 This is for L infinity. Supposing you have, let us try to see what happens if you look at this. Let us look at even equal to 1, infinity we have already defined. So, we want to look at all functions from x to r, such that if I look at the component, this is the x th component. What was r n mod a 1 plus mod b 1 to the power p and they are finite. Some of them, some of the p th powers of the components are finite. In r infinity, it was some of the p th powers of the components, all the components is finite. But if we do not know what is x, how do you sum it up? I want to sum of mod x, I want to add up all these things. We only know we can add up all these things when they are numbers. x is arbitrary z. So, let us specialize x to be equal to for the time being, let us look at it a b. Just to understand, special case, x is a closed bounded interval. Now, mod of f x is a real number. Now, x below belongs to x, which is a b. Now, we can add up all these things. So, what should addition f of x, when x belongs to a b should mean? When x was a finite thing, 1 to up to n, I could write f of x i summation 1 to n. When it is 1 to infinity, we could say those which are finite. So, what should summation of this mean? When it is unbounded, a b, the number of elements x belonging to a b is not countable in finite, it is much more. So, it is natural to put integral a to b. Integral is a sum. Integral is a sum on a b. So, even sum, think of that. So, we should put this to be finite. So, not all functions we can consider, we can only look at say functions whose integrals are finite. So, essentially what we are looking at is, we are looking at all. So, let us look at r a b. So, it is f x to r. What is the meaning of this finite? So, Riemann integral is finite. So, all r a b is the Riemann integrable functions. f is Riemann integrable on a b. If f is Riemann integrable on a b, mod f is also Riemann integrable. We know that. So, this quantity is defined and for every f belonging to R a b, I can define norm of f 1 to be equal to mod f x dx a to b. So, keep in mind, it is same as adding up all the components, absolute values of all the components. Notion of what is addition depends on whether it is finite, whether it is infinite, then it becomes a series. If it is uncountable, then you have to interpret it as an integrable. But basically, we are generalizing absolute value. Absolute value on R, absolute value on R to summed up, absolute value on R and summed up, absolute value for functions summed up, copying everything. And one can show that this gives a metric. So, what is the metric? So, f and g belonging to R a b, you can define integral mod f minus g dx a to b. So, that is the distance d 1 f g. So, this is what is called L 1 metric Riemann integrable functions. And proving it is a metric is because absolute value, so d f gh does not matter. Absolute value of f of g is less than absolute value of f h plus h g. So, integral will be less than or equal to. So, that is not a difficult thing to prove. It is a metric. Same proof works. It is a metric. L infinity, we have already seen, it is a metric. L 1 L infinity, what is left in between is L p. You can define. So, I will not go very much detail into it. So, in fact, for every p between 1 and infinity, you can look at what are called R a b with L p metric or one should not say L p, one should not say p metric. Why to put L p? p metric. So, what is the p metric? For f and g, the distance p f and g is nothing but integral mod f minus g raise to power p dx a to b. And the proof is same as before. Holders in equality, you use Holders in equality for mod that a raise to power p, generalization of a p g p mean. a raise to power p, b raise to power k less than or equal to same, idea works, everything. So, we will not go into it. But, this is just for exposition. I am telling you, because you may come across these things later on in your other courses. So, this is called a p metric. Why we want to define metrics on such complicated spaces? One may think of R 1, R 2, R n. Good enough. Why we should go to functions on defined on something? What is the need for such things? So, the need arises, probably that may be a good way of looking at. So, let us note, let us look at c a b. So, what is c a b? That is all functions on a b to R, f continuous. Look at all continuous functions on the interval a b. Every continuous function is remain integrable. We had proved that. So, c a b is a subset of R a b. In fact, is a proper subset of R a b. There are two things one would like to ask here. I will keep analogy with real line and rational form a subset of it. I am going to motivate this, what I am going to do with these things, keeping in mind Q in R. Now, rationales, we said in the very beginning, between any two real numbers there is a rational and that property we called as a denseness of rationales, which is equivalent to saying if you take any interval that must have inside a rational. That is, rationales intersect with every free open interval. That was denseness or another way of looking that was, if you look at this closure of Q in R, Q is a subset of R. If you look at the closure of Q that is whole of R, that is same as saying that every open set intersects. That means closure of Q is every real number is in the closure. Keep in mind what we had done, limit points close sets. So, closure of Q is R. So, saying denseness Q is dense in R is same as saying between any two real numbers. See, if you say between any two real numbers that means there is a notion of order between any two. But, if you take R 2, then what is the meaning of saying that between any two real, any two points there is a rational, does not make sense because there is no order on R 2. There is no order on R 3. So, how do you write denseness? So, there the denseness we saw it is interpreted as saying every open ball must intersect that set, then set is dense. So, in R n if you look at say Q n, so what is Q n? That is Q cross Q cross Q that is same as all vectors x 1, x 2, x n. So, that each x i is each x i is Q, then this is a subset in R n and Q n is dense in R n. That means what given any vector if you look at a ball around it of any radius, then it must have an element of this inside it. So, that is what denseness means. Now, this is on real line. So, on R n we saw what is denseness. So, you can ask on any metric space given any metric space, can you find dense subsets of it? Are there subsets which are dense? Can you define the notion of denseness on any metric space? So, you can define. So, let us come back to C A B a bit later. So, x any metric space, so we say y subset of x is dense in x if y closure is equal to x. That is the notion of denseness. So, you see that notion of denseness is defined in terms of closure. So, closure can be defined in terms of sequences. Every point of x is approachable by a sequence of elements of y or every open ball intersects y or the closure of y is equal to x. All those are equivalent ways of saying. So, example we said that Q n is dense. There is one sort of trivial example of extending the fact that Q is dense in R n. So, let us call it as 1. So, let us go back to that. So, the question is C A B. So, is C A B dense in R A B? Of course, denseness is with respect to a notion of a metric space. So, let us write in L 1 metric to be precise. The answer is yes and the proof is a bit difficult to give. And maybe in some of your courses later on you will find a proof of that. So, we will not go much into it. I am just keeping that this leads to study of general forms of… There is some general theories of integration which will be required to say analyze denseness. So, we will not go into it. But I am just giving you a exposure which we may come across. You may come across in the future courses. So, let us look at another question. There are many things I want to… Here is another thing. So, let us note on R n, on R A B are all vector spaces or even L p and so on, are all vector spaces with norms. Either it is this norm or this is p norm or we have basically looked at three types of norms. On all these spaces, we have L 1, L p and L infinity. They were norms. So, meaning what? They were like the absolute values. So, norms meaning the norm meaning say norm of x is bigger than or equal to 0, where x could be any one of elements or any one of this and equal to 0 if and only if x is equal to 0 and alpha x. Because it is a vector space, there is a scalar multiplication. And how does it behave with respect to scalar multiplication? That is same as mod alpha times norm x. And the third x, if you look at two elements, you can add them because it is a vector space. So, triangle inequality holds. And we had mentioned that norm induces a metric. Norm gives a metric and what is that metric? So, dx y is norm of x minus y. So, it is a general process. Given a vector space, for every vector, there is a notion of absolute value you can think of. And it has these three properties. And every notion of absolute value gives rise to a notion of distance. Study of such spaces are called normed linear spaces. Such spaces are called normed linear spaces. Such spaces are called normed linear spaces. It is a linear space, it is a vector space. There is a notion of norm defined on it. And norm behaves very nicely with respect to scalar multiplication and addition. So, these are the properties. So, such a space is called a normed linear space. And study of normed linear spaces goes into a topic called functional analysis. So, there is a separate subject normally called functional analysis. So, we have a lot of examples of normed linear spaces, LP spaces. What are its properties? What are its uses? And so on. There is a separate topic called normed linear spaces. We will not go into that, but what we will want to look at is the following. So, we have real line, the starting thing, which motivated us. Real line was a complete metric space. It was a metric space, which was complete. That was the big advantage that we could do everything. So, completeness in various forms. One way of completeness was every monotonically increasing sequence, which is bounded above is convergent. That is sequential completeness. Another form was a Cauchy completeness. A sequence is convergent if and only if it is Cauchy. That was equivalent to it. So, these are the various forms of completeness. So, complete metric space and gave us nice results. For example, the nested interval property was a consequence of completeness. We use completeness to prove nested interval property. Now, given a metric space x d, it may or may not be complete. Given a set x, given a metric, for example, let us look at example. So, what is the meaning of it may not be complete? A metric space may not be complete means what? So, that means there exist Cauchy sequences, which are not convergent. In a metric space, we have the notion of sequence, sequence being convergent, sequence being Cauchy, because we only need the notion of distance. Convergence means closeness. x n converges to x. So, distance of x n comma x goes to 0. Cauchy, distance between x n and x m becomes small as n and m go to infinity. Absolute value to be replaced by the notion of distance. That is all. Closeness. So, whenever you have the notion of closeness metric, you can do all that you do on sequences on the real line, r n and so on. So, there are metric spaces. When you generalize the notion of distance on a metric space, there are notions which are saying that the metric space under that metric may not be complete. So, a very simple example is, let us look at q as a subset of r. So, what do we do? What does that mean? We are looking at q with absolute value as a metric space. Look at the set of rationals. Look at the notion of distance given by absolute value of that number. Rationals is a real number also. So, we are restricting the notion of absolute value function from real line to rationals. And you can real line with the notion of absolute value. So, these are two different metric spaces. q is a subset of r. So, you can say this is a subspace. You can say it is a subspace. It only says that q is a subset of r and the metric of real numbers is restricted to metric on rational numbers. And we had example of sequences of rational numbers which are Cauchy, but which are not convergent. We had given such examples when we were looking at sequences. So, there are sequences of rational numbers which are Cauchy, which are coming closer and closer, but they do not converge to a rational number. Of course, they will converge to a real number by the completeness property. So, this is not complete. But what is happening? Rationals are a subset of real numbers as a metric space. They are not themselves not complete, but they sit inside r as a dense set. If you look at its closure, that is the whole of real line. So, one says, so here is something which we should look at. So, one says r d. So, q dense in r is interpreted as that r with the usual metric absolute value is the completion of q with absolute value. So, what we are saying is there is a bigger metric space. Example r d. There is a subspace metric q inside it. q is sitting as a dense set inside r. q is not complete, but r is complete. So, one says that real numbers is the completion of the rationales. Real numbers are obtained from rationales via completion. All limits are put inside it. They are completed. So, one says that the metric space r with the usual metric is the completion of the metric