is that this is an operation that acts on two objects of this set at a time and spits out another object in that set in general it could be a set here it's a Banach space. So this multiplication operation when you define on a Banach space so of course in a Banach space there's already an addition that's sitting that addition operation is there now this multiplication operation satisfies the following property one there exists an identity of course in the Banach space such that I operated with f is equal to f for all f in the Banach space so there is an identity element in this Banach space. So in order for this Banach space in along with this multiplication operation to qualify as a Banach algebra okay these are the rule basis the first rule that you must have an identity element identified corresponding to that multiplication otherwise tomorrow you can call some operation a multiplication and give it your name and say oh this is a Banach algebra but I'll ask you to prove hang on what is the identity element with respect to this operation that you've named after yourself yeah and then you say oh this is the identity okay I'll say okay first check done what's the second check whatever multiplication operation that you have named after yourself in this supposed Banach space yeah must also satisfy the associativity property so for all g1 g2 g3 in that Banach space you must have g1 g2 g3 so this this denotes the multiplication operation is equal to g1 g2 g3 so basically the order of the operation it doesn't matter whether you do this to first and then this or whether you do this first followed by the concoction of the other two it doesn't matter that's the associativity property we will come across these associativity and all these properties very shortly in our course to in a different context but it's important to know the names this is tantamount to the existence of an identity this is tantamount to the associativity of the operation there is also another property there is also another property which is with respect to the scalars so for alpha is a scalar so suppose alpha is a scalar then for g1 g2 coming from the Banach space we have alpha acting on g1 subsequently acting on g2 is the same as alpha acting on g1 and g2 together so even this is some sort of associativity with respect to the operation of course this is not the multiplication operation because this is just a scalar multiplication we will make distinctions between the multiplication that we define and the multiplication that is already known which is the scalar multiplication it's very important to pay very close attention to these distinctions otherwise you lose the plot pretty soon okay remember this multiplication operation and this is not the same so this is a non-trivial assertion okay look this operation is the multiplication we have defined maybe I should use a different color perhaps so this is an operation that is pre-existing already that is already understood similarly this is an operation that is already understood it says that when you have this acting on this it's the same as you pull out the scalar get this operation done in terms of the new operation you have defined as you have defined and then do the known operation it doesn't matter still okay it's it's a non-trivial assertion it's not very obvious it's not you know something very similar in flavor to this because there are two operations that are going on here okay so this is the third property I presume there's a fourth property which is that because it's a normed space the Banach space so you have under your multiplication that you've defined again very important this is less than or equal to this is the submultiplicativity of the norm so again I reiterate this multiplication operation is different remember this operation is the one that you have just defined this M but this multiplication operation is between two scalars because the norm spits out a scalar right so this is non-trivial okay so this the identity there's the associativity there's a submultiplicativity of the norm there's this associativity with respect to the scalar and finally there's the distributivity with respect to addition so if you have g1 operating on g2 plus g3 you might just as well say that's the same as g1 acting on g2 plus g1 acting on g3 okay of course it is understood I'm not writing it that all of these are part of the Banach space g1 g2 g3 so on so any algebra any operation that you define on a Banach space any multiplication operation that you didn't define on a Banach space which means these conditions results in a Banach algebra and the moment you have a Banach algebra a lot of beautiful things happen one classic example of Banach algebra is for those of you again as I said who have done a course on control theory when you take transfer functions yeah this transfer functions this GSS not just the scalar ones I mean right you have matrices of transfer functions for multi-input multi-output systems those are Banach algebras okay so very very important idea in control theory this idea of Banach algebras right so you have done these algebras although we have not given them these monikers before right but from now onwards since we are doing a little more formal abstraction in this course we shall be delving into the names of these objects trying to understand the definitions clearly