So today we start with the Banach spaces. So we know already what is a Banach space. Is a normal vector space. So B is a Banach, B is normed vector space, complete. See, so this is a generalization of Hilbert. So clearly any Hilbert space, the converse is not true. We have an infinite dimensional exam Remember, we have shown that L1 is not Hilbert. We have shown this using the fact that the parallelogram identity was characterizing the scalar product. So this has a norm which does not come from a scalar product. We used, remember, two functions with disjoint supports. The characteristic function. So, but is Banach. Actually, the class of Banach space is very large, which is infinite dimension. Infinite dimension, what do you have to think? For instance, in R2 the analogy is to have on R2 an ellipse. OK, this is a scalar product. But if on R2 you have any other convex body, something like this, or this, or any convex body, which is not an ellipse, not a polynomial of degree 2, say, or whatever you can imagine. These are norms.  But they are not an ellipse. And so they do not come from a scalar product. Concern in differential geometry, the difference between Banach and Hilbert is the same that we have from a Thinsler manifold and a Riemannian manifold. Riemannian manifold has a scalar product on the tangent space, which changes with the tangent space. A Thinsler manifold on each tangent space has a convex body, changing from the point. Here we don't do geometry, so we have just only one body. So examples, very, very, very useful examples. The examples that we have, well, LP for P larger or equal than 1. So these are spaces of sequences. Simply generalizing the L2. So you can imagine, which is the norm. So is the sum. This is the norm on a smaller P. So these consist of sequences. Any point here is a sequence, x, such that this is finite. For any P. Of course, we also have small L infinity, which consists of sequences, which are bounded. So we have, is the set of real sequences, such that the L infinity norm, which is defined as simply the supremum of the absolute value, is finite. Is finite. All these, one can show that are complete. So one can prove that these are, of course, probably you are more used to define capital LP and capital L infinity. But these are also very useful spaces. In view also of our experience of small L2, we have understood a lot of things of infinite dimensional functional analysis looking only at small L2. So and small L2 is important because it is a standard model of separable in the best space. And for the same reason, somehow these are also quite important. Other examples. So if k is compact, say compact, then the continuous functions on k with the L infinity, with the sup norm. So f on k to r, continuous. And the norm is the supremum, which is a maximum, actually, of f, supremum of f of x, such that, which is a maximum. This is another example of Banach space. OK. Other examples maybe is, another example maybe is this, is continuous functions on small c0. Small c0 is the sequences such that, so infinitesimal sequences, which is maybe also this. So this small 0 here, so in general, in general, say, if you have a set, say topological set, let me call it capital X, then you can define this as the set of all continuous functions going to 0 at infinity, which means that maybe something like continuous, such that maybe for any epsilon positive there exist k compact, such that f of x is less than epsilon for any x, x minus k, something like this. So all continuous functions, such that they are very small out of compact sets. Maybe it's something like to say that they are 0 at infinity. Whatever we can mean by infinity here. But it doesn't matter. This is not very important. I mean, don't be, so effects of topological space, this is not very important. I mean, just this. So this is all continuous functions from n to r. So any sequence. However, with the property that it is infinitesimal. So if you don't like this. Other examples, so c0, l infinity with this notation. So with this notation, continuous functions from n to r. So continuous functions from n to r are just sequences, just a sequence, space of sequence. Because on n we have the discrete topology, so everything is continuous. Any sequence is continuous. So this means, so that l infinity actually, if you want, is at the bounded continuous functions from n into r, if you like. This is just notation. But notation that, however, do not confuse infinitesimal sequences with another notation, which is this. This is a different object that today I will not discuss. What can you imagine? What is this? The set of sequences. What does it mean with compact support? It's finite. So definitely 0. So this is another object, such that xk equals 0, definitely. But you know what I mean by definitely. So after some index, all components are 0. So that maybe, such that xk different from 0, only for a finite number of indices, set of indices, set. So only for a finite set of indices k, xk is non-zero. And then it is 0. So these are the continuous functions with compact support that should not be confused with the continuous function infinitesimal on n. So they are different. Actually, one could show that this is the closure of this. It is clear. I mean, if you take all possible limits, say, of sequences with compact support, then you don't get a sequence with compact support, but at least you get something which goes to 0. This is, I mean, natural. But do not confuse this with this. And indeed, this is another symbol. So it is 2c00, different symbol. So I don't want to talk about this. So we will say something maybe about this, but not about it. So we have seen that L2, so the first theorem that we, one of the first remarks on L2 was that L2 was separable, separable. What does it mean? It means that it meets a countable dense subset. Actually, separability is, so I mean, and L2 is separable and is the model of any separable infinite dimensional Hilbert space. There is this, maybe, remark. So L infinity is not separable. So this is a Banach space, which is non separable. And so this is already one new phenomenon. So that makes things more difficult. Can you imagine why it is not? So the idea is the following. It's not difficult, the idea. I mean, the idea is, supposite is, I'm sure that you have seen the other statement, capital L infinity of omega is non separable, right? But for me, infinite dimensional vector space are so difficult that I prefer to start with spaces of sequences. Can you imagine why this is so? Let us try to understand why. So concerning L2, the idea was to take, essentially, sequence, which are 0, definitely, but with entries, which are rational. Can you remember this? And so using this, since the tails go to 0, it was very easy to show separability. Can you remember this argument? So the idea was, well, you have to control the tails. But the tails go to 0, because the series, the infinite sum is converging. So actually, I more or less forget about the tails. But then I have just a finite number of entries, and then I have rational numbers, which approximate all these finite entries. This was the idea of separability of L2. And this argument was working exactly because of the norm. In L2, it was crucial that the infinite sum was converging, so that the tails of the sum was going to 0. Now, however, the norm is something different. The L infinity norm is something much, much different. Do you remember, for instance, in R2, finite dimension of this L infinity norm? Just to have some intuition, we have already said this in two dimensions, which is the unit ball. Please, think of unit balls instead of algebraic formulas. What is the unit ball? Just the square, unit square. I mean, the square centered at the origin with sides 2, minus 1, 1, minus 1, OK? The only thing that we can say in two dimensions is just that, why this is bad? Well, it is not uniformly convex. It is not strictly convex. Not only that, but it is also no smooth. Because we can imagine no strictly convex smooth sets. We can easily imagine. But this is worse. It is not only, it is piecewise linear, but it is also, it is no smooth because it has convex. But this is the only thing that we can say. It is finite dimension, nothing more. But this is infinite dimension. And so the idea is, assume we have, so let me denote it by qn. This is a subset of l infinity. And assume it is countable. It is countable because n is in n, and is dense. Now the idea is that, by the way, I can find a point x at a given distance positive from any of this. So let me denote it by maybe xn, this. So this is equal to qnn. This is equal to 0, or maybe 2, or 0. If qnn is, say, less or equal than 1, and qnn if is larger than 1, for any n. And so now I have, what do I have? I have x, which is a point of l infinity. It is clear that this is in l infinity because the n's are just 2 and 0. So this is clear an element of l infinity. So assume that I have a countable dense subset qn. Any qn is a sequence. So the index of the sequence is lower, as usual. Or less is our convention. So this is our xn. Now I take the distance between x and any element in l infinity. What can I say about the distance? Well, I have to take the supremum between the difference of the components. Well, if this component, the n component is less than 1 in absolute value, then this is 2. If it is larger than 1, then this is 0. And therefore, this is always larger than or equal than 1 for any capital n. Xn equal to qn plus 1. Is it plus 1? Ah, it's the same. It doesn't mean the same. It's completely the same. This is just to have, this is even easier because this has just two values, 0 and 2. Just so it is in mid. So this, the point x in l infinity consists of either 0 or 2, 0, 2, 2, 0, 2, 0, 2 and so on. Maybe it's easier even, so just two values. So is this clear? And what does it mean, however? It means, well, this is accountable, cannot be dense because I have a point in the space with distance at least 1 from the set. And this contradicts the density. So this shows a remarkable property of l infinity. OK. Now, I read a bit more quickly. We can go through, we can go to the definition. So assume that e and f are Banach spaces. Then we can define this. It's the set of all l from e to f, l continuous, l linear continuous. Our notation yesterday was when a is Hilbert, we have considered, we have used the following notation. And f was r. Our notation was, I think that we agree maybe to use the star. This was the notation of yesterday. So this is a much more general situation for two reasons. One is that e is not only Hilbert, but is Banach. And the second is that the target is not the field of numbers, but is another Banach space. So it's much, much more general object. OK? Fine. So, well, maybe home. So definition, maybe? Definition. So we know what does it mean that it is continuous. OK. This is clear. Because these are normed space, so that there is topology, there is, we know. But definition, what does it mean that l from e to f linear, what does it mean that it is bounded? Well, bounded means, as yesterday, is bounded if it takes bounded sets in e into bounded sets in f. So let me write it in a little bit roughly like this, but it is clear. I take a bounded set in the source, and then must be bounded in the target. OK? Or equivalently, so if it takes a ball, so. Equivalently, there exists a constant c such that l of x less than or equal than c of x for any x in e, which is the difference with respect to yesterday. Well, the definition is exactly the same. The difference is that yesterday here we just have absolute value of l of x. Because yesterday l of x was a number. Today is. So if I don't now specify which kind of norm I'm using, it is clear from the context. This is the norm of e, and this is the norm of f. Even if I'm using the same symbol, sorry, but it is clear. Is it, do you agree? So we don't need to make heavy notation in this case. It is clear. So please check that this is the equivalent way to say that the map l takes bounded sets into bounded sets. And check that again, like yesterday, we have equivalence between continuity and boundedness. So l is bound l linear. Then l is bounded if and only if l is continuous. The same proof, exactly the same proof of yesterday works. So really you do the same proof of yesterday. There is nothing new here. Now let me, ah, yes, maybe, again, I can like exactly, like yesterday, I can give a norm to l. So check that what it is interesting now is to take the smallest possible c for which this holds for any x. So, again, this is, and again, it is like yesterday, the statement that we had yesterday, OK? So now, let us do some exercise, because in infinite dimension everything is not so intuitive. So I think it's good to have some intuition. So we have already observed that we know a lot about a linear map if we know something about the kernel. So maybe an exercise here. So let l into f linear, then l is continuous. So there is, now, so we can recognize, given a linear map, whether it is bounded, looking just only at the kernel. Now this is not, I mean, OK, let us try to, if you want to help me, I mean, that there is another which is immediate. Everybody knows. Well, this is immediate because 0 is closed. So, OK, so now let us try to show this. What can we do? So we can assume that l is not identically 0, because, so assume that l is not identically 0. Yes, so, and then, so, OK, so take x out of the kernel. So we can do this, because now, so now we are assuming that l is not identically 0, therefore this is not empty, and so we can take a point outside. So what he is saying now, since this is point outside, we know that it is non-zero, say, so there exists a ball, so now we know that this is open. So we can take a ball, there exists a ball, say, br of x, center the text of radius r, such that it does not intersect the kernel. So we can now take this, take this. So we can assume, therefore, that, say, either positive or negative, we can assume it is positive. And therefore we have that l of y is positive for any y in br. And now, what do we do? Yes, for the beginning, yes. There is a special case. Sorry, this is the assumptions, f is equal to r. So up to now is OK? OK, now, so l of y is bigger than 0 for any y. So if you, now I take a point z in the unit ball, and then I translate it for the, there, so I have that x plus r of z belongs to br of x. And therefore, I have that l of x plus r of z is bigger than 0. So again, I am using the structure of the vector space. So this is clear, right? And this is because of this. Now, l of z is what? Is l of x is l of x plus, is 1 over r, l of x plus rz minus, no, I prefer to change sign. Sorry, I prefer to add and subtract l of x. So l of x minus l of x plus rz divided by 1 over r. So it is 1 over r l of x minus l of x plus rz. Now, what do we know? Well, minus, there is a minus. You are right, there is a minus here. Thank you. There is a minus. It's OK. And now, what do I know is that this is positive, therefore this is negative. And so this is less than 1 over r l of x. And with the plus l of z is now equal to 1 over r l of, I think, is enough. Because now, I take also minus z. And therefore, I also have that l of z is less than or equal than 1 over r l of x. Because if this is true for z in b1, also for minus z. So once I have for z, I have this inequality, then I take minus z, which is still here. This does not change. And therefore, putting minus z here, I have this. But this and this gives me that the absolute value of l of z is less than or equal than 1 over r l of x. OK? So taking minus z gives this. Now, this together give me that the absolute value of l of z is less than or equal than 1 over r. And this is enough. This is enough to show that it is bounded. Because it takes the unit ball into a bounded set. And once you take it unit ball in a bounded set, so this means that l of b1, 0 is bounded. And this is enough to show boundedness. So this is the first remark on linear maps. Linear functional. Actually, I use the word functional because this is scalar valued. Otherwise, maybe I can use linear operator or linear map. Some other words. So the kernel is a subspace. It is very important to understand whether it is closed or not. In any case, it is important to understand what is a subspace. So let me try to make another exercise, which says the following. Assume that s is contained in e. And assume that s is a strict, is proper subset. Is not a wall of e. And assume that s is a subspace. OK? Is a subspace. Then the interior of s is empty. OK, so assume. So this says something. I mean, difficult to imagine. But you have a proper subspace. It can be infinite dimensional. Could have say co-dimension 1, in some sense. But still, it is empty interior. OK, so assume it is not. So assume that this is not true. And so there is a point, let me denote it by x not into s. And then there is a radius such that the ball, the open ball, the open ball centered at x not of radius r is all contained in s. The idea is then, the idea is quite simple now.  So you have a ball in your subspace. OK. Then you translate the ball at the origin, remaining inside your subspace. Then you have a ball centered at the origin in your subspace. But then any point of the space can be reached as a homotetized of a point of the ball. And then the wall space is equal to s. Because the homotetized point still belongs to s. Because s is a vector space, subspace. This is the idea. So first translate at the origin. So you have an open ball centered at the origin inside s. However, once you have this, any point of the ambient comes from some point in this ball. But then this is a vector space. So also that point must belong to s. And therefore, s is equal to e, which is a contradiction with this. So this is the idea that we write it. So br of 0 is equal to br of x not minus x not, which is an element of s. Because s is a vector space. So differences of elements of s still lie in x. So this is the set of all differences. This is a subset of s. So this is an element. Is this clear? Because s is a vector space. So actually, the assumptions here really do not need any completeness. This proof now works for any normed vector space. Just normed, it seems to me. Because s is a vector space. So completeness, I think that completeness here is not the point. So you have this. And now take any point x into e. And then you scale it back, dividing. So you just take x. So assume it is not 0. Then you just scale it to the boundary, say, of the unit bowl. I cannot use the word boundary here. But it is clear what is this. And then I multiply by r. So that I'm going inside the boundary of this. And then I multiply, say, by r over 2, just to be sure. So this. And then I multiply by lambda. So this is just inside r0. Because it has distance r over 2 from the origin. And lambda is just the inverse of this. No, lambda is a number. I mean, it's just given x, lambda is a number. So lambda is just the inverse of, OK. So I'm simply writing x as follows. What I'm doing is simply this. It's very clear. This is br0. I have a point x. And then what I do is this. I isolate this point at distance r over 2. And then I re-multiply by this. So this point is in the bowl. And each line, the line pass into the point. So this point is in the bowl. And therefore is in s. But this is a vector space. So all these lines belong to s. And therefore, x belong to s. So x belongs to s. And therefore, this is true for any x. And therefore, s is equal to e. So what I'm using here is just to have a distance. So the norm is just to have the vector space structure. I don't use any completeness in this kind of line. So you see, still difficult to imagine. There are infinite dimensional subspace. Doesn't matter. They at least have always empty interior. OK. So now theorem. So theorem. Now it is clear that lef. Now let us come back to lef. It is clear that this is a vector space. We can sum two linear maps. We can multiply by a number. And it is also clear that this is a norm. The supremum of this is a norm on lef. And so the theorem says that this is actually complete. Actually, this is complete. So is complete. And it is interesting to remark that so this is actually lebanac space. But it is interesting to remark that the proof that I will do now, the assumptions are just e normed. So you don't need to have that the space of all maps, linear continuous maps from e to f, to have completeness of this. You don't need that the source is complete, but just need that the target is complete. OK. If what I am saying is true, in particular, so in particular, there is something which is improving when I pass to the dual. I mean, assume that I have a normed space. Well, this is a normed space. I can define star, the dual space. Well, then star is banac. Even if e were not banac. So there is some improvement in doing this star operation. Is it clear? This is because this is and r is complete. So we improve in doing this star. So it is maybe doing star and star and star several times. Actually, two times is enough. Doing two times star sometimes. So now, so we have to take a Cauchy sequence of element. Let me denote it by ln Cauchy sequence. Cauchy sequence of in f. I have to show that this converges with respect to the norm we are considering. Converges to an element of lef. So what does it mean? It is Cauchy. So it means that ln minus lm. That's denor equal. So maybe in inequality that I've never remarked. But this is immediate, however. But it is important maybe for the first time to say, is that if you have l from e in f l, if you have an element of script l, then it is always true that l of x, this is immediate. Is it OK? Because this is just the supremum of all these quotients. So this is the supremum of all l of x divided by norm of x. So this is large. So this is something that I will use even without addressing it. Now, the ln is a Cauchy sequence. And we have, therefore, this inequality. For any nm, we have Cauchy sequence. Moreover, for any x in e, we have this inequality as a direct consequence of this. Therefore, this means that for any x, hence for any x in e, the sequence of element of f is Cauchy in f. Do you agree? Yes, it's clear, right? Simply because of the Cauchy assumption here and this inequality. So if this is small, definitely in nm, then for any x, fixed x, this is small, definitely in nm, which is exactly to say that for a given x, this is Cauchy. But now, as I said, I have that f is complete. So f in complete means that for any x, now f is complete by the completeness of f, so we can define the following. We can give the following the definition. We can define l of x. This is a definition for any x in e. We can define l of x as the limit. This defines a functional, a map. So I have a map x in l of x. Do you agree? This I can do. Simply because this is Cauchy and therefore it converges and therefore given x, I take the limit and I call this limit l of x. So of course, what do I have to show? I have to prove is linear. L is continuous. And also that ln converges to l. And also this. Is it OK? Do you agree? So I have defined a map, a sort of abstract map, because it's a limit. But still it is a map. And so I want to show that this is linearity. Well, linearity is almost immediate. Because ln of x, we know that for any n, for any x, for any y, we have this. So simply in passing to the limit in this relation implies that l is linear. I mean, you can multiply by a number alpha and still you can pass to the limit and so on. So it is clear that l is linear. And now, remember that we have, for any epsilon, that exist such that we have this, for any x. For any epsilon, for any x, we have this. Then we can pass to the limit here. I want to fix one of these indices. So remember that this is also larger or equal than, so remember, this inequality may be a minus b. We have this inequality, so I can use it here to get this, which says that, so this is actually even larger than without the absolute value. So ln x minus lm x, so without the absolute value. And so I have now put in together all these inequalities. I have ln of x less than or equal lm of x plus epsilon x. This is true. For any epsilon, there exist n bar, such that this. So in particular, I can fix one of the indices. So for any epsilon exist n bar, such that, for any n bigger requirement bar, for any x, we have this. Passing to the limit as n goes to infinity, gives me lx less than or equal, then ln bar x plus xx, which gives me the boundedness of l. OK, so we have that l is bounded. We have it is linear. Therefore, we have shown at least, so what remains to show is now that, so what remains to show is now that, ln converges to l in norm. So remember that we have, I have already erased that, for any epsilon, for any x, there exist n bar, such that for any n bigger requirement bar, for any x, we have ln of x minus lm of x less than or equal than epsilon of x, we have this, we have this. Now, before we have fixed just one index, now it is not sufficient, we have to pass to the limit as n bar, now it is not sufficient, we have to pass to the limit in this as m, for instance, or n, one of the two goes to infinity. So sending m to infinity, we obtain that for any epsilon, there exist n, such that for any n bigger or equal than n bar, x ln of x minus l of x is less than or equal than epsilon, this, and this is exactly to say, therefore the supremum of x, the say different from 0, ln of x minus l of x divided by x, so passing to the supremum with respect to x in this inequality is less than or equal than epsilon, and this is by definition, is the difference of the norm. So first pass to the limit in one index, use the definition of l, and then take the supremum with respect to x, and then use the definition of norm. So we have shown that for any epsilon positive, there exist n bar into n, such that for any n bigger than n bar, then this is less than epsilon, which is like to say that ln minus l goes to 0 in norm, do you agree? OK, so we have that this is a Banach space, so from now on we will always know that the dual of a Hilbert space is actually complete, is Hilbert, the dual of a Banach space is a Banach space, and more generally the set of all linear bounded maps from one norm into a Banach is Banach, a proposition. So now the point is that we would like to understand more or less what is the dual of some space of sequences. So let us now start from C0, so let be an element of C0 star. There exist x in l1, such that l of x, l of y. By the way, sorry, notation, sometimes l of x in the books is denoted by lx sometimes. Just a notation, notation. x equal xk, let me denote it by x, no, let me denote it by l, sorry, I change notation, it is better. So that l of x is equal to, now I have this lk, and then this xk from k1 infinity. Moreover, sorry for denotation, this l, so it is clear, this is a space of sequence, this is an element of this small l, say small l, small l, this is small l, this is l1, I hope the denotation is clear. So this is a sort, remember, this reminds a little bit the Ritz theorem. The Ritz theorem was saying, then if I have a linear continuous map on l2, then I can find h0 in l2, so that this was true and that was an isometry. So what remains here? Well, you see, this is c0 star, this is l1, this is Banach is not Hilbert, again this is Banach is not Hilbert. The duality between l of x is still represented by this, and again I have that the norm is preserved. So if I associate to l, this small l, I know that the norm is preserved. This existence and this is also, what we will see next is what happens now if I start from l1 star, I will get an element in l infinity. And so at the end, we will conclude something about the star of the star. But anyway, so let us try to prove this assertion here. So two assertions maybe, so there is another assertion, l1 star here, then there exist this here, then this is in l1, and this is l1 star, and this will be l infinity. So we will try to prove these two assertions. So let us for the moment start with the white one. So now concentrate on white. Ok, so we use the usual notation EI equals 0, 0, 1, 0, et cetera. This is in the height position. Ok, this is clearly an element of C0. Ok, and also of course the supremum is equal to 1. Ok, so now take any point x in C0 and we use also this notation. So we use xm, take a point x, take an index n, and then we take the first m elements, so the first relevant part, 0, 0, 0, et cetera. This is a notation, ok? Just a notation for any x notation. Ok, so what do we know? Xm is just also, I can write it also as the sum from 1 to m, xk, xk, from i, k, xk, k. So now is it true or not, the following crucial point is this, is this true or not as m goes to infinity? Why is this true? First part is 0. The element x is in C0. So this means that it is an infinitesimal sequence. So if I remove from the sequence the first element, then it remains the tails and the tails as a supremum which goes to 0. Therefore this as m goes to infinity goes to 0 because x is in C0. So this is very important because it says that we now have an element which is continuous on C0, right? L is continuous. And this says that xm is an approximation of x. Therefore we expect L of xm to be an approximation of L of x because of this property. So L of xm, well, xm, this is an element, of course it is, this is an element, so this is just bilinearity of some numbers, L of xk, e k. Maybe this, just a word here. This is maybe another way for who of them that know what is a shoulder basis, this is more or less says that this is a shoulder basis for C0. That doesn't matter, doesn't matter. OK, so we have this. So let me denote it by what I am looking for in the sequence. So this is, for me, is Lk by definition. OK, so this is k from 1 to m, small l. Lk, now L and E are very similar. OK, Lk, e k. Now, as I said, this is going to 0. L is continuous. Therefore, this converges to L of x, because L is continuous. So L of x is the limit of some, which we, so this is a limit, this is real, these are real numbers. OK, just the usual limit is real numbers. So this is xk, e k. There is something wrong here. Xm is that Lxk, which I didn't know. E k, yes. No, no, there is something wrong here. This is OK, this is OK, OK. I have simply reversed the role. Sorry, this is a number, this is a vector. That simple problem, sorry. As usual, when you do linear algebra, so this is Lk, this is, sorry, this is xk. Sorry, these are real numbers, real numbers. Real numbers and real numbers, not elements of C0, so xk. And then, sorry, this is xk. This is just convergence in real numbers, OK? Sorry, so this is the standard image through a linear map of an element of the basis. So let me call it Lk. The notation Lk is OK, but there was a mistake here. OK, so L of x exist, and this is already something in the statement of the theorem, right? Because the first statement of the theorem was saying, let L be a linear functional, then there exists Lk. So what do I have to prove now? Problem. Well, I have to show that L, so now let me indicate by L this, I have to show that this is in L1. Not only this, then once it is in L1, so problem this and this also. This is what remains to show. But at least we have this representation, realization, representation of L. So this is true for any x. OK, now the idea is to choose special x in order to make the series of the abstracts. So the point here is to show that this is finite. I have to show this, OK? And I have to show this, and I will choose special element x, that I call now y, yn in C0. OK, now yn in C0 is cooked exactly in order to make this product the absolute value of Lk. How can I do this? Well, this, OK. So yn, ynk is equal to the sign of Lk if 1 less than or equal to n0, else, OK? Well, so this is plus minus 1 or 0. Therefore, this is converging to 0, is infinitesimal, OK? And therefore, this is an element of C0. So I can put it here, and it gives me, so it is allowed to put a yn here. It is allowed. And so L of yn is exactly what I want, because it is the sum k1 to n. Now here I have the sign plus, if I have the sign plus Lk, positive Lk, if the sign of Lk is positive, it is negative, and so at the end I have just the absolute value, but summed only up to capital N, OK? So this gives me the absolute value of Lk. Not only this, but also maybe I should observe that yn in absolute value, no, not in absolute value, sorry, in C0 is equal to 1. These are unit vectors. And this is the sum, OK? OK, now L of yn is this. So L of yn is this, which is less than or equal than the normal by the usual inequality that we discussed before. This is always less than or equal than this, OK? But this is 1. Therefore, this is equal to L. And this shows that for any capital N, this is less than a number. Therefore, this infinite sum is converging, OK? So this implies that the sum from 1 to infinity of Lk is less than or equal than L. And in particular, this problem is OK, hence L is in L1. Not only is in L1, but also this is less than or equal than this. So, I leave you to continue the proof in the sense that you have to show now that you have to show this, OK? This we will do tomorrow in the next lecture. So please, for the next lecture, try to show this and try to show also the green part of the proposition. So, realization of the dual of small L1. And I can say that the proof is very, very similar. So it is doable. So, conclusion summarizing, you have to show this inequality at home and then to show, to prove the proposition concerning L1, small L1, the dual of small L1, OK? OK.


Think Tech Hawaii. Civil engagement lives here. We're back 4 p.m. on a given Monday, wow. Isaac Kim, our old friend Isaac Kim, the grandson of Hock Kim. Yes, sir. Who's the best barber in all of Moiliili. Thank you, thank you. And who came back to take care of his granddad here, and he's been doing that for a while, and that's great to see that, great to see the structure of that family and of your family, of course. Thank you and thank you for having me again. I'm remarkable that you have achieved things that most people your age do not. You worked for L'Oreal for a while, you went to Columbia, your wife went to Columbia, and then you started this business, pilot men's business, and we talked about that last time, shot pleas of the pilot men's product, okay? And it's really good stuff. It's nicely designed. You made this happen. Thank you. The world doesn't know a boundary for you. We don't know how far or fast you're gonna go, but it's gonna be impressive. Thank you, thank you. Yeah, no, it's a lot of work, but I think hopefully it'll pay off. And what's interesting is, you know, when you get to a point in a given entrepreneurial experience where you don't have to be in the city where the places where the headquarters is, you could be anywhere, which is essentially what Isaac is doing. Exactly, yeah, nowadays, I mean, as long as I have the internet and the computer, good connection, you could really do a lot. Yeah, and you're earning a living, making a company, it's really remarkable. But when I met you last week, we ran into each other, you told me a story that I wanted to cover here on the show because I thought it was very important. Oh boy. In the confluence of the world as we know it today. And the story is your friend, John. Bomb. Bomb. Oh, bomb. Which I haven't seen him in a few days. I really wanted to ask him what he thought about, you know, sharing his story and all that, but I'll have to. You can call him John if you want. Yeah, let's call him John for anonymous purposes, no. So, you know, the title of the show, incidentally, is Dealing with Homelessness, One Person at a Time. And I was very touched by what you told me about John. Yeah, John. So tell everybody what's going on with John. Who is John and what is John doing and how did you run into him and what are you guys going with John? John, you know, he, I met him just completely randomly. He's actually been sleeping right in front of my grandfather, grandfather's barber shop for almost 10 years now. And he leaves really early in the morning, but we get there, you know, at six. So sometimes he's there, sometimes he's not. But over the past month or so, just slowly getting to know him. He's saying hi in the morning. All the business owners love him. I mean, he does no harm. He's a super nice guy. He cleans up the parking lot without anyone, you know, paying him or anything like that, just because he's so grateful that he has a place. It's his way of payback, yeah. Exactly. And it started with just, you know, giving him a real cheap blanket, like an airline blanket that I just so happened to have with me in my backpack. And from there, it really developed into building a great relationship. He's a wonderful, wonderful man. He's 58 years old. He's, you know, he's an army veteran. He has a son.  And, you know, through our conversations, I got to know him a lot more.  Some guy donated his bike right away. All I had to do was go pick it up, which is just real great stuff. And right now we're just trying to work with him because, I mean, he's had jobs. I mean, he's not hopeless by any means. I mean, he's totally functional. Just a wonderful- Rational. Rational. You can have a conversation.  He knows what weather's coming. I mean, he's listening to it all day. And yeah, so over the last few weeks, we really just got to know each other. And I think just being there and just having a conversation is a huge part of just helping somebody. Yeah.  You have to be able to talk to people if you're left in isolation and it makes you a little crazy. Yeah. And that's why I asked you if he's rational because everybody tells me that a lot of people who are homeless are no longer rational. Some people believe they were never rational. That's why they're homeless. I rather believe they were rational, but when they went homeless, they were isolated and the lack of human contact and compassion makes you a little crazy. You've got to have people around you. Yeah, I would think that that's got to be the case.  They used to be our neighbor. They used to be, you know, our friend or all sorts of things. So I don't think that's the case. And I think that stereotype needs to be a little altered. Yeah, yeah. So what does he do though, the mechanics of it, you know? I'm afraid that most, that myself included, most people don't know how a homeless person actually conducts his life or her life. I mean, what does he do to shower? What does he do to take a leak? What does he do to brush his teeth?  I mean, unfortunately, they probably couldn't perform them as much as they would like. I think what's unique about Hawaii is at least there's certain showers and a bunch of water fountains and different ways you can get creative versus, you know, let's say in New York. So in that way, it's good.  They're all retired Army, Navy officers. They're not homeless. Yeah, they have kids, they have house cars, they're comfortable.  And I totally know that all day long, I mean, he's worked, he's, you know, he wouldn't accept change from people. He's never held a sign asking for money. He's out there collecting bottles, recycling, doing things that most people just, a lot of people don't do, period, but he's there cleaning up after them. And that's actually his source of income. Unfortunately, in talking to him, I didn't realize, and now my, you know, now I really notice it, is when you see people, at least here in Hawaii, since I've been here, it's a competition for these cans. It's not easy. I mean, you think it's just a matter of going and finding all the bottles and going to the bars and whatnot, but everyone is doing this, every homeless person. Yeah, so there's only so many bottles. Yeah, so he has to go out certain times at night where it's not so safe, because that's the only time where he can find the largest amount of bottles or cans and whatnot. And what's really cool to see is also, I mean, maybe just because he is such a nice guy and just all the business owners within the whole plaza that my grandfather's in, a lot of them save up their cans, their bottles and stuff from home. And once a week or so, they bring it to him and they give him a banana or some juice. So in that way, I guess maybe he's received a little bit more love than some others would. But again, I think this problem is so widespread, but if you just make a little bit of the effort, I mean, it can make a huge difference, even just listening to how he does it. And I would have never really thought that there was so much competition, but when you think about it for half a second, it totally makes sense. How's his health?  So that bike is so crucial. And so he was so bummed when he lost that bike. I mean, to carry all those bottles, he's 58 years old now. I mean, he doesn't have the best healthcare by any means. So being able to transport things, just something that you would think take for granted, I mean, is a huge part of it. Yeah. Well, the fact that he lost the bike when he took the bike away from him goes to the question of security. Security of property, whatever minimal property a homeless person would have, but also physical security of getting beat up. And there have been a number of stories in the paper about how some really nasty people would attack the most vulnerable of all, the homeless person and injure him or kill him. I wonder if this is a concern for John, Bob. Absolutely, absolutely. In fact, it's funny. The night that his bike was stolen was the only night where he slept with the bike outside away from the wall. There's the wall, there's him and then the bike, and that was the night it was stolen. And he had a concern about that. In fact, it was my fault. I actually asked him if he could, for one night, put it on the other side because I just painted the outside and it was still wet. Oh no. So I repainted my- I'm feeling guilty, eh? Yeah, yeah. So then, so he put it on the other side and literally that one time he put it outside of his, you know, not against the wall, not in between him and the wall, it was stolen by some kids. I mean, and who knows why they stole it.  Yeah, or- It was his only possession, effectively. Yeah, basically. Yeah. What about going to the hospital? I mean, if he gets sick, what does he do? You know, what are his options? Honestly, I don't know. I don't think he really has any.  His family doesn't invite him in. They don't offer him a place. No, apparently, well, unfortunately, the last time he was, you know, he was still had three jobs and he actually moved to Las Vegas for six months to help out with his mother and uncle. And once they passed away, he came back here and things were a little too expensive and before he knew it, he was on the streets and it's been about 20 years now. He's not local. He was born in Pennsylvania, actually. But he came out here when he joined the army. So he's been here pretty much ever since with that exception of going back to Las Vegas for a little bit to help out his mother. You know a lot about him, Isaac. Yeah, I mean, I've been spending a good amount of time. I mean, he's a great, great guy and I guess what really struck me is his kindness, his just minimalism that he's totally fine with.  I try to help out in different non-profit organizations. But, you know, that's in a mass sort of way. And that's, you know, technically it's kind of easier to do. It's kind of out of sight, out of mind. And yeah, we're doing this, we're doing that. But here's a time where you can really develop a relationship with an individual, a person who just so happens to be homeless. And you've done that. Yeah. But I guess, I mean, I would ask you why have you taken the time?   You could do other things. There's lots of other things to do in the world, but you have dedicated your time to this one person. Who knows exactly why, but maybe it kind of just fell on my lap whether that's God just putting him there, right? He literally has been in front of my grandfather's shop for 10 years. And, you know, my grandfather doesn't speak the best English, but has left him alone.  And, you know, I guess that struck me and ever since, you know, if I see him in the morning, we'll just have a great conversation and talk about the news, talk about whatever. And honestly, I may have encouraged him some, but I can honestly say that he's probably encouraged me more or at least opened my eyes and just made the problem closer to home and more real. What's the future for John? I mean, do you know enough about him? Can you project and predict the future? What's it going to be like? 58 now, next year 59 and so forth. In fact, it's been hitting him, especially now that he's been in Hawaii for the better part of the last 40 years and he's been thinking how he's getting older and that it's getting more real, the end, I guess, unfortunately. And he's been talking about going back to the mainland and that's something that, you know, would be great.  It'd be completely irresponsible to just do that and leave him right back in a whole new foreign place where he has to go make it again. And I mean, it's very dangerous. There's all sorts of other challenges who have to figure out where to sleep.  Simple thing like, you're going to have to figure out where to sleep, where it's safe, where he won't freeze in the middle of the night. Exactly.  It's hard to pick yourself up when you've been down for so long. That's a lovely thing you're doing. That's Isaac Kim. It's a lovely thing he's doing and it's part of his worldview, I think. I don't know him that well, but I sure like what I see. And when we come back from this break, I want to tell you about a movie that I saw last night. I promise that I would. Sure. It's called, it's by Ai Weiwei, Chinese activist and movie director. And it's called something about, it's called something about flow, human flow. That's the name of the movie, Human Flow. It's an Amazon movie. When we come back, I'll talk about that and see if we can't find a larger truth in all this, a larger sea change of humanity, which you're touching one person at a time. We'll be right back. Hello, everyone. I'm DeSoto Brown, the co-host of Human Humane Architecture, which is seen on Think Tech, Hawaii every other Tuesday at 4 p.m. And with the show's host, Martin Desbang, we discuss architecture here in Hawaiian Islands and how it not only affects the way we live, but other aspects of our life, not only here in Hawaii, but internationally as well. So join us for Human Humane Architecture every other Tuesday at 4 p.m. on Think Tech, Hawaii. Aloha. I'm Marcia Joyner. And I'm Beatrice Gantejo. And we have come in this series, young and old alike, to take a look at our past, your past, and the past that's not seen history books. History books are his story. And what we refer to as mirrors of the past, but we as colonized people, indigenous peoples and people of Kola, look into the mirror and to not see ourselves that. On the Ties that Bind, we will examine those underlying causes. Please join us with the Ties that Bind on Wednesdays at noon, twice a month. We look for you there. Aloha. Aloha. It's a huge part of it. I've been learning a lot. The thing is that when you're homeless, you're up against it, you know, it's threatening you. And being threatened like that gives you a sort of a heightened sense of awareness, I think, or it could, a heightened sense of awareness in the world. So when you talk to him, it's quite remarkable because you're talking to somebody who's willing to share that heightened sense of awareness and there's a benefit there to talk to him both ways. Absolutely, it's huge value. Yeah, it's great. So anyway, so I wanted to tell you about Ai Weiwei. Okay. And it's a human flow. Uh-huh. And what it's about, it's a remarkable move. It's a beautiful art film, among other things, the production values are extraordinary. And Ai Weiwei is in the movie from time to time. You see him, you see him engaging with people. And he goes from displaced person camp to displaced person camp, refugees all over, wherever you can find him. He goes into the Middle East. He goes into the southern and northern parts of Europe. Goes into Africa. And he goes into, I think, into the Mexico and South America. I don't remember exactly where, but point is that he covers everything. He goes into the Rohingya issue in Myanmar. Yep. So what he's trying to tell you with a lot of wisdom is that we have a problem. Earth, we have a problem. We have something over 60 million people who are in these camps living and he has photographs, including drone photographs of huge camps of one kind of tent or another, where all these people live in this kind of imprisonment situation for not years, but decades, and even more, more decades. And there's no hope for them. There's nobody gonna take care of them. There's nobody gonna, they will allow humanitarian groups to come in, NGOs come in and pass out food. But they're not gonna let them out. Okay, they're not gonna make a home for them. And it's very tough what's happening. And it's more and more all the time. So we have a problem, Earth. Mm-hmm, thank you, George. A problem of humanity, human flow. And I think it sort of puts it together. If you watch this movie, you'll see, it's not just one place. It's not just one country or migration pattern. It's everybody. And somehow we have created two worlds here. We have the world of people who live within civilization, you and me, and the people who live outside of civilization. And the reason I put it that way is because John is living outside of civilization. He happens to be here, and you talk to him and all that, but he's not in our civilized communities. He's outside. And so we have this, I'm gonna call it separation. This dichotomy, this polarization of human people all around the world as never before, even after World War II, when there was so many displaced people, it's bigger now than it was. So what do we do, and what have you learned? I mean, I know this is a hard question. What have you learned from John that maybe we could apply to these many tens of millions? I mean, that's a loaded question, and by no means do I have any real answers to that. But again, just from my experience and even dealing with John or certain other individuals, I think it's important to remember, and I think the simplest way to break it down would be to remember that they are just like you and me, and they're totally fine. They may have had bad luck here, bad luck there, something happened to them. You never know their story, and it's important to ask. It's not just that they were crazy. I mean, a lot of them aren't, period, even now, but it's about understanding and realizing that everyone has a story and not just discounting them right off the bat. And kind of like what you're saying in terms of these huge camps, I mean, that's heartbreaking. I mean, that's self-perpetuating. I mean, how many genius children are probably in there who could add so much value to the rest of the world yet they're not given the opportunity as so many of us squander away even. Yeah, we don't even appreciate how critical it is. And Ai Weiwei runs around and shows you that these kids never go to school. They just don't go to school. Oh, another example is Gaza. You know, Gaza has been, I don't know how many decades, Gaza has been locked up that way, and nothing's happening in Gaza. They're not supposed to have weapons. That's why the Barricade, the Blockade, but in fact, they have a lot of weapons, and the government is useless. Their government is useless. Hamas, whatever it is, useless to help them. They don't have any agriculture, they don't have any medicine. I mean, they live into mouths in this community, this overcrowded community, and what can we offer them? What can we do for them? It's more, they get handouts. They get a lot of food comes in. 80% of the food in Gaza comes from the outside. That's how they live. There's no economy to speak of.  And I think the homeless is kind of related to that, don't you think? And so, you know, is it that we have to have a lot of Isaac Kims around to go out and talk to them?  Is it that you invite them in your home? If everybody in Hawaii did that, if everybody in Hawaii did that, we wouldn't have any homeless, would we? Exactly, and I think the answer is yes to kind of all of that, but at the same time, that wouldn't be much easier if we just remember them. Literally, I mean, it's so easy to put it out of sight, out of mind. It's so easy to be busy and caught up with your day. It's so easy to stress out about something that really isn't that stressful, relatively. And just remembering that there are other people out there that are much more in need, whose struggles are much more dire and urgent than yours. And yeah, just have a conversation. It might totally open up your mind. I mean, there's so many people who say they care and they do, I don't doubt that they do, but once you really actually put yourself in a situation where you can't have a conversation or where you can just love on them a little bit and just see- You're a fellow human being.  I mean, it could happen to anybody, really. So I think the first step is just being, not only aware, but being present and being willing to just understand that they're totally like you and me. Well, government doesn't do very well at this. And I think if we find this to be a global failure, it's because government just doesn't know how to handle this. It hasn't been faced with this challenge, not to this degree before, and it doesn't have the tools or for that matter, the conscience, the morality, do something about it. Only exception to that, at least in this movie, was Germany. Very interesting, in the airport in Berlin, they've dedicated a number of hangers, large buildings, and they have very nice little areas for the homeless and they helped them out. It is really a statement of tolerance and love and affection as no other place in Europe that I know about that was in the movie. But here's the thing, human conduct is best affected by incentives from government, incentives and disincentives. So we know you and me sitting here together, we know it's unlikely that government really, in most places, is gonna be able to reach out and do, for example, what Germany is doing to try to actually affirmatively take care of them, which doesn't always work in Germany's favor, by the way, and creates political problems. But maybe government can do something less than that. Maybe government can do something to facilitate people like you, to facilitate people who are well-thinking, people who might help if they were just given a little nudge, some kind of nod where you say, hey, Isaac, why don't you do this? And we're gonna somehow incentivize you to do it. Do you have any ideas about that? I'm not sure exactly on how to incentivize, but all I know is that, I don't know, I don't care what excuse anyone gives. I mean, there's enough money to go around. There's enough super rich, even if just the super rich people just donated, whatever percentage. I mean, this could be pretty much solved, at least temporarily. I mean, I think long-term, this is an issue that's been going on forever, and it's a tough one to really break down. There's so many different assets or different factors involved. But back to that point, I think that people need to be incentivized, I think, but it shouldn't be that hard. It really shouldn't. I know that, in talking to John, that he said up until 10 years ago, there was a certain, and I haven't done the research, I don't know his name, I need to ask him, but there was a governor who came into power here, who was elected here in Hawaii. And up until 10 years ago, he said almost every church in the whole area in all of Hollywood would have a certain day or a certain time where they welcomed homeless people. And it was just a safe spot. No one created trouble. It was just a safe spot for them to hang out for a couple hours without worrying, to have one meal, to talk to each other, to just be normal. And he said up until 10 years ago, that was almost every church, and then once this certain governor, I don't know who it is, he apparently told churches or disincentivized them or gave them an incentive not to help discourage them. Or he went out of office, or he went out of office. Or something, yeah, so. And he said 10 years ago, that all stopped, and now he can't find a place to do that. And this is a man who wants to go to church. He wants, if a church won't do it, who will? I mean, maybe the government, but we've seen that. And again, so I think there needs to be the proper incentives. And I'm sure the governor was thinking, hey, this might help or encourage homeless people to get up off their feet, do something else. But I don't think that's the answer. I think you really have to care and really look at the person as a person and see their needs individually and just try to do what you can. Do you have any idea what happens? Do you have any thought about what happens if we as a society do nothing? If there is nobody, but Isaac Kim out there taking these steps to deal directly with people like John, what happens to us? I mean, I think that can only hurt society as a whole. By no means do I think I'm doing anything that great. Honestly, it's just one person. But if everyone cared, I mean, the world would be such a better place. I mean, these people can bring value. They can add value. Shoot, I know that John could definitely have a job at a parking lot, doing the little math, 10 minus four. He can definitely do that. Just whatever it might be, if we just include them and realize that, hey, there's something for everyone. Find a place for him. Exactly. Yeah, yeah. So last question, Isaac. Yeah.   But are you gonna continue to do this with John or other Johns going forward here or on the mainland? Is this something that you've built into your system where it's part of what you do? I would hope so. I mean, even just starting out with my company, I plan to help the homeless. In fact, we specifically designed this one product so that we can give out that it'd be the most useful. But I think even before all that, I mean, sitting here now talking to you and having gone through this experience, I think it's even closer to home. And a business can't just be for profit. It just, it can't, I don't think any great company, I mean, there are plenty. But to be really a great company, I mean, you have to have a deeper purpose. And I think that was my purpose. I think it's been solidified and even more highlighted and even more important now that, you know, I'm closer to the cause. I've met a few of his friends. And I mean, this is important. This is not something we can just ignore.  It's only gonna get worse and worse and worse. Especially when you have these families and all these huge camps who aren't going to school, who don't know what a normal civil, how can we expect them to learn or to know that later? It's gotta be kind of nipped at the butt as much as it can be. But I know that's a loaded, loaded topic and a tough one to solve. But I think we need to really think about it. We can't just do nothing. Yeah. So before we close, why don't you show the product that you like to give to John or other homeless people? In fact, we have, he's gotten two bottles. He hasn't showered enough with it yet. And that's what I keep bugging him and the other store owners. But here it is, it's a Castile soap. So it's 100% natural, biodegradable, responsibly sourced ingredients and sustainably sourced. So it's all good stuff, but more importantly, they can use it to wash their hair, their face, their body, they can wash their pet. They can do their laundry even with it. I mean, it's just an overall good soap to use. So I'll integrate it. Your granddad, John. Kind of unplanned, unplanned. Everything comes together. Yeah, it's beautiful that it has the way it does. But I hope that it continues to perpetuate and grow even bigger. And that's truly the hope, because it's hit home a lot harder. I mean, I would love to hopefully maybe some other time bring John on or something, because he's a lovely person to talk to. And he's got a good head on his shoulder. And it would just, it means so much for other people to learn about his story or to anybody, anybody else. I hope you can do that. Yeah, that would be great. Isaac Kim. Hey, thank you so much. Great to have you. Yeah, so good. Always good to see you. Thank you. Yep.


Hey guys, what is up? Welcome to my channel. For today's video I'm going to be doing a shop or a drop where I go through trend moods Instagram page and I go through all of the new releases that I just feel like talking about and I tell you my thoughts on them whether or not I will be shopping them or dropping them from my list so if you are interested then just keep watching. Hi, today's one of those days where I kind of am in a bit of a funk. I'm feeling a little unproductive. I haven't done so much. It's four o'clock. Usually I'm filmed and edited by now and I'm just getting started. I don't know. It's just one of those days I guess. I put on makeup so I'm feeling a lot better. That's always a good therapy for me. That's good. Yeah, trend mood I feel like has just been so blah lately of course given the current situation. Lately though there has been some things popping up on my feet so I figured while I'm in a funk let's just talk about makeup to get myself out of it. You know also just the things I've been seeing I'm like I gotta film a shop or drop. I'm not sure if I talked about this in my last shop or drop but you know I apologize if I did but whatever. Alright so the first one I want to talk about is Shanta Kai's new summer collection. Now this is currently available. There are two cheek, cheek and highlighter duos and then you also have what looks like four lipsticks. Anyways this looks so beautiful again. Shanta Kai and their packaging just so incredibly cute. It is a very lux brand and I'm really starting to get into Shanta Kai. I'm enjoying their products a lot. I seriously was going to shop this product but I did decide to drop it from my list only because I am on a low buy right now and as much as I would love to buy these products and review them they are so expensive that I feel like the payback that I would get just wouldn't be enough like they're really expensive. Even though I allow myself to buy things on my low buy if I'm going to review them for me it's just not as valuable I feel as other products could be so I'd rather invest my money in other items so I am dropping that from my list but oh my gosh you guys I am so interested in it. It looks really gorgeous. Those cheek duos look phenomenal and also I'm a big fan of Shanta Kai lip products in general so definitely hurts me to say no but I do think that's a beautiful collection so it is available right now though if you were interested. La Mer is also coming out with a radiant skin tint. This has SPF 30. Right now it does not seem to be available unless I scroll up and I missed it. I'm not going to be purchasing this. I'm dropping it from my list. I really enjoy my La Mer the soft fluid foundation. I think La Mer has really beautiful products but I also think that they do have a mix of products that you're kind of buying for the name. This actually looks really good but I just feel like to pay a lot for just a skin tint I'm not too sure about so I'm not going to get this. I'm dropping this from my list. My Dior Summer Skin Foundation is actually going to be here tomorrow so that's a skin tint that I did get behind. I don't think I'm going to purchase the La Mer but it is available right now in the UK if you would like it and I'm actually very intrigued by this product. I definitely would not mind watching some reviews on this product. Also something that reminds me of that but it is definitely way more affordable. It looks like L'Oreal has come out with a tinted water cream which again a lot of brands are coming out with these tinted products because of the time of year and this says it's coming soon online but somebody saw it in stores and honestly this is a product that I would ponder shopping if you guys were very interested in a review. I don't really do too many reviews on more affordable drugstore products but this is a product that I do see myself purchasing as I'm at the store and I see it not on my lo-bi. Again if I'm not doing a review on it I'm not going to buy it. If the lo-bi weren't happening I would purchase this because I like a lot of different drugstore foundations. I really do and I'm testing out a lot of the skin tints and like tinted moisturizers in my collection right now because I do want to do like my favorite skin tints lightweight base products for the summer so that would be something I would be interested in trying. Just gonna pass on it and then it looks like L'Orex is coming out with a Luxe Diamond Golden Hour Collection. Looks really gorgeous. It is actually available right now on the L'Orex website. It will be coming to Ulta and you have a palette which is highlighters. You have some glosses, some eye gloss trios, really glowy stuff. I think L'Orex is really trying to revamp their brand, come out with more fresh products that are more trendy and kudos to them because they did need to do that. Nothing in this collection really screams out to me so I am going to be dropping it from my list. It does look pretty though but nothing exciting and nothing that we haven't seen before honestly and I'm going to talk about this really quickly not really too much but obviously Charlotte Tilbury has released her bronzers and the new quad as well for the eyes. I did do a full review on everything on these products so of course I did shop them. You guys know I love Charlotte Tilbury. I know you guys love Charlotte Tilbury and these products are beautiful. Make sure you check out my review to see my full in-depth thoughts on them but if you were curious about them they are very very good and then of course I had to talk about this. Pat is coming out with the Divine Rose 2 palette. I am so excited so this is actually going to be available June 15th at Sephora and it will be available I think June May 29th on the Pat McGrath website.   I really really do and this palette might be in my new favorite like this just by looking at it. There is potential that this could be my favorite palette you guys. It looks so gorgeous. I like it way more than the original Divine Rose and the original one is really beautiful but from what I can tell like I have a good feeling about this palette. I'm so excited for it. Also in the collection there are going to be some lip treals and I think looking through it I had most of the items already so I won't be getting those but the palette absolutely heck yeah and then also House Labs has come out with a new eyeshadow palette and everything in the past that House Labs has come out with I have not been interested in it at all to be completely honest so this was available May 19th so it is available now. I think this is a very interesting palette. I think it's a nice play on different colors. It's really fun. I'm attracted to this. I really am unfortunately I just haven't had much interest in House anyways and I just don't want it so I'm dropping this for my list but I have to say in my opinion this is the best thing that Lady Gaga has come out with in her line but I don't know like I just I'm not interested in the line at all. I did also get a lot of questions about the cremated palette and my thoughts that Jeffree Starcutt's Medics has come out with. It is available at this point though I do believe it sold out and I was going to purchase this because I do review his products sometimes on my channel. I'm a cool tone fan. I know all of my subscribers who also like cool tones you guys are like so are you gonna get this unfortunately I think they're people were acting crazy like I'm gonna be honest I always get crazy comments whenever I talk about a Jeffree Star product or review it and you know what that's fine everybody's entitled to their opinion I understand both sides absolutely but people are like threatening my job with this one I don't know if I was getting crazy messages before I even purchased the palette I could only imagine the crazy messages I was going to get after the palette I don't really know the situation you guys honestly like I don't pay attention to that crap I just know I was getting messages where people were telling me I was immoral and I was going to lose my job if I reviewed this palette and I was like first of all I'm not but anyways that's why I'm not reviewing it because people are crazy okay and then I also did get a couple questions about the lunar beauty collection that came out from Manny MUA I passed on this I dropped this from my list I don't need another blush palette or anything in this collection but that blush palette is absolutely gorgeous though I can see why people would want to purchase it and why you guys were asking me about it I love blush I love the look of this palette again lo-bi not gonna get it but it is a really gorgeous palette and I can see myself definitely seeing this at like a morphe store one day and deciding to pick it up but for now I'm passing on it I don't really have any interest in reviewing it bobby brown has also come out with the summer glow collection which I think is weird that they came out with this because they came out with that flower girl NYC collection and that would never really fully released and I was very interested in purchasing that and then they came out with this and this really pretty honestly like I have more interest in the flower girl NYC products but this does look really gorgeous so this is available online and they have a highlighting powder in warm gold this looks like it would be way too deep for my skin tone so I wouldn't get that bronzing powder duos those I'm interested in the lighter shade I feel like would be beautiful and then oil infused glosses which are so in I wouldn't get any of those so the only thing that really interests me is a duo and that's the last thing I need so this is a beautiful collection I think they did a nice job with it it's very summery I'm dropping it from my list but it is very gorgeous did you guys see this none of us really knows what this this is but whatever it is I like what they're doing I can actually see this though being very difficult to actually be workable it's just pretty to look at but I think as far as being functional this doesn't have it um yeah I just see a lot of issues with application for this one but it's really really pretty really summery and whatever they're putting down I would pick it up so I actually really do like that but it's probably not the most functional thing and then Mark Jacobs has a collection and a new brow product this is online I think well obviously I didn't buy it it says it will be available June 2nd on Sephora I gotta be honest this eyeshadow palette I'm digging it a lot like I'm thinking about picking that one up they have a new brow product which what is this one side is a soft powder like pencil on the other side is a tinted gel honestly I haven't picked up a brow product in a long time I wouldn't mind picking that up we have some eyeliners I won't get any of the eyeliners because I have so many and they're so expensive and then a mascara I might have to shop that brow product and that eyeshadow palette those look really really pretty you guys I'm into this I like the colors of those they look very warm and bronzy and just like something I could get down with this summer so I will be picking that up well not a hundred percent sure I want to sit on it first but those are catching my eyes to where I'm like I really like would not mind to pick those up you know Urban Decay is coming out with the ultraviolet palette and I appreciate that they're really mixing out their color stories I really do but over the years Urban Decay has just not been doing it for me and it really sucks because I'm just so bored with this they need to do something different something innovative back in the day I remember Urban Decay was one of my very first eyeshadow palettes and they had this old school cool original vibe to them and they just don't have that anymore and I just remember the brand being so fun and it like where did it go and I'm a purple palette lover as you guys know and this should excite me but it doesn't they need to change it up they really do I just I don't know get over it with the naked palettes do something different please I'm actually considering this you guys elf cosmetics is coming out with a collaboration with J. Kessa and it is just this completely basically rainbow palette and it's going to be 20 dollars and it will be available yesterday okay so May 25th it was available for early access elf members and May 27th it will launch online at the elf website and then the 29th at Ulta May 31st in stores honestly I might have to pick this up I like the collection a lot won't use those colors a lot obviously but I do want to support her I do really like her so I need to actually write this down I'm gonna pick this up tomorrow because I want to support her and I like how she went really bold with this collection I like the different textures that she put in here I'm into this I really like this you can get really monochromatic looks okay yeah I'm getting that I wasn't going to but in this moment I definitely decided to okay and this one is what kind of sparked this video for me where I was like huh that's new I need to film a shop or drop so YSL is coming out with these palette clutches and they are so cute you guys now the color stories maybe not I don't know I'm definitely think I'm gonna get that bottom one that's the one that's color story really speaks to me these are coming soon no launch date but I think I'm gonna shop that bottom one with the yellow and the purples and the pinks I love the packaging on this it's so lux and beautiful I don't have too much from YSL beauty and I don't think I have a single eye shadow from them so I think this could be a fun video let me know what you think but I I think I'm gonna shop that and the last item that I see is from Milani and it looks like they're coming out with the make it last setting spray with sunscreen in it that's pretty cool that's something I'm going to run out and buy but I did like the original I think it reminded me of Urban Decay the all nighter setting spray and it's affordable and I mean I don't think this is something like I said I'm not gonna rush out and pick it up but if I needed a setting spray I would pick this up in the store I don't need a setting spray so I'm dropping it from my list but the original is good so if you can add some sunscreen and SPF to it why not all right guys that is it that is all I have for this time of a shopper drop I hope you guys enjoyed this video like I said not too many crazy things have released but still fun to talk about them so thank you guys so much for watching this video if you aren't subscribed to my channel already I hope you take the time to do so and I will see you guys in the next one bye guys have a good one