Češče, da smo dobro izgledati, da smo način v teori. Prvom, da je nekaj remač. Zelo smo videli, da je izometri, da je zelo, da je zelo, da je izometri, da je zelo, da je izometri. in to je in izometri. V sveče in inilber spes, da bomo, da bomo, identifujemo to v to, da bomo. In tukaj, prišli do starov, da bomo, da bomo, identifujemo h s duvala, h starov. In včetno, da imamo tudi banak spes, tudi je tudi natučnja mapa, včetno je tudi in tudi b-dual. Tudi je tudi definicija tudi v dual. Kaj je tudi definicija tudi v dual? If I have a point x, then j of x is an element of e star star. Therefore, by definition, is a linear map sending star. Proste, da je tudi, kaj je tudi J of X, kaj je tudi, kaj je element istar star, je tudi, kaj je tudi, kaj je tudi, kaj je element istar star. Zato vzvečenje, da je vzvečenje, je to. Vzvečenje, zato vzvečenje, je to vzvečenje kajte dvej. Vzvečenje, je to vzvečenje, je to vzvečenje, kajte dvej. Kaj je to? Zelo, da v zelo, da jih je vse obježen, vzelo, da jih je vse obježen. Zelo, že jih je vse obježen, zelo, da je vse obježen. Tako to vidi to. A čekaj, izgledaj se vse element stripped star. To je vse element strip, to je vse element strip, to je vse element obi izgledaj. Parti celo je duvalitvanie v drzostih da je duvalitvanie v komod za glasboke. me vse imamo the same symbol for a different duality, is the duality between the dual and the original space. So, to be more precise, one should distinguish maybe using different symbols, and therefore one should write something like if you want something like this to remember that it's duality between different spaces. Zato vidiš, da v svoj nekaj vse je jazem jazem vse. Vse značimo, da jazem jazem vse. Vse je jazem vse star star. Vse je, da je vse suprimum, jazem vse f, kaj je tukaj nekako ni rukaj, ni nekako ni 1. Bejdevaj, jah se ki je sate z k ε, odunje se je k samo roasti a je x 1 plus x 2. Tukaj je tukaj f comma x 1 plus x 2, z kaj sem dreven. Now, what is this? By definition, the definition of this map is this, so this is fx, such that. But we have seen yesterday one of the corollaries of ambanak was this, that this is exactly equal to ex, if you remember. This is corollary of ambanak. What does this mean? Well, it means that the map J preserves the norms. This means that this is an isometry. J, what does it mean? Partial isometry. Isometry to some subspace for e double star star? Yes, double star star. So this way you can look, so this is a way to, so we can identify e with J of e, if you want. And this is an element of e star star. So at least we know that e, if we want, can be embedded into e star star. The point is that in general, this is not surjective. So in general, this is a proper subset of e star star. And not necessarily, so you cannot, in general, identify e with e star star. But just you can embed e inside e star star isometrically. In Hilbert is different, because in Hilbert we can, even more, because you can also identify with h star. So there are some spaces, so in general this map is not surjective. We have already almost seen this. When we studied space of sequences, we saw that if we have C0, then we pass to L1, and then from L1 to L infinity. Remember? I mean the dual of C0 was realized as L1, and the dual of L1 was realized as L infinity. And therefore the b dual of C0 was realized sort of L infinity, which is not C0. It's larger. So in general, the t is not surjective. This map J is not surjective. So in general, surjective. When it is, the space is called reflexive. When it is surjective, of course this kind of phenomena you don't see in finite dimension. In finite dimension you can always identify a finite dimensional vector space with its dual, and then with its b dual, and so on. So this fact that this is not surjective is a fact that you see in infinite dimensions for suitable Banach spaces. There are examples, as we already remarked, that of non-reflexive Banach spaces. But there are examples also of Banach and not Hilbert reflexive spaces. Do you have some examples in mind of non-Hilbert Banach reflexive? I don't think so. I don't think so. Even if at the moment I don't know exactly who is the dual of the measures, because you pass to the space of measures, for instance, continuous with compact support, you pass to the random measures. If they have, now if you take continuous with zero in infinity, it depends. It's a space of measure, but then taking the dual is much more difficult. I think it's more simple. Lp with p is strictly larger than 1, as strictly smaller than infinity. You know that when you take Lp, then the dual can be identified with Lp prime, p prime the conjugate exponent, and then you read dual once more and then you take the conjugate and the conjugate and then go back. Another exercise. No, this is not an exercise. This is far from being simple, but it is interesting to have this map J. Now an exercise, say a Banach space, and take two points, different points, then there exists L, which separates these points. The duality between L and X. Remember that in my notation, I'm switching notation for convenience, sometimes I use this, sometimes I use this. But this is a notation. So there are sufficiently enough linear continuous functionals on a Banach space, at least to separate points. So do you have some suggestions for doing this? Yes, you take X minus Y, you take X minus Y, which is different from zero, then you take the linear span, the linear span, let me call it maybe G, span over R, over X minus Y, and then you can define some this, say G of lambda X minus Y equal to, for instance, lambda times norm of X minus Y for any lambda into the real. Also probably it works, also it's OK. We can take also equal to lambda. OK. So this is linear continuous functional on small g, sorry, on capital G, and therefore by Han Banach, this can be extended into F, so there exists a continuous linear functional on the whole ambient space extending G.  So you see what happens to, and then one X that then L of X minus Y is non-zero, because L of X minus Y coincides with G of X minus Y, which is this norm, which is non-zero, and therefore L of X is different from L of Y, because this is linear. There are several linear functions. Other exercises before going to other consequences of the Han Banach. So what is hyperplane in a normal space actually? So in normal space. Well, we can give the definition in two ways of hyperplane. Let me denote it by M is a hyperplane. One would like to say that it has co-dimension. It is a vector space of co-dimension one. One should make this rigorous. Co-dimension one linear space. So there are two possibilities. One is the following, is that N can be written as M times R, let me denote it X naught. So there exists one point out of M such that this. Or, well, it is an exercise to show that equivalently, in equivalent ways, is to say that there exists linear continuous, so assume we are in Banach just for simplicity. Just for simplicity. And there exists such that M, there exists L linear such that there exists a linear function N such that M is equal to, and there exists alpha in R such that is the set where L is equal to alpha. Hyperplane. Maybe not the vector space, but also in a fine hyperplane. So translation of the vector space far from the origin still is an hyperplane and therefore it is a level set of a linear function. So this is the definition of hyperplane and L star is non-empty. The star is non-empty. Let me see. Well, this is true in normal space, but also in normal space. I think that I am just using, well, no, I am using the continuity. So I am using the topological version of Banach space. So the question is that can we say that this is true in more general. So the proof of the topological version of Banach was do we use completeness or not? Maybe not. No. So this is okay also in normal space. I wrote Banach for simplicity. So, well, this is a hyperplane, sort of hyperplane like in finite dimensional space. It is something which is affine, could be also a subspace that is not so important. I mean if it pass through the origin or not. The important point is to say that it has co-dimension one in the sense that the outside of m, say, is one dimension somehow. This explains the word hyper in front of plane. Now, of course, this hyperplane is closed. Another exercise, which, however, we have already done essentially, is that this hyperplane is closed if and only if L is bounded. Do you agree? We have already seen this. Not exactly. We have seen that the kernel of a linear functional is closed if and only if the linear map is continuous or bounded. This is a little bit more. It says not only L equal to zero closed, but for any alpha L equal to alpha closed. Well, they are homomorphic. L equal to alpha is homomorphic to L equal to zero. So, exercise, for any alpha L equal to alpha closed if and only if L is continuous. So, you add the word here bounded. If you add here bounded, then you find that L equal to alpha is closed. And vice versa. Now, examples of, so these are, for what concerns, hyperplanes and example, other exercises maybe. Now, let me go to a subspace, general subspace, assume that S is contained into some Banach, if you want. And S, finite dimensional, then you should prove that S is closed. So, this is just to have some intuition of what are subsets of infinite dimensional spaces. We have seen something. Remember, we made an exercise on the interior part of a vector subspace. The interior is empty. So, like in finite dimensional space, if you take a line in R2 or a plane in R3, they have empty interior. A plane in R2 have co-dimension 1, so it's a hyperplane. And if a finite dimensional vector space, subspace is closed, this is an exercise. And then the closure of S is a closed subspace. So, this is item 1, item 2. Of course, here there is no assumption on the dimension of S. Take any subspace, maybe could not be closed. Why? We don't know, could not be closed. If it is not closed, we close it. We take the closure. And then the exercise, they says that this is still a subspace, say Banak, simplicity. This is an exercise. Now, it is interesting, one could wonder, are there non-closed subspaces? Because this suggests that there are, right? So, are there non-trivial, non-closed subspaces of a Banak space? So, let us try to see if we can do an example. So, this is our homework. Now, exercise. So, take the Banak space smaller than 1. And then take the vector subspace defined as follows. V is the linear span over R of the following vectors. So, E, N. What is this? So, as usual, we use the standard notation. So, EI is 0, 0, 1, 0, etc. At the height position. Then we have a vector space. Because this is the span. So, by definition is a vector space. Finite linear combinations of elements of this sort. Finite linear combination. With real coefficients. So, this is a vector subspace. Vector space contained in a 1, for instance. The point is that it is disclosed. Is this closed? It is not closed. Because there is an N. Is not. Do you agree? I don't want to say what is this. But I just want to say that there is one element here. Which is outside here. This is enough. Sorry, sorry. No, okay. Sorry, sorry. Is V closed? Sorry. You have to erase this. Sorry. And so, it is not. So, I just show that there is one element in the closure. Which is outside V. It is enough. And yes, exactly. E1. E1 is not in V. Because it is not in V. On the other hand, it is easy to see that E1 minus this difference obviously goes to zero as N goes to infinity. So, there is a sequence of elements of V. Elements of V converging to E1. And so, E1 is in the closure. So, E1 is not in V, but is in the closure of V. And therefore, as a consequence V is a subspace, which is not closed. Is a proper subspace, which is not closed. I think that this phenomenon we don't see in finite dimension. We don't see it. Any subspace in finite dimension is closed. Okay. So, maybe it is useful in functional analysis to isolate what kind of statements are really due to the infinite dimensions and what they are still true in finite dimension. Maybe just to start, right? So, this we don't see in finite dimension. It is written here. And so, let's see if there is something else. Yes, maybe this one. Exercise. Let V be contained in E. Hyperplane. If V is not closed, then V is dense. So, we know that there are non-closed subspaces. Now, here there is a little bit more. Is a hyperplane. Well, it is not a subspace because maybe I find. Here is a hyperplane. Assume it is not closed. Then you close it. The closure is still a subspace, right? But the hyperplane is called dimension one. So, if you close it, you have all the space. This means it is dense. So, try to make this rigorous. But the proof is exactly this one. So, if it is not closed, you take a point outside. And then you close it. This is still a subspace. Remember the definition of hyperplane. And then you cover all the space. So, this is a home. Let me see if there are other exercises. So, now we go back to the ambanak. Let me see other consequences of the ambanak. OK, so maybe I can state the theorem. So, assume that E is a banaq space. And let A be a subset of V. And B is a subset of V with the following properties. So, assume that A is open convex. B is convex. Assume that the intersection between A and B is empty. Then, so you have two convex sets. Maybe this is a convex set. And then you have another convex set like this. So, this is A. This is B. Then, the theorem says that there exists a closed hyperplane with the following properties. Notice it is closed. This means that we have to look for linear continuous functional. But the point is that there is a closed in the picture, something like this, say, which separates the two convex sets. It says that one convex set is in a half space and the other convex set is in the other half space. How can I say this? OK, so this is L equal to alpha, maybe. L equal to alpha, the green line, closed hyperplane separating A and B. What does it mean separating? It means that, namely, there exists, that is, edesta, exists a linear continuous linear bounded, that, well, you have to think of, you have this, the graph of, you have the graph of your hyperplane and this means that this is in sub-level alpha and this is in the super-level alpha. So, there exists linear bounded such that say, for instance, A is contained in L less than or equal to alpha and B is contained in this super-level, alpha super-level. Or, in other words, for any A in L of A is less than or equal to alpha and for any B in B, L of B is larger or equal to alpha. This is equivalent to this. So, you have to think about, yes, simply the word separating is OK. The point is that here there is this closed. So, you have to look for something which is continuous. So, it's not trivial. And so, this is one version and then there is, before doing the proof as yesterday, I think that we will state the theorem, then we make some corollaries, comments and try to prove it. So, this is, so now, crucial assumptions here are this and convexity and of course, they must be disjoint, otherwise, in general, you cannot. OK, so let me write, so this is theorem one. Now, let me state theorem two. So, in Banak, A and B, contained in E, assumption one, two and three. So, one, A is non-empty and is convex. Two, B is non-empty and is convex. They are disjoint. So, here, you see, I have not written open. Now, I add closed here. And compact here. So, what could happen in this picture? In this picture, concerning here, I have to know, I will state that this is, but in this picture it could happen the following. I could imagine this open convex set, little bit larger, but open, still open. OK, this is A, which is open. So, the boundary in finite dimension, the boundary is not in A. And then, I can have another convex set, like this, just going to this point. OK. Still, we can separate. Since we have here the large inequalities, we can still see that this is still reasonable, even in this condition. When this point is exact, this goes here, say, this point belongs, say, to B, but does not belong to A, because A is open, so that they are still disjoint. And still, we can imagine the green line. Now, here the situation is slightly different, because now you are assuming, say, that A is closed. So, A is closed. Assume, you are in finite dimension. So, A is closed, now you add also the boundary. But B is more, now in finite dimension, B is even compact. And they are disjoint. So, now this situation is not allowed anymore, because this endpoint would belong both to B and to A. So, one imagines something like, OK, this is closed, this is closed, OK, maybe it's infinite, convex. I don't know, but closed. And then this is compact, however. Now, the idea is that, well, under this condition there is a little bit space in between, and so we can gain some epsilon. What does it mean? It means that there exist positive epsilon, and there exist linear bounded. Such that now is strictly separating. I mean, one is in the alpha minus epsilon sub level, and the other is inside the alpha plus epsilon super level. So, there is some space, such that A is contained in the, is contained, something like this, OK. So, this is the statement of theorem one. Of course, here compactness will be crucial in some sense, to get to have this small space. You see, infinite dimension, what could you say infinite dimension? You could say, well, I have a closed set, a compact set, they are disjoint, then they are at positive distance. Well, here we are not infinite dimension, we are in infinite dimension, but at least in finite dimension there is some distance in between this and the other one, OK. So, we gain a small space, and still here one gains, in this sense one can have this small space, so in some sense I could enlarge a little bit one of them, and fine, I could enlarge it, take points of distance, say epsilon, and then still have this statement for the enlarge set. Now, however, we will see the proof, but before then, before doing the proof, I need some, I would like to show you some corollary of theorem one and theorem two in the order, so, ah, well, this is the first corollary, corollary density, I will tell you now, in which sense this is called density criterion. OK, so assume that f is a subspace of E, subspace of E, and assume that this is not, this is strictly contained in E. Then there exists, let me call it L in a star, this is a linear continuous function in star, such that L of x is equal to zero for any x in f, and L is not identically zero. This is the density criterion. Now, in which sense it is a density criterion, then we will prove it a density criterion. Density criterion, this is usually as follows, assume that you want to show that some subspace is dense, assume that we want to show, this is not the proof of the density of the corollary, it's just consequence of the corollary, then I will prove the corollary, OK? How I use this corollary? Assume that we want to show that some subspace, some subspace f of E is dense. This means this. Then it is enough to show, then it is sufficient to show that given L in star, L x equals zero on f for any x in f implies that L is identically zero, OK? This is very useful. So, in general, assume that you have a subspace, you want to show it is dense, while you take any linear continuous function vanishing on f, and then you prove that this, if you prove that this vanishes everywhere, then necessarily f was dense. Because if it is not dense, then you contradict this statement. Is it clear? OK? So, this is usually used as follows, this corollary. OK. Now, how can we prove this corollary? So, let us go through the proof of the corollary now. So, we have proof of the corollary. Take a point, so if f is a subspace, but we know that by assumption this is strictly contained in e, so we can take a point outside. We can take a point outside, and now we apply theorem 2 with the following choice. I don't remember which was the compact and which was the closed set. B was compact. B is equal to x naught and A is equal to f bar.  OK? In particular, f bar is convex. So, f bar is convex, closed, non-empty. Well, non-empty subspace. Add non-empty here, OK? So, f bar is in the assumptions of theorem 2. B is a compact convex non-empty set. By this inclusion, they are disjoint. So, the intersection of B and A is empty. So, we define the assumptions of theorem 2. Therefore, we can find the linear continuous, there exist alpha, let's say there exist l linear and continuous, bounded, linear and bounded, and then there exist alpha, let's say l of such that. I am strictly separating. So, I even don't take epsilon, but I have the strict inequalities here. So, l of A is less than alpha, less than l x naught. And this for any A in A. I even don't need epsilon here, but I just need the strict inequalities, OK? I have not used one assumption, well, not completely at least, but now I remember that A is a subspace, because it's the closure of a subspace. A itself is a subspace. So, this inequality is not possible, unless l is identically equal to zero, because this is less than a number, but then I can change sign to A and I can multiply it by any real number, 1000 and minus 1000. This inequality is not possible, you see? This is a subspace, I can change sign. I can multiply for any big number here, positive and negative. OK. And therefore the conclusion is that, unless this is vanish identically on F, this is the only possibility. This equal to zero. If it is not zero, I can multiply by big positive or negative number. So, the only conclusion that it is possible, so this is impossible, unless this is equal to zero, which is the conclusion. So, this is one assumption, sorry, one corollary, and then let me show you another interesting corollary. So, this is the density criterion. So, assume that star is separable, then e is separable, and the converse is not true. Remark. E is separable. It does not imply in general that the dual is separable. Sorry. Why is this so? L1 is separable, L infinity. Sorry. L1 is separable. We know that we can realize the dual of L1 as small L infinity. This is an exercise that we have made. But L1 star is not. On the other hand, we can use corollary. So, this is corollary 1. This is corollary 2. The claim is that we can use corollary 1 to show corollary 2. And therefore have this interesting conclusion on separability of corollary 2. So, if it is separable, star separable, we can choose, implies that we can choose a countable dense subset of the star. So, there exists, let me call this fn contained in star countable dense subset. Now, for any n, now fix epsilon positive, take for any n, first take xn, take for any n, take for any n, xn into e with the following properties divided by xn larger than or equal to 1.5. So, I take for any n in correspondence on this, the definition of norm of fn, which is the supremum of this divided by this, you can always choose this. Remember in the definition. Remember definition of, do you agree with this? Do you agree? It is the supremum of this divided by this. And therefore I can. So, now I have a family, countable family of xn, and then I can take the span who can consider the linear span with rational coefficients of xn. So, we can consider the span of xn. It is, of course, yet countable because I'm countable, countable contained in e. And it is also dense and dense in the span over the real. So, let me call this v. So, if I show that v is dense, I am done. We agree? Because then this is dense in this, this is in dense in e, and therefore the span over q is dense. Is it okay? No? Which is the point? Are there questions on this? This is okay? So, I have chosen for any n one point of your space. Okay. Satisfing this inequality. Then I would like to show that, now I'm looking for a countable dense subset of e, right? Because I want to show that it is separable. So, the idea is, okay, I take all linear finite combinations of this with rational coefficients. And this is this, and this is yet countable. Since q is dense in r and these are finite sums, this object is dense in this. And I call this v. Now, I try to prove that v is dense. Try to prove, let us try to show, let us show that v is dense. If we show that v is dense, then this is dense also. Because this is dense in this, this is dense in v, and v is dense in e. And therefore also this is dense in e. Okay? Fine. So, we have to show this now. Let us try to show this. And now we apply the corollary one. The density criterion. We want to show that some subspace, because this is a subspace, v is a subspace. The closure of a subspace is a subspace. We want to show that this is dense, okay. And therefore we apply the density criterion. I mean the consequence of the density criterion was, I want to show that some subspace is dense, so I take any linear function on v and I show that this is zero. Vanishing on v, then it is identically zero. Okay, so let, let me denote it by L. From e into R, L linear and bounded. And assume that I know that L, Lv is equal to zero for any v in v. And then we have to show, okay, so now fix epsilon positive. So now we know that fn is dense. Therefore, there is fn minus L is less than epsilon. Okay. So there exist n, depending on epsilon, such that fn minus L is less than epsilon. Okay, this is by density because fn is dense in vista. Now, no, I mean this is just to say that given epsilon there is some element of fn which is epsilon close to L. So fix epsilon, in the ball centered at L of radius epsilon you take one fn is dense. Now we want to use this inequality and this assumption here. So we proceed as follows. So we have that epsilon is larger than fn minus f which is larger. This is a norm and so it is larger than fn minus f, fn minus f comma xn corresponding to this. So to any fn, so we have chosen this n epsilon, right? So we have fn epsilon, maybe. Let me use n epsilon and given n epsilon we have xn epsilon. So let me use this xn epsilon here. Maybe it is better to use n epsilon here. So xn epsilon, fn epsilon minus L. I am making some mistake in the notation. This is capital L, sorry. This is by definition of norm of course which is the supremum of this quotient. So what is this now? So we know that xn epsilon, remember xn epsilon is in v. Yes, of course. So xn epsilon is important to remark that xn epsilon belongs to v. So this is equal to what? We have fn epsilon comma xn epsilon minus L comma xn epsilon divided by this. And now we are going to use this. So if in place of v, we take xn epsilon, this is equal to zero. Because xn epsilon is in v. This means that this is equal to zero. But let me go slowly. This is zero, right? This is equal to zero because of star and star star. Where star is this? And star star is this. So this is simply this xn epsilon divided by xn epsilon. And now we use this assumption. So by this assumption, by this choice, we have that this is larger than 1 half the norm, as he was saying exactly this. So you see interestingly we have found that in the end of this inequalities fn epsilon for any epsilon positive exist an epsilon such that fn epsilon is less than 2 epsilon. So from one hand this says that the longest subsequence fn epsilon is going to zero and the longest subsequence it is also going to f because we have this. So take epsilon say equal 1 over k and 1 over n. Then you can find a sequence of indices such that along this subsequence fn converges to l. At the same time epsilon is 1 over n and so this converges to zero. So the limit f is equal to zero necessarily. l is equal, so sorry, I am confusing f with l. So sorry, I repeat. Along the subsequence this is going to l. This is going to zero. Therefore l is equal to zero. And this says so we have to show that l is equal to zero. We have shown that l is equal to zero. Therefore necessarily v was dense. As a consequence of corollary 1 of ambanak. So this is a quite interesting result of structure of this strange infinite dimensional spaces at the end. I think that I have said all consequences that I want to say on theorem 1 and 2. So now we have to show those theorems. So proof of theorem 1. So you remember the proof of theorem ambanak theorem of yesterday was made into two steps. The first step was how to define the extension at the point on the linear span of one point outside keeping the constraint and then there is the zorn abstract zorn machinery. Now we want to use the theorem also in this case but still we start with the computational part assuming that now I don't remember if a and b maybe a was open convex of a was so assume that b is just a point for simplicity. So assume for the moment now we start to do the proof in the simple case that we have just one point outside. So you want to separate an open convex set with a closed hyperplane separated the open object and the point outside which is a convex set. So here there is no time today to conclude the proof. Of course we have five minutes. What I can say just to make a picture to convince you which is the strategy. Ok. So let me try to draw some picture in finite dimension sorry R2. Ok. This is a convex a and this is a point outside x0. Now the first thing that we will do is first of all a is open so a is convex and open so by translation it is very convenient to put the origin of our vector space inside a. So we translate the origin so that maybe the origin was here. Ok. Now we translate everything so that now the origin is here. Ok. Somewhere I don't know here. Origin. So by translation we can do the following interesting assumption that 0 is in a. So remember that a is open so a coincide with its interior. So now we have a convex set with the origin in its interior. Convex set with the origin in its interior. Ok. This is the origin. Now we want to apply so there is a general principle when you have a convex set with the origin in its interior this corresponds essentially to a semi norm it is essentially the unit ball of a semi norm and conversely if you have a semi norm then take the unit ball where the unit ball is a convex set contain the origin in its interior so let me make just some comments here. Ok. So find a dimension. You have a convex body. Ok. Like this. The origin is here. Well, this is a convex set with non-empt interior and the origin is in the interior then it is not difficult to define a one homogeneous function thinking of R2. Now this is in R2. It is not difficult to find a norm essentially having this as a unit ball. Not quite a norm exactly because this is not symmetric with respect to the origin, right? The origin is here and this is not symmetric. Symmetric would mean that if you have a point here then you have also the opposite inside but you have a point here the opposite is outside. So this is not symmetric. If it were symmetric then it is also convex and bounded convex and bounded then it is easy to find, to look at this as the unit sub-level set of one homogeneous of a norm. And then I will tell you this is called the gauge that there is a way to define a function in a natural way such that this is to construct the norm starting from the unit ball. Conversely, it is much more easier. Assume that I have given you one homogeneous convex function cut at level 1 that object is convex that contains the origin in its interior. So cut at level 1 put it in the horizontal plane this is the one level subset containing the origin somewhere. So now which are the difficulties? Well first of all this point is not this object is not symmetric with respect to the oil. This is the reason for which in the Banach theorem of yesterday we have written lambda positive and not modulus of lambda. Remember? We have written p of lambda x equal lambda p of x but only for lambda positive and this was to include non-symmetric sets because if you put this for any real then essentially you are symmetric. So if you want to include the cases in which you have no symmetry you cannot ask for this strong assumption just ask for this for positive lambda. So first difficulty is this second difficulty in finite dimension just to understand that you can have a convex body but this say could be infinite. So this this creates some degeneracy in your norm it's not really a norm actually because when this is infinite then here there is a direction essentially where you pay nothing I mean it's degeneracy here in the norm. So for instance the norm coming from this convex set is not equivalent to the Euclidean norm. So you cannot put yes of course you can put a Euclidean ball but you cannot put a Euclidean ball outside. This you cannot and so you expect that if you want to have an inequality inequalities between this norm and the other you can have just only one inequality and not two inequalities. They cannot be controlled one each other because this is unbounded. So I mean asking convexity of A and open says that you can assume that zero is in the interior but you cannot assume that in finite dimension for instance that this is bounded we can do the proof in any case but keeping in mind this this now which is the definition the natural definition of passing from a convex object to a norm or semi norm or whatever I mean even less than a semi norm but inside the statement of unbounded. Ok so if you have C in general so in general you have say C convex with zero in the interior so with non empty interior in particular then there is a function defined in the ambient space which is called maybe the gauge function of C itself and is the what do you do the idea is the following you take the definition what is it it is exactly sorry ok I am changing the name for maybe I am wrong Minkowski functional Minkowski functional also gauge gauge in France maybe minkowski let us call this Minkowski so what we are doing here we are doing the following so we have a convex body this is the origin we want to define at a point x a function then you do this you take well first you should prove that this set is absorbing points it absorbs points in the sense that if you omotetize it sooner or later you will include any point x so first you have to show that this is obvious in finite dimension but this is true also in infinite dimension once this has the origin in its interior it is convex then it absorbs points namely given any point you can scale your C so large with an alpha so large so that alpha times C becomes so large that includes x like a ball then you take a point outside then you take one thousand the radius of a ball sooner or later you include the point x and then you take the smallest alpha such that you have included x so you start from outside there is a big alpha and at some moment the smallest alpha you can have then this is the value of B at x so we will show on the next lecture that this is exactly the function such that its one sub level set is C and and you see now the idea is the following well I have a point outside here above this I have a linear sort of linear function now I define also say g of x on the linear span equal lambda so that the g of x is one it is rather clear that this is outside remember the idea that if you have one convex state inside the other and they take the linear functions then the linear functions corresponding to the smallest set is above the linear function corresponding to the larger set and therefore here it is rather clear that this p should be above this g well if this is true then we are almost in the condition to apply hand banach because we have a seminorm so almost a seminorm p linear function g and so on this is more or less the idea of the proof we cannot do this in 5 minutes we will do this, this is just the idea


Ladies and gentlemen, good morning. Welcome to this session, China's Evolving Business Context. It seems that the guests on the stage are quite influential. I very often went to the winter session at the World Economic Forum, and from time to time we saw the number of guests on the stage outnumbered the audience, but today we see such a big audience. So this session is focused on China's Evolving Business Context, a quite big title subject, but unfortunately we have only one hour, so we have to focus our discussions on Chinese businesses and the re-engineering of micro-governance structure of Chinese companies. Yesterday there was a closed-door session at which Premier Li Keqiang delivered a speech. During his speech he said, more than 10 years ago there were only less than scores of Chinese companies ranked among the top 500 companies in the world, but today there are more than a hundred Chinese companies in the name list of the top 500, so obviously Chinese companies have been developing quite successfully. And for those top 500 Chinese companies, most of them are state-owned enterprises. So this is the first message Premier Li Keqiang delivered to the audience. Second message of him is what should be the next step that China shall take to reform, particularly in terms of economic reforms. Premier Li Keqiang said, the most important focal point will be how to have a proper limit on the power of the state of the government and to give a bigger play to the market. SOEs play a very big part in the Chinese economy, for instance Chairman Fu Chengyu who represents SinoPak. His company is a typical example and also Madam Dong, her company Ge Li is also a state-owned enterprise. So the first question will be why China must have reform on state-owned enterprises and after that we will have free discussions. I was told to brief the audience of the meeting approach. I think this session is televised and if you're interested in this topics you are welcome to raise your questions on the internet and I will convey your questions to the four VIPs on the stage. So we have wireless terminals available for you to raise questions but first and foremost I will give the floor to the three gentlemen and one lady on the stage they all of them represent a very famous companies Madam Dong chairman chairperson Dong chairperson Dong Ming Zhu is chairperson of Greek electric appliances well-established and very respected brand in China and the company's history can be dated back to the 1980s. Second Madam Fu Chengyu chairman of China Petroleum and Chemical Corporation and his company is of course definitely one of the top 500 and the third VIP is Missed Klaus Kleinfeld chairman and CEO of Alcoa. Look at Mr. Kleinfeld he represents a very strong American business he has been trying to promote his company his company's products in the meeting room and the fourth gentleman is Mr. Mohamed from SAPIC also a very big and influential investment company. Okay I think lady first I will first invite Madam Dong to say a few words about why China must reform its SOEs in the international community people always see SOEs as a part of the government so when they look at SOEs they look at them with a quite a subtle and special perspective this is quite unique secondly SOE is something formed with very special historical reasons but as China is opening itself wider to the rest of the world it is of course necessary and natural for China to reform its SOEs to make the SO Chinese SOEs more adapted to international business community to allow them to manage themselves from a more internationalized perspective I think this is the most important reason okay Mr. Fu what's your thought why why do we have to restructure Chinese SOEs any serious problems with SOEs well when we are talking about this round of Chinese economic reform we need to think about what happened back in the early 1980s when China started the market-oriented reform at that time the entire economy was a hundred percent I mean the companies in China were a hundred percent state-owned enterprises there were none private companies in China so at that time people believe the advantage of China to compete is to restructure its economic institutions at that time China started to practice a mixed ownership of companies with the state ownership as the dominant structure and coupled with the cultivation of private companies and in the past three decades the SOEs survived the economic reform and prospered and also private companies survived and prospered we see bigger and bigger community of private companies and private entrepreneurs and they are the new opportunities the new vitality of the Chinese economy and they represent the competitiveness of China in a new era in late 1990s and also at the turn of the centuries Chinese SOEs it experienced a very painful process of restructuring and in about three years time many SOEs almost died but fortunately after a very painful process of market-oriented restructuring they survived flourished prospered and competed in the market and now they are much stronger than before and even in the international business community these Chinese large SOEs function and perform very well so this represents the power the economic power of China compared with the former Soviet Union the biggest difference of China is as follows in 1980s when the Soviet Union practiced economic reform the practice of Soviet Union was similar to what China did namely market-oriented reform however in China we had a mixed ownership with the state ownership as the dominance and the supplementation of private companies but the Soviet Union didn't do that so the outcome was quite different the Russian SOEs lost their competitiveness but at the same time we didn't see the formation of a very competitive private companies and private business people in Soviet Union now come back to China both Chinese SOEs and private companies are developing themselves SOEs have gone through the period of survival and the target in front of them is to improve the efficiency of themselves as well as the efficiency of the Chinese economy as a whole so both SOEs and private companies in China are very much committed to enhancing their core competence and in that process reform is definitely the pivotal to to make Chinese companies more market-oriented to reduce the influence of the administration on the market on the business community and in that process the key lies in governance restructuring corporate governance the improvement of the quality of assets and of Chinese companies so the purpose of the SOE reform is not to get rid of them from the market rather on the contrary we want to build them bigger and healthier and this is very clearly written in the document of the third plan of the 18 CPC Central Committee the purpose of Chinese economies that the purpose of the reform of the Chinese economy is to improve the governance structure and institutions of Chinese companies and make them modernized that's the fundamental goal and every discussion about SOE reform must be based on that fundamental guide line in the document so number one we need to maintain the status of state ownership as the dominant power in the Chinese economy SOEs must be more competitive more vital more dynamic and the key is to reduce a diminutive influence on them thank you very much Chairman Fu and Madam Dong Mr. Fu said that market orientation and de-administratization shall be the key of SOE reform but I what I want to ask you is how can we realize the goals of making SOEs more market-oriented and make SOEs less influenced by the government by the state it might be a little bit sensitive or difficult to you Madam Dong you when Premier Li Keqiang organized a business forum to solicit opinions from business people you said on that occasion something very interesting Mr. Li of Premier Li asked what the business community wants to have from the government any support and your answer is that we don't need any support from the government you just leave us alone and that's the biggest support for us okay Mr. Fu talked about the unique features of SOEs my company is to some extent a state-owned company but we run the company as a free private market-oriented company the other three the three gentlemen represents three top 500 companies but my company is not yet ranked as a top 500 but we are working on that we are close to the threshold but not yet in the club yet air condition is quite highly competitive market Mr. Fu his industry is quite monopolistic industry my industry is quite different so for industry like air conditioner what we really need is of environment for free competition instead of the support from the government what I want to see is that what we really need is to gain the market by using our state art state of art technology if we really rely on the government support and the administrative support from the government I don't think our enterprise can be developed in a system system ability way so we really need to think about the long term and the sustainability of our corporation if you look at the fortune 500 we can see many of the previous fortune 500 has existed from the club I so I don't think that becoming a fortune 500 is such prime importance to us I think what I really want to focus is that whether we can establish a long term sustainable corporation that's really what I want to focus if we want to talk about the reform and the reshuffling of the SOEs the focus is that we need to set up eternity of enterprise which is we can have a long live enterprise now we would like to move to the international perspective now mr. Klaus from alcohol alcohol in your view I mean how you have been dealing with the state on enterprise all along so how do you view China's state on enterprises from your own point of view well look I would come from a different perspective and I would come from a perspective of China has roughly 1.3 billion people right a lot of those people are very intelligent and China is known for being very entrepreneurial right when you now move to the west coast in the US and look who is engaged there in the Silicon Valley you see a lot of Chinese you know and they are very entrepreneur they are involved in building new companies and it's just a simple fact of life that you have to let the market forces flow and give people of any age and certainly young people the opportunity if they have an idea to find the money to find the support and to build those right so I think that the question is rather I mean how do we have to reform the state on enterprises rather I mean we have to give that segment of the society better access so that they that they have the same type of chances than the state on enterprises right and we know that the financial system has not been as such and that it actually was was and is still very difficult and very expensive you know if you want to build your own company so I would start with this and I would yesterday we had the private session as chairman who said I mean with Premier Lee and I mean you see the reforms and the understanding is very very clear there is a desire to have a more balanced economy to have a more sustainable growth so in reality we need those type of reforms he was talking about yesterday about the financial reforms this is great let me tell you an anecdote that shows you that something has happened that I think we need to reverse I remember I've been coming to China for many many years and I remember and typically once a year I go to university and talk to the students and I remember when I first came as always nice and full and a great debate and now they are all speaking even better English you know fantastic and they are typically very intelligent and I would hire most the most of the time would love to hire almost everybody I still remember in the first years I would say the first ten years when I asked the question of what do you desire where do you want to go and I asked it in categories I mean who of you wants to join an international firm who of you wants to do a startup you know and who of you wants to know a state-owned enterprise it was very clear the desire of the best and brightest in the university was I want to join an international firm because I want to get it into education and I want to have an opportunity to understand how the global a global market works not so in the last I would say ten years if you go around to the top universe today and ask the question it's kind of scary because most most of the young talents as well I want to go to a state-owned enterprise when you say when you ask them why do you want to do that they say because that gives me better opportunities to grow inside of China right now there is something that's wrong with the allocation of of what I consider the most important resource for any economy and that's talent right so I think we got to put it back together let it let let opportunities grow and also see the opportunities that Western international firms bring because on the question of I mean what is wrong with state-owned enterprises first of all monopolies are never good I agree with that point that that point because they make you slow right I don't by the way I think that there are quite a number of state-owned enterprises that are very well-run also very very well run and have excellent excellent talent they have avoided this this lethargic lethargic move but at the same time you need to make sure that that the talent has opportunities to find its own way okay yeah thank you I'll pass out the question to mr. Armandy or he's one of the 500 you know just biggest company in the world but your company is also a state-owned right I mean so how do you operate a state-owned company in in Saudi let me let me share some of our beginning we started with almost two billion dollars that was the beginning the government gave us two billion dollars and say you know you have to start petrochemical that initial capitalization initial capital how many years ago now that was in 1977 today this company is 130 billion dollars and it's listed as the number 120 from the five fortune 500 so this didn't come really by chance it came by a really good strategy by the government it's a hands-off strategy and it put the company on the on the road and then it's up to the company to continue but we have not been tested you know if we had made a mistake or something whether the government is going to interfere or not we have been successful all the way so hopefully this strategy will continue we have a board from government and then later the government divested 30% from Sabbath to the private sector we gained some experience from the private sector we gained through our acquisitions we acquired the view companies around the world the likes of GE plastic and Dutch state mind DSM and Huntsman and we gained experience from those company from global company in the company especially in the compliance area which is very critical to us we gain experience from the media like you because you you keep the transparency of these companies under scrutiny so media compliance and government's hands-off policy really resulted in a good company we have joint ventures here and in China with the my friend here Mr. Fu and in the petrochemical area and we have gauged that the competence of state on companies in Sinopik and we don't find really much difference as a matter of fact we really appreciate what they do and the safety and the environment and other things which for our two companies was really a win-win so the most important thing is really have a long-term strategy and execute on that long-term strategy the compliance is of the essence and also I think some participation of the private is important but the muscles of the state must be there really to drive forward and bring the strategy to completion okay yeah so why do we need to reorganize or restructure the SOE the four speakers has shared with us their insights and their perspectives how can we just reform the SOEs what kind of methodologies can we use to really implement it the third plenary of the 18th CPC session has raised out the mixed ownership structure concept the so-called mixed structure concept is it really a very good methodology to have the reform of the SOEs our friend from the Arabian world has told us they are also SOE they're also developed from the 1977 the fund of 2 billion to current days of 130 billion so if we can solve the ownership structure can we solve all everything so I'm going to raise several questions since we are going to get some questions from the internet and then we are going to open the floor for Q&A session if you have any questions or any comments please just raise it madam don the mixed ownership structure do you think that is the good solution for the SOE problems I think the main issue is still about such a mixed ownership reform can be regarded as a kind of medicine if we take a good medication and then the disease will be well treated however we are worried about any kind of other complications it may bring about when we had the medicine the most important thing is that we need to follow the market rules the corporations will operate in a market oriented environment without the interference of the government if the government is still thinking about since I'm the biggest shareholder or least I'm one of the shareholder and still holding the old rooted concept of the administrative interference I don't think the mixed ownership structure can be a good solution I have been thinking about it for quite a long period of time since I've been serving as the delicate successively for three times and I have been raising the idea that no matter your mixed structure or listed company or SOE every company should have a uniform and standardized modern governance structure which will also create a level playing field for all the enterprises no matter what kind of ownership are you and during the development process of your corporation no matter your SOE or private sector I believe the decision makers which means the senior executives decision are extremely important what kind of road are you going to take your corporation to 20 years ago for the very first 10 years we haven't fully aware of our corporate social responsibility that is to say for the first 10 years we are thinking about how to make a profit or how to survive that is our objective of our corporation however during the process of our evolvement the CSR seems to be more important than making a profit the environmental damage that has been brought about by the operation of our corporation or not being able to able to operate in an honest way all these things are not the things that we want from a modern corporation with CSR so one more we have realized we need to have better operational ideas and use optimal technologies to ensure that we can have a good corporation it is not really about whether we can realize a mixed structure which will just ensure our success so you are talking about from the shareholders or stake structures perspective if these types of ownership structure wouldn't be able to create a miracle without our modern governance for example in the board of directors there should be a platform for open exchanges and etc if we are only taking some fund from the private sector it won't solve everything right yes indeed grain electronic appliances at the very beginning it is a wholly owned SOE owned by our provincial government at the very beginning they have an investment of 30 million currently we have given them great return which is more than 300 times than the investment at the very beginning so they are getting our return with and they are trying to receive our generosity however if there is anything they will just interfere from the administrative perspective even though they have been gaining great returns from us so we need to battle with them as I said our industry is a fully competitive market without a market-oriented business approach we would have already died in the market so let me rephrase what you said the government comes to you when there is good things but when bad things happen they just run away yes that's true this is what what we have experienced in the past this is the truth they the government is sometimes we want sometimes the government comes to us that C agree is a company just to lend some money to this subsidiary of a Chinese of a government controlled company because that company needs equity injection but we always said no no way because we was a power agree is a public listed company we can't do that okay mr. Fu mixed ownership do you think it's the is that the solution I think our discussion as I said should be based on the document of the third plenum of the 18th CPC Central Committee mixed ownership is not everything about SOE reform and it is even not the main part of the SOE reform it is only one part or one of the many targets objectives of China's SOE reforms in the past actually mixed ownership is not something new ever since 1984 mixed ownership has always been the buzzword of the Chinese economy we've been walking along this road for more for 24 for 30 years already so today when we talk about mixed ownership our attention is focused on the ultimate goal of the reforming of Chinese economic institutions it is merely a two so as I said when we talk about SOE reform this topic actually involves many diversified thoughts modalities and so on the in my view the fundamental solution is what I said earlier how to make SOE is more market-oriented how to reduce the a domestic administrative interference in Chinese companies so great is a local government-controlled company but to my company Sino-Pak is a central government controlled company to be honest I think Sino-Pak is has a bigger degree of a freedom do you necessarily mean that the central government has less influence or interference in you compared to Madame Dong's company yes that's true in there is no no people designated by any central government ministry to sit in our boardroom you see this industry is somewhat a monopoly industry in China but when we compete we have to compete in the international market for instance for Sabic we work we cooperate with them but in some cases we also compete with them so for companies like Sino-Pak our eye is on the international market we compete in the international market instead of domestic market we have invested 90 billion Yuan and this 90 billion Yuan is not completely from the Chinese government or from Chinese banks actually we borrow from the international capital market from foreign banks because the borrowing cost is lower in the international capital market compared to domestic market so and there is no government approval process when we make investment decisions in the international market but only one exception exchange foreign exchange when we make outward investment we must get the approval from the state of administration of foreign exchange so as Premier Li Keqiang said the government also needs to change its mind and as far as this point is concerned first and foremost is the approval system and also the rule of law mentality so Mr. Li Keqiang said yesterday the Chinese government must delicate its power to the lower level government or even to the business community bigger freedom should be given to Chinese companies to make their independent business decisions of course some other topics shall be considered such as a safety or environmental protection but as far as the profitability as far as the bottom line of the company's concern this is the business of the company it's not the business of the government in China within China apart from tax we have to pay many kinds of fees fees are very special phenomena here in China apart from tax you also need to pay a kind all kinds of fees to the government last year we paid over 300 billion yuan of taxes and fees to the government every day I pay 800 million yuan every day to the Chinese government covering all kinds of taxes and fees this is unimaginable for my international peers because Sino-Pak is a state-owned company we have to fulfill our corporate social responsibility so the biggest thing we pay to the government is not tax actually it is fees this is very unique in China but was that said of course Sino-Pak has to improve its competitiveness to make it more make it stronger in the international market so we we are focused on the international market we need to be more open more transparent more standard as far as corporate governance is concerned only when we can do that can we develop into a stronger business can we have a bigger capability to undertake our corporate social responsibility for instance climate change big companies play a big role in this process we commit to the government that in a three year period we need to spend about 28 22.8 billion yuan to improve our environmental responsibility at the same time we cannot cut our workforce significantly this is also why the profit per employee of Sino-Pak is lower than our international peer so cutting the headcount is not the possible way for us to improve our profitability so our only way out is to run more efficiently so Sino-Pak is a huge company we have a big responsibility we also have big burden for the society for the government but still we managed to grow at a very remarkable remarkably faster pace year by year so as I said cutting employees is not the possible way compared to some of my international peers the number of Sino-Pak's employees is about ten times of them so Chinese companies have to find their own ways we shall learn from the west but that's not enough we learn the western concepts the western skills the western expertise but when it comes to the management the day-to-day management on the field we must take into consideration the Chinese unique Chinese characteristics so Chinese SOEs and Chinese private companies it's the same thank you now I'd like to give the floor to the two gentlemen from outside China maybe for for them the mixed ownership might be a little bit difficult to concept for them anyway it seems that there are many questions raised on the website the internet recently Chinese government decided to to set a ceiling on the annual salary of a C-level executives of Chinese state-owned enterprises and this involves many industries banking and many other industries and the ceiling is 600,000 or seven six 600,000 to 700,000 this is the ceiling for their annual salary so my colleagues from America and Saudi Arabia many of these Chinese state-owned companies are already top 500 companies they compete not only within China but also outside China when such a ceiling salary ceiling is set on them namely around a hundred thousand US dollars every year do you think it's a good way for them to stimulate or incentivize them in their day-to-day work well what's your what's your comment Mr. Klaus perhaps you can yeah I mean Mr. Fu I'm sure as a salary reduction he he's leading one of the largest oil company in the world he cannot make over his salary probably cannot be over a hundred thousand US dollars I mean so what's your take on this you like with the previous discussion on on co-ownership I mean I think the real question is what do you want to solve what's the problem that you want to solve right so if you if you reduce the incentives of talent going into a certain industry by cutting the salaries is that what you want to solve I personally believe that talent is the foundation for everything right so you want to attract the best and brightest to go into the places where you need the most right so that's one way to look at it I mean and and on the other point I mean if you have somebody co-investing with you I mean again what do you want to solve with this I mean I personally think that if we want to have a broader sustainable growth in the in the society and I think we have to focus on getting a level playing field first right and a level playing field requires a financial reform so if somebody has an idea and wants to wants to start their own company they have to have access to to money and to funds in the same way then state-owned enterprises have to have it because otherwise this is not going to fly on the discussion on governance you know again I can see what you solve with better governance I mean if you say I want to have a more global perspective in the board obviously that's a great thing I personally love the board that we have and the discussion that we have and every time that we are together I learn a lot and there but again it's a different perspective I mean there it could be could make a lot of sense for Chinese companies that say we want to globalize so we want the global discussion also to happen more strongly inside of the firm and we start with the board so we bring on board I mean a number of respected international voices so that you can have them reflected you know if you want to solve the the the whole aspect you touched upon it on and you too on environmental aspects I mean unfortunately not every company here in China as we know also from recent events is very safe you know and not every company here in China it really is very environmentally friendly otherwise we would not have these gigantic issues that we are currently struggling with here and here in country so there the issue is you you you establish not only standards we have established standards but you enforce the standards right so again I mean depending on what you want to solve the answer is different on the bureaucracy side there's thousands of ideas how you can solve for because you want aspect you mentioned I mean if you live in a province and the provincial leader who is basically also giving you the finance you know or gives you tax reliefs you know it tells you you can't fire any people well that's a sheer recipe for bureaucracy right and certainly there's no way if you have those type of restrictions that you ever are going to be competitive on a global scale okay yeah well what they are trying to solve there is this our central government that I'm party they feel that those stay on can enterprise the chief chairman's CEOs and that they are making too much over the society a tolerance level I mean that's what they're trying to solve I mean they feel that they're making too much I mean the simple rule of life is I mean like with taxes you know I mean people go where they feel their benefits are higher you know so in the end I mean the consequences people will go somewhere else that might be a good thing at this point in time you know and in general I mean these these discussions on who should make what are very difficult societal discussions I mean that have really multiple facets and I've always seen that they only come up in reality because very often they are a substitute for different discussions a substitute for discussions of what societal standards do we have I mean what how how clean is our society how do people get treated how much social responsibility is shown in a in a region usually this is the real discussion and the rest then becomes the proxy okay so you're not going to talk about conspiracy theory so to already I mean I'm not going to ask what you make I mean but could you imagine like a mr. fool here I mean they cannot make well is a hundred thousand US dollars a year I mean is that to you I mean it's too much to make or too little I mean could you make well let me go back a little bit to the mixed ownership question just if you divert little bit and they come back to to the compensation issue the mixed ownership is okay if it if it has the representation on the board without representation is meaningless as far as I'm concerned so having members from the private on the state owners right they will defend they will defend the company and if there is interference from government we always go and get the help from our independent people on the board and and the other way around I mean if there is much pressure here we can go to the government so you can you know get a balance of governance there so if you fix the the salary of the the there is no incentive for entrepreneurial ship and into the company the best way is really pay for performance if the company doesn't perform he doesn't get much if the company do a fantastic performance on a global basis he should be paid he should be rewarded not only him but all of the executive management on all of the employees so really we have a performance culture in the company that drives efficiency as mr. full said you cannot drive efficiency without performance and performance require you have to pay okay yeah now this one day jingles them and I'm sure we're going to find a way I would like to come back to the two Chinese speakers madam so on this competition scheme clearly involves you because it is closely relevant to you actually since we are listed public company for so many years we have been govern our corporation according to the rules of the public listed company so this question doesn't have so much relevance with me so I do not you do not have a ceiling of your salary right and actually it is not a new rule after the reform of the mixed ownership structure it is basically targeted on the central SOE is probably it has more relevance with mr. fool why would we just give the floor to mr. fool to share with us your opinion your idea probably it is more pro appropriate for me to address this issue because I'm one of the state coders real state coders actually personally all from countries perspective we are going to reach different conclusions and from the international trend you can also reach different conclusions and results from a personal perspective the reduction of a compensation wouldn't be extremely happy for a person however if you leave a person aside and if you stand under the level of our country you will believe it is something we should do why do I say so because our country is different from the western nations in a sense that we want to pursue a collective richness so this is the national and communist party policy of pursuant of the collective wealth of everybody and we have been paying great attention to the idea that our mass public and our workers stuff are the real masters of our nation this is a national strategy and also another thing is that we have a widening gap of the income distribution how can we have a harmonious development of our society and to benefit the majority of the society is that of only benefit the minority of a society for example the person like me who is the minority of a society I think from our country's perspective we should bring benefit and welfare to the mass public and the majority of the population however our reality is that we have been witnessing the widening gap of the income distribution our central SOE especially in the manufacturing industry is much better than our other industries however our salary is still much higher than that of the ordinary staff of our workers the more they pay me the better I the better it is however how the next question I'm going to ask is that how can we incentivize our stuff if I'm getting extremely higher salary compared with my stuff I'm not trying to sing high praise of myself or anything I want to share with you some of the examples of the early 98 years I began to engage the international exchanges as the chief representative of the Chinese side we are working with the foreign joint venture side including my secretary they are given the salaries from the foreign shareholders side they are getting a salary of more than 1,000 Hong Kong dollars and I'm getting 300 RMB and my secretary who is given a salary from the foreign side is getting a much higher salary so I'm used to that I'm also trying to share with your example of the working experience of working for Synoc which is the offshore oil company that is to say our company is a red cheap public listed company because we are listed in Hong Kong that is why we are called red chips so at that time foreign investors said that you are getting such a low salary and I'm really worried that you wouldn't really devote to your work so according to the international practice and the tendency we have designed the so-called the Hong Kong compensation scheme including the options that we can get as a senior executive so at that time our compensation scheme in Hong Kong in the year of 2001 should be around 3 million to 4 million and later this increase to more than 8 million the options added that we will have annual salary of more than 10 million Hong Kong dollars it is approved by the Board of Directors and it is also approved by the MOF the Ministry of Finance so it is totally legal and legitimate legitimate for us to get such a high salary but at that time I'm talking with my senior executive that we cannot get such a high salary the reason that we cannot get the compensation is because if you take such high salary you wouldn't be recognized by our workers to be the leader of this team because our Chinese history is different from the western world their history in the western world are accumulated by the process of privatization however our ownership is totally government monitored such a structure so if you want to give us such a high salary then we will just donate it and we are still getting the salary from the Chinese SOE but it doesn't has any influence to the performance of our company at that time our market cap is about 6 billion when I leave it has reaches more than a hundred billion Hong Kong US dollars all the investors said that I'm a happy investor so because the reason that they are happy investors because they have been making huge profits so when I come to Sino-Pak I have attracted the investors from the center could to Sino-Pak they begin to buy the shares of Sino-Pak so it doesn't really rely on whether how much I can make it really rely on the performance and the efficiency of a company so we cannot leave the Chinese contest to make a comparison with the salary with our firing counterpart after 2008 the financial crisis the whole world is criticizing the extreme high salaries of the city executive and so the Chinese have been following that concept even before 2008 if you want to get a high salary then you don't need to work for SOE since you are so capable you can work for other private companies where you can earn a high salary however all value is not shown there because I can gain respect by my peers because I can lead a Chinese SOE to be competitive in the whole world that is much more important than the personal salary and the personal income so Mr. Fu you mean that for the Chinese SOEs you will have to depend on your man morality right you will have to be very ethical you need to stand together with the country actually you should have a proportional kind of distribution of income with your staff if in the western world if this proportion should be dozens of hundreds of times if it is a corporation based on private ownership that is reasonable however it is if it is a public ownership as in China then you wouldn't be able to work in China when I leave SINUC my options measured by Hong Kong US dollars Hong Kong dollars it is reaches more than 400 million Hong Kong dollars I donate all of them so the Chinese leadership do not pay great attention to the income when they announce the new policy of the reduction of the compensation for the senior executives we are not complaining at all because of the time limit I would only allow one quick question I would like to see whether there are any questions from the audience please would you please be more specific because we are we don't have the time to cover a very big topic I'm coming from Tsinghua Technology Park we have already cultivated more than 2000 corporations in our park however there are a few private companies in China can become a fortune 500 so I think there should be a mutual and bi-directional investment between the private sector with that of the SOE sector how can SOE support the private sector corporations development because of the time limit we would only allow one minute for each speaker to answer this question how SOE support the private enterprise I think we shouldn't just divide the private and the private public in such a clear and crystal way because grain corporation our company it should be sad to be a mixed ownership we should say if we really want to have such kind of divide to be divided into big or small enterprise if you have developed yourself into a very big scale we would pay attention to the development of the SMEs this small and medium enterprises the SMEs should try to have your own reflections on how to create the state of art technology while at the same time the big enterprises would also give chance to the SMEs it is not such a simple relationship between public and the private to be very simple is the SOE is obliged to support the private enterprises for each enterprise we cannot cover all the supply chain so obviously the big enterprises will be followed by chains of SMEs in our petrochemical industry we have more than hundreds of enterprises working with us so we in the future the more we want to say is that when we go global can we just create such a supply chain conglomerate to go global well we don't have time left before the thing so now today we have a very good discussion about the SOEs we would like to ask everybody to have a round of applause to thank all the speaker on board