 So, let us recall what was Holder's inequality. So, we had a number p between 1 and infinity and q such that 1 over p plus 1 over q is equal to 1. Then for every a1, a2, some an belonging to R and b1, b2, bn belonging to R, sigma of mod a i b i i equal to 1 to n was less than or equal to sigma mod a i to the power p raised to the power 1 over p i equal to 1 to n, sigma mod b i raised to the power q raised to the power 1 over a. So, this quantity we had called it as norm of the vector a, bth norm and this was called the norm b, the norm where you consider the vector a to be a1, a2, an and b is the vector a1, b2, b. So, this was the Holder's inequality, which is a generalization of Cauchy-Schwarz inequality, which is when p is equal to q is equal to 2. Using this, we proved Minkowski's inequality of course on Rn, namely norm of a plus norm of b, p is less than norm a. So, that gave us as a consequence of this, this gives a metric on Rn, namely distance p, the vectors a and b is equal to norm of a minus p, for a d belonging to Rn. So, that gave us the notion of distance, generalizing the notion of the Euclidean norm when p is equal to 2. So, for every p between 1 and infinity, one gets a notion of a distance. What we want to do is, we want to extend it further than Rn. So, we had started doing that. So, let us consider R infinity. So, that is all sequences. So, we can consider this as the space of all sequences, real sequences. Now, of course, if you want to copy this notion of the norm, which may not make sense, because the number of terms in that summation become infinite. So, one has to restrict as we saw. So, we look at what is called Lp. So, that is all x in xn, all sequences size that norm of mod xi to the power p, 1 to infinity is finite, that this sum is finite. So, let us just put the claim, which we had already looked at. But anyway, let us prove it again that Lp is a vector space, real and for every sequence xn belonging to Lp, define i equal to 1 to infinity raise to power 1. So, we are just copying the norm of Rn, the p norm in Rn and of course, we have to restrict it to all sequences, which are, it must be power. This is a series of non-negative numbers, which is convergent and it should be finite. So, we will also look at series, soon convergence of series, but for time being. We are saying that the partial sums converge. So, this is finite. So, the claim is a vector space over R. So, that means what? If we define this, then 1 for x, y belonging to Lp. So, what does it mean? We want to say that Lp alpha x plus beta y also belongs to Lp and the triangle inequality holds. Namely, alpha x plus beta y is less than mod alpha times norm of x beta times norm. So, basically the idea of the proof goes as, in the case of Rn, first step should be to extend holders inequality from Rn summation 1 to n to 1 to infinity and then using that proof, Minkowski's inequality, because that proof does not require anything else other than the holders inequality. So, let us just prove holders inequality. So, this needs the holders inequality for Lp. One should say not Lp for R infinity. Let us write. So, what does it mean? That means for sequences x, xn, n. So, what does holder inequality in R infinity mean? It is the following in R infinity given two elements such that this x belongs to Lp and y belongs to Lq 1 over p plus 1 over q equal to 1. Then sigma mod xi yi i equal to 1 to infinity is less than or equal to the corresponding norm. So, that is mod of xi to the power p sigma i equal to 1 to infinity raise to power 1 by p and sigma i equal to 1 to infinity mod yi raise to power q raise to power 1 over q. So, that is perfect generalization. Namely, this is x to the power p and y to the power q, pth norm and kivoth norm. So, this is same as that. So, the idea is how do you extend that inequality? For Rn, we already have it. So, let us note. So, note. So, proof of this for every n, if we just take the sum from 1 to n mod xi yi, then this is less than or equal to by the holder's inequality on Rn, this is less than or equal to. So, this is by holder's inequality when the sums are finite up to n, then this holds. Now, this is something which is done very often in analysis. The left hand side here is less than or equal to the right hand side for every n. So, on the right hand side, let n go to infinity. Keep the left hand side n as it is. So, let n go to infinity in right hand side of star. That means, that sigma i equal to 1 to n, that is the essentially means, it is less than or equal to sigma 1 to infinity, raise to power 1 over p yi, raise to power q, raise to power 1 by q. Essentially, it means this is adding up non-negative numbers. So, we increase n, that will be less than or equal to the right hand side. So, sums will increase and for every n, it will be less than or equal to this quantity. Now, this holds for every n. Now, let n go to infinity in the left hand side of the inequality. So, on this side, left hand side, let n go to infinity, implying that sigma i equal to 1 to infinity mod xi yi is less than or equal to this quantity. So, sigma i equal to 1 on the right hand side, which is as it is before. So, the idea is that essentially, we are letting n go to infinity on this inequality, holders inequality for R n, but the justification comes from the fact that we can let the n go to infinity on the right hand side first and these quantities are finite by the given hypothesis. So, for every n this holds and we can let n go to infinity. So, that proves holders inequality. So, this is equal to… So, that proves holders inequality for R infinity in the sense that if you got sequences, say that the sequence x is p th power summable and y is q th power summable where 1 over p plus 1 over q is equal to 1. Then the corresponding result for holders inequality holds and using this, so as a consequence of this, one proves Minkowski's inequality in LP, 1 less than p less than infinity. Proof is same, there is no change at all other than writing using holders inequality at appropriate place. So, we will not repeat the proof and that says for every x, y belonging to LP, x plus y also belongs to LP and is less than or equal to… So, we get on LP a metric, so we get LP, namely for every x belonging to LP, you have x and y belonging to, you want to define a metric. So, let us write for x, y belonging to LP, define D of y to be equal to this y p. So, Minkowski's inequality says this is precisely a metric, it has triangle inequality property. So, what I am trying to show is that whatever you do in R 2, you can do the same thing in R n and same thing is in R infinity. There is a quite interesting mathematically to ask the question, so we had the real line, we had R n and we had LP, which is a subset of R infinity. So, we had the notion of absolute value, we had the notion of norm of x to the power p and R also, we had the norm of x to the power p. Basically, here it is 1 to n and here it is 1 to infinity. The interesting thing is one can go beyond this and what should be. So, the idea is this R infinity treat. So, here is the R infinity as the set. So, x, what is R infinity? That is a set of all sequences. Let us interpret each sequence slightly differently and you will see how this change of interpretation. So, let us look at all functions defined on natural numbers taking values in R and what is f of n is denoted as x n. So, a sequence is treated as a function from the set of natural numbers to real numbers. So, every n goes to a point in R n that you call as x n. So, when is a function known completely? If you know its images, so knowing a function f is same as knowing x n for every n. So, that is the interpretation for sequences treating a sequence of real numbers as a function on the set of natural numbers. And here this infinity, so infinity is equal to how many elements are there in natural numbers? They are countably infinite. So, that is the infinity. I hope you all know what is called countable infinite and the set of natural numbers. You want to say how many are there? You assign a number to it which is called LF naught and it is denoted by this symbol which is called LF naught. LF is a Greek letter and naught is, so this is in some sense the first infinity. You count 1, 2, 3, 4, n, go on counting and you reach something, visualize something infinity. So, that is counting and going on, not stopping. So, this is called, so n is countably infinite. So, one says L is countably infinite and the cardinality of it or the number of elements is LF naught. So, this is, you can think of as infinity. So, instead of R infinity, we like to write it as, it is better to write it as LF naught. So, this is a better notation for R infinity. And this LF naught is the cardinality of the set of natural numbers that is the indexing set and that is the domain air coming. So, this gives us a way of extending. So, define, instead of natural numbers, let us put something, any set X and what is this? So, these all functions X to R, f of X belongs to R for every X. So, we are just R to the power infinity, sorry, LF naught or this is same as R to the power n, natural numbers, if you want it as a set. So, generalize it, just replace. So, this is the definition, what is R to the power X, where X is any set. So, one can interpret that way. For example, if X is equal to R, so what is R to the power R? That is all functions from real line to real line. What is R to the power AB? That is, all functions f from interval AB to, so there is another way of saying what is this object. So, now, let us look at this R X. So, consider M s is equal to R to the power of X. So, trying to illustrate the way mathematicians think and generalize. So, R infinity, we had and we have defined R to the power X. So, these are all functions f from X to R. I want to copy that idea, L P, L 2. I want to copy that ideas. We had the notion of L 1, we had L 2, that is the ordinary Euclidean distance, we had L infinity, that is the supremum. You can try to copy all of them on this set now. So, let us try to copy this. So, let us first one for a function f X to R. I want to define what should be this thing. For X n, for the function defined on R n, so what is R n? That is same as f defined on 1, 2 up to n, 2 R, n components. What was this thing defined for a function, for a vector? We looked at the supremum of the components. For a function, what are the components? The finite, they are as many as. So, you can treat for every X belonging to X, f of X as its component, X th component, you can think it off. So, if you think it a function as a vector with as many components as the number of elements in the set, then for every X, f of X, the value is the component. That is what is happening in sequences, that is what is happening in vectors. So, look at the X th component, look at the mod of that and we want to take a supremum. So, let us take the supremum of this, where X belongs to X. So, copying that supremum thing, but the problem comes, this supremum may not exist, because we know the completeness property of real numbers says, every non-empty subset of real numbers, which is bounded above will have a supremum. So, this set may not be bounded above. So, 1 has to restrict now, instead of R X. So, restrict. So, look at all functions X to R, such that supremum X belonging to X mod f X is finite. Now, you see, automatically those similar conditions we had put earlier, when sigma mod X i square is finite, X i to the power p is finite. So, for functions, we should put this condition. So, what are such functions? If a function f X to R, whose supremum exists, that means it is a bounded function. That is same as this is equal to set of all bounded functions on X. So, one just writes m X R. You can write any notation, you can write B here to indicate, let us write B instead of m. Let us write B X that may look like ball of radius something. So, let us write some funny B called script B. How do you write script B? Script B X R is all functions f X to R f bounded. And for any function f belonging to B X R, we can define to be equal to supremum X belonging to X mod f X. And this becomes, this is a norm on B X R giving a metric. So, it gives a metric. So, what is a metric? As well, once norm magnitude is defined, the metric is, so d infinity f G is equal to norm of f and G belonging to B X R. So, basic idea is defining a norm absolute value for