AI Assistant
Transcript
00:07:100Paolo Guiotto: I'm open.
00:15:280Paolo Guiotto: Okay, good morning.
00:18:840Paolo Guiotto: So today, we are going to… Talk about limits.
00:27:970Paolo Guiotto: of sequence factors. So we start with the definition of what is the limit that looks similar to the limit we have for sequences of numbers.
00:39:850Paolo Guiotto: So let… Z… equipped with a certain norm, huh?
00:47:590Paolo Guiotto: B.
00:48:860Paolo Guiotto: E. Norm.
00:53:420Paolo Guiotto: faces.
00:57:270Paolo Guiotto: Then we take a sequence, F, N,
01:02:520Paolo Guiotto: of elements of V, vectors of V, called, as you like.
01:08:880Paolo Guiotto: We say that,
01:15:540Paolo Guiotto: So, at least the limit.
01:18:820Paolo Guiotto: in N of the sequence FN, and this is F, an element of the space.
01:25:630Paolo Guiotto: belonging to V, or equivalently, we are right that… well, this notation does not emphasize the fact that, as you will see.
01:36:940Paolo Guiotto: The existence and the value of the limit depends on what kind of norm we have on the space, so perhaps it is sometimes better to write something like this.
01:50:750Paolo Guiotto: So, it converges to F according to the norm.
01:56:180Paolo Guiotto: of the space, if… well, if you want, we could give exactly the same definition. URL for limits of sequences of numbers, so for every epsilon positive, there exists an initial index. Well, let's write in red.
02:12:980Paolo Guiotto: for every epsilon positive, there exists an initial index, capital N, such that the distance between Fn, and the limit F, becomes smaller or equal than epsilon for every index sufficiently large.
02:28:960Paolo Guiotto: If you take the absolute value, you have the definition of limit for a sequence of numbers, of finite limit.
02:36:850Paolo Guiotto: Or, equivalently, If you look at these, these are… these quantities are numbers, no?
02:46:430Paolo Guiotto: They are norms of something, so they are numbers, positive numbers, so they are actually greater or equal than zero.
02:54:880Paolo Guiotto: And so what this property says is that that quantity, so the distance between FN and F, becomes very small, arbitrarily small, provided you take N is sufficiently large.
03:07:790Paolo Guiotto: And this is equivalent to say that that quantity as a sequence of numbers goes to zero in R. So, or, equivalently, the distance between F, N, and F, as here a numerical sequence, okay.
03:23:790Paolo Guiotto: It's not a sequence of vectors in the space V, it is a sequence of numbers in R goes to 0,
03:30:570Paolo Guiotto: when n goes to plus infinity.
03:36:820Paolo Guiotto: Now, this is the definition.
03:40:980Paolo Guiotto: A first, apparently innocent remark is the following, proposition, little proposition.
03:50:250Paolo Guiotto: That, if… limit.
03:55:250Paolo Guiotto: of FN exists, Then, it is also unique.
04:07:990Paolo Guiotto: So there cannot be two limits for the same sequence, and this is the standard argument.
04:14:440Paolo Guiotto: Because, we can say that, if the sequence FN goes
04:21:709Paolo Guiotto: According to this norm, 2F, and at the same time, it goes to G,
04:28:520Paolo Guiotto: then we would have that for every epsilon positive, so writing this, that would be an initial index capital N such that the distance between Fn and F would be less or equal than that epsilon for every N greater than this initial capital N.
04:47:830Paolo Guiotto: Okay, but the same would be for the second property, so I should write, and also…
04:55:520Paolo Guiotto: For every epsilon positive, there exists an initial index that is not necessarily the same in principle, so let's call it NCLDA.
05:05:580Paolo Guiotto: Such that the distance between F and the second limit becomes less equal than epsilon for every n greater or equal than n tilde.
05:16:510Paolo Guiotto: But if we take an N, which is larger than this 2 capital N, so 4…
05:23:360Paolo Guiotto: And larger than the first one and the second one.
05:28:510Paolo Guiotto: They actually take the maximum,
05:31:220Paolo Guiotto: You have that, both are verified, so… Both… stop.
05:39:610Paolo Guiotto: true.
05:41:950Paolo Guiotto: And this means that, in particular, if I computed the distance between F and G,
05:48:470Paolo Guiotto: Now, I add and subtract inside the norm FM, doing this.
05:55:570Paolo Guiotto: And I reorganized this sum in this way.
06:00:180Paolo Guiotto: Applying now the triangular inequality, this is less or equal than the distance between F and FN.
06:07:410Paolo Guiotto: plus the distance between F, N, and G.
06:10:980Paolo Guiotto: Of course, the distance between F and FN is the same of the distance between FN and F, no? Because then all… we can factorize the minus 1, carry outside modulus of minus 1 is 1.
06:24:180Paolo Guiotto: And since the two are both less or equal than epsilon, I have that everything is less or equal than 2 epsilon. So I would have that the distance between F and G is less or equal than 2 epsilon.
06:41:470Paolo Guiotto: You see that there is no N here.
06:43:890Paolo Guiotto: And you have to remind that this epsilon is an arbitrary number, so I can say for every epsilon positive.
06:51:950Paolo Guiotto: But then, that quantity cannot be positive, no? It is greater or equal than zero because it is a norm, but it cannot be positive, otherwise I can always choose epsilon small enough in such a way that
07:04:270Paolo Guiotto: this becomes impossible. So from this, I deduce that norm of F minus G is equal to 0.
07:14:360Paolo Guiotto: Now, it is this last step which is interesting, because, of course, we know that if norm is a norm, we have vanishing that says that norm can be zero if and only if the vector is zero. So, from this vanishing…
07:30:660Paolo Guiotto: we have F minus G equals 0, so F equals G, and that's the conclusion.
07:36:860Paolo Guiotto: Now, we know that for some important spaces, we do not have the vanishing in this strong form. So, what happens for remark?
07:51:760Paolo Guiotto: What happens,
07:58:860Paolo Guiotto: for… norms… like, that's a very important class, the Pinorma.
08:10:580Paolo Guiotto: where P is between 1, and we have seen that there is also the L-infinity norm on the space LPX, so there is an underlying measure, and so on, no?
08:25:730Paolo Guiotto: Well, as you can see, this argument applies until this step, no?
08:32:419Paolo Guiotto: So, I can… I could say that,
08:36:39Paolo Guiotto: If a sequence FN goes to in an LP normal 2F,
08:42:620Paolo Guiotto: and it goes in the same LP norm to another function, G, then I will have a norm of F minus gnorm of this equals 0. And this does not imply that that is 0. So, in particular, it does not imply that f is equal to G.
09:00:600Paolo Guiotto: So this could be a little bit disturbing, because it means that F minus G is equal to 0 almost everywhere, so F equals G almost everywhere.
09:13:820Paolo Guiotto: So, in other words, the limit, strictly speaking, is not unique.
09:18:980Paolo Guiotto: No? But, we can say that,
09:22:990Paolo Guiotto: it is, let's say, a unique modular measure zero set of points. So if you forget of a measure zero set of points, which are irrelevant for the measure, we will never be able to distinguish anything on measure zero points.
09:38:680Paolo Guiotto: So we can say that the limit is unique, okay? So we accept this little ambiguity for which we can have, formally, infinitely many limits in principle. If the measure has infinitely many sets with measure 0, as in the Lebang case.
09:56:560Paolo Guiotto: No? Any finite set, any countable set is a measure zero set, so you could have infinitely many limits. But the point is that all these limits differ on a measure zero set.
10:09:60Paolo Guiotto: So, at the end, still, we will have to compute the quantities connected to integrals, etc. It's like if we do not feel any difference, okay? So that's the unique,
10:20:670Paolo Guiotto: particular picture.
10:22:800Paolo Guiotto: Okay, let me start showing an example, simple example.
10:28:620Paolo Guiotto: Just Google.
10:31:250Paolo Guiotto: Get confidence with this. Because most of our spaces here are spaces of functions.
10:40:810Paolo Guiotto: So… are more complex than finally dimensional spaces, more complex than R, RN, etc.
10:50:320Paolo Guiotto: But not particularly… not extremely general, because we deal with the spaces of functions. So we will focus a bit on what does it mean, the convergence with respect to a class of norms, like, for example, the P norms, or other norms that we have normally on these typical spaces.
11:09:590Paolo Guiotto: But, before to do that, let's show an example, which is, this. We take a space V,
11:17:630Paolo Guiotto: the, now, let me take… let me modify a bit the space of continuous function.
11:26:140Paolo Guiotto: on C… on 01, huh?
11:29:910Paolo Guiotto: We know that, we have a natural norm here, which is the infinity norm.
11:41:00Paolo Guiotto: But we consider also the one norm.
11:45:980Paolo Guiotto: Which is well-defined here, okay?
11:48:870Paolo Guiotto: Now, let's take this sequence of vectors. Now, these are functions, so I have to say how they, how they work on the point X. This sequence is the sequence of powers X to the n.
12:04:00Paolo Guiotto: Now, let's discuss…
12:11:190Paolo Guiotto: convergence.
12:13:240Paolo Guiotto: of this sequence, FN.
12:16:740Paolo Guiotto: With respect to… The infinity normal?
12:22:550Paolo Guiotto: and the one norm. This is just to show you that convergence is not independent of the norm.
12:33:20Paolo Guiotto: Okay?
12:34:200Paolo Guiotto: in… let's say that the definition seems… well, it is obvious, because norms measures how mass am close to the limit. So if I change the way to measure that distance.
12:45:930Paolo Guiotto: It seems reasonable that if someone is close to the limit according to some norm, it might not be close to the limit in another norm.
12:55:590Paolo Guiotto: And this, of course, suggests that there is some connection with the concept of equivalent norms, stronger norms, that we will see immediately after the example. But let's work out the example.
13:10:350Paolo Guiotto: Now…
13:15:80Paolo Guiotto: The problem now is that if you look at this definition, this is the classical definition of limit. When you give a definition of limit.
13:25:30Paolo Guiotto: you give automatically, in the definition, the limit. So, the definition is not a practical definition to compute limit. You would never compute the limit with this definition, because no, you have a sequence, you don't know, like, what is the limit.
13:40:790Paolo Guiotto: Okay? Now, you have maybe used the sequences where Fn is explicit, then I have certain rules of calculus, if you take two numerical sequences, and I can somehow complete the limit.
13:54:990Paolo Guiotto: But normally, in doing modeling problems, this FN comes out from a certain numerical scheme, a certain approximation scheme, so it's never explicit. And therefore, it's not easy to understand if that sequence has a limit, and who is the limit.
14:13:120Paolo Guiotto: Normally, for example, sequences are used in approximations to build an exact solution of your problem, which is the limit F.
14:23:970Paolo Guiotto: But then you don't know exactly how to find out of solving a certain equation better.
14:29:480Paolo Guiotto: So you, maybe you are able to show that you can solve the equation with certain simplification, approximations, etc. So you build an F and, but you're not able to exact solve the F, otherwise you wouldn't need to do the approximation, no? You could solve that, no?
14:46:330Paolo Guiotto: So the idea is that normally, when we have a sequence.
14:50:50Paolo Guiotto: We are now to, we need now to a reminder that it's not easy to understand, to identify the limiter. And the definition that you see here, requires that you are grading domain.
15:03:550Paolo Guiotto: In fact, in concrete.
15:05:710Paolo Guiotto: Take this example. I have to prove that this… I have to prove… I have to discuss that if this sequence is converted according one of these nodes. The first thing we have converted to Y,
15:19:30Paolo Guiotto: Because if I want to apply the definition, I need to estimate the distance between D set at N and the candidate screen.
15:26:600Paolo Guiotto: So we need a candidate limiter before we can say, this is for Gordon. So that's the point. Let's… so for the moment, we don't have a general method, so let's do things a bit intuitively, since here we can plot
15:44:980Paolo Guiotto: the functions. Let's see what happens. Of course, you understand that this is a very special situation. Now, these functions are, let's… let's take an greater or equal than 1.
15:58:700Paolo Guiotto: So, F1 is X, so it is this one.
16:05:680Paolo Guiotto: This is F1.
16:07:760Paolo Guiotto: Then F2 is X square. So this square, it is something like this.
16:13:270Paolo Guiotto: This is F2. F3 is X cubed. Since X is between 0, 1, no, powers are smaller.
16:23:690Paolo Guiotto: the higher is the exponent, so X cubed will be a little bit below of X squared, so this is, this one is F3, and so on, no? Of course, we cannot plot all of them, they are infinitely many.
16:39:20Paolo Guiotto: But what, what happens here? For large N,
16:44:310Paolo Guiotto: The typical function for large n, since we have to take n and going to infinity, we are interested in what happens with large n.
16:53:920Paolo Guiotto: So what can be said about these functions? These are the power X to the n, so they are positive, they are increasing, no? If X is between 0 and 1, the values at 1 are always equal to 1, at 0, they are always equal to 0.
17:12:30Paolo Guiotto: And for a big n, so a bigger exponent, X to be n, when X is not 1, but strictly less than 1, is small.
17:21:839Paolo Guiotto: So I expect that this block will be very close to the x-axis, and when I approach one, it will go out in this way. So typically, I will see something like this.
17:38:850Paolo Guiotto: That would suggest When we send them to infinity, at least if we look at pointwise limit.
17:49:160Paolo Guiotto: What happens for the point wisdom? We already met this, but we can,
17:53:900Paolo Guiotto: If we look at the quantity Fn of X for X fixed.
17:59:270Paolo Guiotto: So X to the N, this is equal to 1 when X is 1, and therefore it goes to 1.
18:06:620Paolo Guiotto: No?
18:09:80Paolo Guiotto: for X strictly less than 1 greater liquid then 0, this quantity goes to 0, no? It goes to 0. So, I could say that there is a function
18:21:780Paolo Guiotto: that comes out when I send them to infinity.
18:25:540Paolo Guiotto: And this is a point-wise limit, has nothing to do for the moment with the infinity or one norm. But this green function could be a candidate.
18:37:280Paolo Guiotto: Now, as you notice, For this space, there is a little problem.
18:46:200Paolo Guiotto: Because that vector, that green F, is not an element of V.
18:52:850Paolo Guiotto: So, if this is the limit, it cannot limit in that space.
19:00:150Paolo Guiotto: Okay, so perhaps I have to change this one. Let's put L1. So, we make… these…
19:09:290Paolo Guiotto: Now, L1, what, okay, let's put to the…
19:17:430Paolo Guiotto: Yeah, the infinity norm, actually, maybe we should observe this. The infinity norm works also on L1, because,
19:30:30Paolo Guiotto: No, I'm saying something.
19:32:720Paolo Guiotto: Wronga.
19:37:20Paolo Guiotto: I want to have at least one…
19:41:620Paolo Guiotto: Now, let's keep continuous, let's say.
19:45:270Paolo Guiotto: Because otherwise I have problems. Okay, let's take a continuous function. So, let's say that
19:51:300Paolo Guiotto: Apparently, if there is a possible limit, this limit should be that green function, but that green function is not a function of the space.
20:01:590Paolo Guiotto: So, I should say… Perhaps this sequence is not converging in any of the two norms.
20:10:160Paolo Guiotto: Actually, for the one normal, I may think that that function, the integral, considers as the same of this one, but maybe let's do it in another color.
20:22:370Paolo Guiotto: Yellow.
20:25:880Paolo Guiotto: As the function constantly equal to zero?
20:29:210Paolo Guiotto: So the green and the yellow function for the one norm are the same function, you see? No?
20:36:40Paolo Guiotto: And so, I could say, perhaps, these functions could go to zero, the function constant equal to zero.
20:45:730Paolo Guiotto: Okay? So, I have now a little gas.
20:52:330Paolo Guiotto: So, about the, the convergence,
20:56:510Paolo Guiotto: in one norm, I could expect that this sequence FN converges in one norm to zero.
21:07:490Paolo Guiotto: And then, what about the convergence in infinity norm?
21:16:570Paolo Guiotto: Well, let's first try to see if this is true, the one norm convergence.
21:22:350Paolo Guiotto: Now, to show that this is true, I have to estimate the distance between F, N, and 0 in one normal. So, when I subtract zero, of course, this becomes the norm.
21:35:650Paolo Guiotto: of FN, the one norm of Fn, which is the integral from 0 to 1 of modulus FNX DX.
21:46:80Paolo Guiotto: This means that we have to integrate from 0 to 1, FN is X to the n.
21:52:440Paolo Guiotto: it is positive for x between 0 and 1, so I can just write x to the n. I have to compute this integral. Now, this integral can be easily computed because it is the derivative of Xn plus 1 over n plus 1.
22:08:570Paolo Guiotto: That has to be evaluated for X equals 0, X equals 1, and then you do the difference.
22:15:90Paolo Guiotto: When you plug x equals 1, you get 1 over n plus 1. When you plug x equals 0, you get 0.
22:21:40Paolo Guiotto: So, we obtained that the distance between Fn, and 0 in one norm is exactly equal to 1 over n plus 1. That goes to 0 when n goes to plus infinity.
22:36:380Paolo Guiotto: So, you see that the gas is correct. This sequence FN… Conclusion?
22:46:150Paolo Guiotto: This sequence FN converges in one norm to zero.
22:53:90Paolo Guiotto: Which is an element of the space.
22:56:640Paolo Guiotto: Now, what can be said about the convergence in infinity norm?
23:02:900Paolo Guiotto: Can I say, for example, that it converges to zero?
23:06:620Paolo Guiotto: Maybe the limit is the same, huh? So…
23:10:230Paolo Guiotto: Is it true that FN converges in infinity norm?
23:15:810Paolo Guiotto: to zero.
23:18:10Paolo Guiotto: Let's see if this happens. Now, to do this, we have to estimate the distance between FN and 0, now in infinity norm, which means that the norm of Fn, infinity norm.
23:31:320Paolo Guiotto: Here we are… we deal with continuous functions, so this is the maximum of the modulus of DFNX.
23:40:600Paolo Guiotto: when X is in 0, 1.
23:44:380Paolo Guiotto: Now, our FN is X to the N.
23:47:980Paolo Guiotto: Which is positive, so this is the maximum of X to the N for X between 01.
23:57:140Paolo Guiotto: Now, do you see what is the maximum?
24:00:900Paolo Guiotto: Yes, it is 1. If you look at the plots, you see that this function attains the maximum at x equal 1, and the value is 1. So that's constantly equal to 1. So here, we have that the distance between Fn and 0, infinity, is constantly equal to 1
24:19:820Paolo Guiotto: So, certainly, it does not go to zero.
24:23:880Paolo Guiotto: So what does this say?
24:26:170Paolo Guiotto: It says only for the moment that this sequence does not converge to zero.
24:30:570Paolo Guiotto: But perhaps it converges to someone else.
24:34:00Paolo Guiotto: I don't know yet, okay? So this says, conclusion, that this FN
24:42:610Paolo Guiotto: Does not converge in infinity norm to zero.
24:47:790Paolo Guiotto: Now, is this sufficient to say that the sequence is not convergent?
24:53:520Paolo Guiotto: No, because in principle, it could converge to some other limit that I don't know yet who it is.
25:01:660Paolo Guiotto: I cannot say it converges to the green function, because the green function is not continuous, so I cannot even… I know that that function, of course, is a well-defined function, but it's not a vector of the space, so in that space, that green function simply does not exist.
25:19:30Paolo Guiotto: So… Is it possible?
25:26:230Paolo Guiotto: that FN converges in infinity norm to some G,
25:35:60Paolo Guiotto: Clearly, G will be… G will be different from 0, because we know that FN cannot converge to 0.
25:43:490Paolo Guiotto: Is it possible that this happened?
25:47:890Paolo Guiotto: Now, of course, I cannot try with all possible Gs, and that's the distance, because that's clearly impossible.
25:55:620Paolo Guiotto: So… so for the… at this point, we do not have really too many tools in hand, but we have one tool.
26:04:480Paolo Guiotto: For this specific, example.
26:07:960Paolo Guiotto: You remind that we already considered this space with these norms, no? We have taken this as perhaps the first example of a finite dimensional space
26:18:850Paolo Guiotto: with the two different norms defined. And maybe you remind that we use this example to show something which is different with respect to the finite dimensional spaces.
26:34:400Paolo Guiotto: And finally, dimensional spaces or norms are equivalent.
26:38:490Paolo Guiotto: But in infinite dimensional space, this is no longer true, and this example was an example. In fact, maybe you remind that the infinity norm is stronger than one norm, but not vice versa.
26:50:210Paolo Guiotto: Okay? Well, it is this factor that now it becomes interesting here. So, we recall that
27:01:980Paolo Guiotto: the infinity normal… is stronger.
27:09:290Paolo Guiotto: than one norm.
27:15:640Paolo Guiotto: What does it mean, this? It means that there exists a constant, C, such that I can control the one norm through the infinity norm.
27:26:430Paolo Guiotto: This for every F in the… in that space.
27:31:80Paolo Guiotto: Now, you see that, huh?
27:33:80Paolo Guiotto: This would be, of course, part of a general argument. You see that. Now, if FN
27:43:190Paolo Guiotto: would converge to, in the infinite norm, to some G, whatever is this G,
27:50:750Paolo Guiotto: This would mean that, equivalently, the distance between Fn and G in the infinity llama would go to zero.
27:58:450Paolo Guiotto: But then, if you apply this, this control.
28:03:590Paolo Guiotto: to FN minus G, you would have that the distance between FN minus G.
28:09:310Paolo Guiotto: The distance between FN and G, so the norm of Fn minus G in one norm should be less or equal than constant distance between FN minus G, between Fn and G, infinity norm. And if guy goes to zero, necessarily this goes to zero.
28:27:900Paolo Guiotto: You see?
28:29:410Paolo Guiotto: This is the squeeze theorem.
28:32:670Paolo Guiotto: So, the conclusion would be that if this happens, so if you have convergence in the infinity norm, you would have convergence also in one norm and to the same limit.
28:50:580Paolo Guiotto: Okay.
28:52:290Paolo Guiotto: And therefore.
28:53:820Paolo Guiotto: remind that the one norm on continuous function is a true norm. We do not have the vanishing with the weak form. You remind this.
29:05:510Paolo Guiotto: So the one norm, when restricted to continuous functions, it's a true norm. So in this case, I would have limit is unique… uniqueness.
29:19:20Paolo Guiotto: The one normal is a true Norma.
29:28:60Paolo Guiotto: So, with vanishing.
29:39:210Paolo Guiotto: on… The space of continuous function.
29:43:790Paolo Guiotto: On some closer than bounded interval.
29:47:740Paolo Guiotto: So uniqueness applies, and I have a unique limit. So that G should be equal to the limit we proved above, that is 0. Should be 0. But that is not possible, because we just proved that Fn cannot converge to zero.
30:08:30Paolo Guiotto: So, we would have that FN should converge in infinity norm to that G, to zero. This is impossible.
30:18:610Paolo Guiotto: So, what is the conclusion?
30:20:730Paolo Guiotto: If assuming convergence in infinity norm leads to a contradiction, means that the sequence cannot be convergent in infinity norm. So, conclusion…
30:32:820Paolo Guiotto: FN… is not… convergent.
30:39:610Paolo Guiotto: In infinity.
30:42:520Paolo Guiotto: No.
30:45:830Paolo Guiotto: Okay, so this is about the example.
30:50:40Paolo Guiotto: Now, this part is, of course, a general argument. As you can see here, we use the fact that one norm is stronger than another. So we have this proposition, that if you… if you want, you can
31:05:50Paolo Guiotto: write down the proof, which is basically the generalization of the argument we have seen here. So, if,
31:13:620Paolo Guiotto: We have, two norms.
31:17:60Paolo Guiotto: Norm and star norm.
31:20:760Paolo Guiotto: R.
31:22:440Paolo Guiotto: norms.
31:25:240Paolo Guiotto: on the same space V.
31:28:60Paolo Guiotto: Weed.
31:30:570Paolo Guiotto: The star norm.
31:35:270Paolo Guiotto: Stronger.
31:39:90Paolo Guiotto: Boom.
31:41:670Paolo Guiotto: the North Star Norma?
31:45:860Paolo Guiotto: Okay? So this means there exists a constant C such that norm of F is controlled by C times star norm of F for every vector F in D.
31:59:390Paolo Guiotto: Then… Whenever you have that your sequence is convergent in the stronger normal, to something.
32:09:430Paolo Guiotto: It will converge, also, to the same vector F in the weaker norm.
32:17:580Paolo Guiotto: So this, this, is important to know, because sometimes we can use, as in the previous example.
32:26:280Paolo Guiotto: So, I suggest you to write down the proof, it's just written above for that special case.
32:35:370Paolo Guiotto: Okay.
32:38:590Paolo Guiotto: Let's say that, There are just another two… only two general facts that can be said about…
32:48:750Paolo Guiotto: convergence. One is, both are necessary conditions. One condition is the following.
33:01:370Paolo Guiotto: So, this is to say… is there a way to… the question is, is there a way we can say if a sequence is convergent or not? So, is there… without knowing the limit?
33:19:650Paolo Guiotto: Okay, so a major question is…
33:33:710Paolo Guiotto: Is there… And ye… Condition… on the sequence FN, not involving…
33:52:750Paolo Guiotto: Any possible limit.
34:01:710Paolo Guiotto: such that, That… well, let's say that…
34:10:580Paolo Guiotto: Ensure… Ensures convergence.
34:16:929Paolo Guiotto: So is there any test that we can do on the sequence itself without knowing the limit, that can tell us this sequence is convergent, this sequence is not convergent?
34:29:570Paolo Guiotto: Well, what can be said about this?
34:33:130Paolo Guiotto: The first factor that we mention is this information, which is actually only a necessary condition.
34:44:179Paolo Guiotto: that says that if a sequence FN Ease.
34:49:650Paolo Guiotto: convergent.
34:51:880Paolo Guiotto: Notice that here we do not define the infinite limit. It would be possible, maybe I will tell you something, but it's of no interest, because normally we are interested in looking for a limit vector, okay? So, for a vector of the space. So, when we say convergent, we mean convergent to some vector of the space.
35:10:460Paolo Guiotto: So, if Fn is convergent, then necessarily FN must be bounded.
35:19:560Paolo Guiotto: This means that there exists NM such that the norm of Fn is controlled above by that M for every N natural.
35:31:870Paolo Guiotto: So, in particular, you can say, you can use this as a…
35:37:780Paolo Guiotto: attest to disprove existence. If the sequence of norms is unbounded, it cannot be convergent, okay?
35:46:880Paolo Guiotto: So… If the sequence of norms
35:54:00Paolo Guiotto: So this is a sequence of numbers, positive numbers, is unbounded, Then… there is no limit.
36:07:130Paolo Guiotto: for the sequence F and.
36:11:30Paolo Guiotto: Now, it's not particularly…
36:15:800Paolo Guiotto: complicated this, this argument, because assume… notice that I avoided to write any limit, because this is a property of the sequence.
36:27:700Paolo Guiotto: You don't need to know the limit to show that it is bounded, because bounded means that you are able to bound the norm of vectors by a universal constant with respect to index. So this does not require that you know the limit. But in the proof, as you will see, it appears…
36:49:800Paolo Guiotto: we use this just to put the conclusion, okay? So the proof says that if the sequence F ends.
36:59:370Paolo Guiotto: goes in norm to some vector F of V, then, by definition, we know that for every epsilon positive, there exists an initial index such that the distance between Fn and F is less or equal than epsilon for every n greater or equal than capital N.
37:18:920Paolo Guiotto: Now, fix an epsilon. For example, take epsilon equals 1.
37:27:520Paolo Guiotto: This says that the distances between Fn, and F are bounded by 1, for example, for every N larger than some initial capital N.
37:38:980Paolo Guiotto: So, intuitively, you have your limit somewhere here.
37:44:30Paolo Guiotto: Saying that distances between FN and F are less frequent than 1, it means that your FN is turning around F at distance no more than 1.
37:54:230Paolo Guiotto: So you imagine that this would be a sort of neighborhood centered at death and radius 1. So that if that radius is 1, we would see DFN here. So these are DFN for N greater or equal than capital N.
38:10:930Paolo Guiotto: How many of them? Basically, all of them.
38:14:190Paolo Guiotto: This first capital N, L, actually capital N minus 1.
38:22:340Paolo Guiotto: It is not said where they are, huh? We know that F capital N is here, F capital N plus 1 is here, F capital N plus 2 is here, and so on. All of them are there, so definitely they are inside that
38:40:700Paolo Guiotto: these, yeah, no? And, the others…
38:44:770Paolo Guiotto: We don't know where they are.
38:46:820Paolo Guiotto: But, they will be somewhere… the point is that they are a finite number of vectors, so we have F0, F1, F2, etc, FN minus 1, they are around, but they are only a finite number of them.
39:02:140Paolo Guiotto: So, when now I have to bound the norm of a fan, I notice that from this I get that norm of a fan, alone, without F,
39:12:640Paolo Guiotto: I introduce F, so FN minus F plus F.
39:17:20Paolo Guiotto: triangular inequality, less or equal than FN minus F, norm, plus norm of F.
39:24:810Paolo Guiotto: Now, when n is larger, this quantity is controlled by 1.
39:30:220Paolo Guiotto: So, less or equal than 1 plus norm of F, these for every n larger than capital N.
39:38:350Paolo Guiotto: So, as you can see, I have a bound, which is independent of N, for the norms of the FN, not for all the N, but for all the N larger than some initial N.
39:52:670Paolo Guiotto: What about the first N-1? Well, if I define if m is now, by definition, the maximum.
40:05:350Paolo Guiotto: between the norm of F0, the norm of F1,
40:11:630Paolo Guiotto: etc. We arrive until F capital n minus 1, and this bound here, 1 plus norm of F,
40:21:610Paolo Guiotto: Now, this number M, you see that it is bigger than all this, in particular, this one, so this bounds definitely all the norm of Fn, or n large, and it bounds also the norm of F0, F1, and F2, FN minus 1, so it bounds everything.
40:41:590Paolo Guiotto: It turns out that norm of Fn is less or equal than this M for every N in naturals, and that's
40:51:600Paolo Guiotto: That's all for this quote.
40:55:30Paolo Guiotto: Okay, so this is, as we said, Very mild information.
41:01:800Paolo Guiotto: It does not mean that the sequence is bounded, then it is convergent. This does not happen even in R, okay?
41:09:880Paolo Guiotto: Remark, or warning.
41:16:970Paolo Guiotto: Vice versa.
41:22:310Paolo Guiotto: is false.
41:25:770Paolo Guiotto: But this is not because we are in a vector space, we are in finally dimensional space, no. It is because just take V equals the real liner.
41:35:970Paolo Guiotto: Take, norm equal the absolute value.
41:40:520Paolo Guiotto: And take this sequence FN equal minus 1 plus 1, minus 1 plus 1.
41:46:230Paolo Guiotto: That sequence is bounded, no? Modulus of FN is constantly equal to 1, so it is even constant, but
41:56:160Paolo Guiotto: There is not limiter.
41:59:90Paolo Guiotto: in this norm for the sequence FN. And actually, since the space is finite dimensional, that would not be any limit. There is no norm for which this has any…
42:09:730Paolo Guiotto: Okay? So, you cannot say the sequence is bounded, I know that the limit exists.
42:16:480Paolo Guiotto: So it's a very weak condition, and the most interesting part is actually the consequence of this. So if I prove that the sequence is unbounded.
42:28:660Paolo Guiotto: So the norm cannot be bounded, because maybe it goes to infinity, okay?
42:34:640Paolo Guiotto: then I can disprove the existence of the limit. So it's a test for non-existence, in fact.
42:42:310Paolo Guiotto: Well, again, is there any intrinsic condition? Actually, this condition exists.
42:51:570Paolo Guiotto: It is not 100% a characteristic condition, but the nice thing is that for all the spaces we work with.
43:02:390Paolo Guiotto: This is an characteristic condition, and it is called the Cauchy property.
43:08:320Paolo Guiotto: So, let's, say, the second proposition.
43:12:170Paolo Guiotto: That, for the moment, we state as, again, a necessary condition, not sufficient.
43:20:60Paolo Guiotto: So… Lights, FN… Be a sequence in V.
43:28:990Paolo Guiotto: Be such that, Let's say… UV… bye.
43:40:640Paolo Guiotto: convergent.
43:43:510Paolo Guiotto: In a certain norm.
43:47:30Paolo Guiotto: Then… FN… fulfills… the so-called Koshi.
44:00:990Paolo Guiotto: property.
44:05:50Paolo Guiotto: Or, equivalently, we say that it is a Cauchy sequence.
44:09:420Paolo Guiotto: This property says it looks like the definition of limit, see if you look at the property. But the point is that it does not involve any limit, so it's an intrinsic condition on the FN. So for every epsilon positive, there exists an initial index, N,
44:29:240Paolo Guiotto: Such that… well, the condition is that the distance between any two elements of the sequence becomes as small as you want, epsilon, provided these indexes are sufficiently large. So, for every n and M,
44:45:10Paolo Guiotto: greater or equal than this capital R.
44:47:480Paolo Guiotto: So, as you can see, this condition
44:50:180Paolo Guiotto: Does not require any action, does not require any known.
44:56:600Paolo Guiotto: Nice.
44:57:780Paolo Guiotto: And,
45:01:780Paolo Guiotto: It really turns out that in most of, well, I'd say in all the spaces we consider normally, this condition is also sufficient. So, it's a characteristic condition of convergence signals. It's important, because this is something that, in principle, you could verify
45:20:660Paolo Guiotto: Even if we don't know what is the limit. This is how, in, let's say, when you do a model, a model, to solve some problem, and you introduce an approximation scheme to find out the solution.
45:41:630Paolo Guiotto: is built as a limit of a means. You hope that these functions are in the right space where that condition becomes sufficient ratio converters. Then what to do is to check that this condition is verified.
45:55:70Paolo Guiotto: So you… the problem is the escalation of distances between two elements of the approximation.
46:02:410Paolo Guiotto: Okay, so you do the hard work is to do that estimate, you get that this condition is 25. Once you know that these functions belongs to a space where this condition is also sufficient, you are done. And fortunately, this works for, basically all the spaces we normally consider.
46:23:150Paolo Guiotto: So we continue until, at the end, okay?
46:28:470Paolo Guiotto: So, let's see the proof.
46:31:410Paolo Guiotto: Also, this proof, of course, involves the limit, but the condition itself does not require the limit. So, if we know that Fn converges to some F in normal.
46:47:170Paolo Guiotto: Then, let's write again the definition. For every epsilon positive, there exists an initial index.
46:56:640Paolo Guiotto: such that distance between Fn and F is less or equal than… well, if you want to get the epsilon there, you have to put the epsilon half here, for every…
47:09:300Paolo Guiotto: N larger than this capital N. Now, if you take two indexes, N and M, larger than that capital N, and you assess the distance between Fn and FM,
47:23:850Paolo Guiotto: To show that it is smaller.
47:26:240Paolo Guiotto: Well, you introduce the limit vector F, so you do FN minus F plus F, minus FM.
47:35:630Paolo Guiotto: And then you split this by regrouping this sum in this way.
47:41:690Paolo Guiotto: and using the, triangular equality. So this comes less or equal than distance between FN and F,
47:49:450Paolo Guiotto: plus distance between F and FN.
47:53:630Paolo Guiotto: But since both are less or equal than epsilon half.
47:58:240Paolo Guiotto: The total is less or equal than epsil.
48:01:480Paolo Guiotto: And as you can see, this argument holds whatever N and M are larger than this capital N, and that's the conclusion. So, as you can see, it's a very short proof of
48:12:740Paolo Guiotto: It's basically an immediate consequence of the definition. But the nice thing with this is that this is an intrinsic test.
48:20:810Paolo Guiotto: Okay.
48:22:80Paolo Guiotto: Now,
48:24:400Paolo Guiotto: This says, this very delicate, important point, that if this property holds, then… so, if the sequence is convergent, then the Cauchy property holds.
48:37:500Paolo Guiotto: Now, is this a characteristic condition?
48:41:320Paolo Guiotto: So, if… is it true that if this condition is verified, that we can say that the sequence is convergent? Unfortunately, no.
48:50:210Paolo Guiotto: So let's see an example, which is, I think, quite instructive example.
48:58:380Paolo Guiotto: Well, let's say… Question.
49:03:570Paolo Guiotto: Is… Koshy… sequence.
49:10:370Paolo Guiotto: Also, convergence.
49:14:630Paolo Guiotto: In general, the answer is no.
49:18:960Paolo Guiotto: in general, the… I swear… is… no.
49:28:700Paolo Guiotto: And let's see an example.
49:31:880Paolo Guiotto: Well, this example is an infinite dimensional example. That's a specific reason, because if you try to do this in R, this condition is sufficient.
49:42:280Paolo Guiotto: So for numerical sequences, if the Cauchy property is verified, they are convergent. So you cannot find an example in R that is a Cauchy sequence, but it is not convergent.
49:56:910Paolo Guiotto: Okay?
49:58:190Paolo Guiotto: The same happens for RD, the finite dimensional RD or CD, no? The same arrays of complex numbers, or arrays of the real numbers. All them have this property, so whenever you have a Cauchy sequence, it is convergent.
50:17:600Paolo Guiotto: So it's not easy, I'm saying it's not easy to find a counterexample of this, and we have to find inside these kind of spaces we are dealing with. And the space will be…
50:29:670Paolo Guiotto: This Saturday, The space V is the usual, well, let's take a symmetric…
50:45:400Paolo Guiotto: C minus 1, 1.
50:47:630Paolo Guiotto: And I take the norm here, the one norm.
50:57:330Paolo Guiotto: Okay? So I will show you an example of a sequence which fulfills the Cauchy property according to this norm, but it is not convergent to anything, okay?
51:12:980Paolo Guiotto: Now, the example is, the following. I, I do a graphical first,
51:19:400Paolo Guiotto: example, then we don't need, actually, to write analytically what are these functions. So take the interval minus 1,
51:28:440Paolo Guiotto: And we take functions, the idea, let's say, is that I want to have as a limit this function here.
51:38:230Paolo Guiotto: So the function which is, say, 1,
51:41:660Paolo Guiotto: from 0 to 1, and minus 1 from minus 1 to 0. It doesn't matter what is the value in 0. Now, if this is the moral limit, it cannot be in the space, because it is not continuous, okay?
51:54:610Paolo Guiotto: So what I do is to approximate this red function through continuous functions, and they are these ones. So take a function which is constant equal to minus 1,
52:08:700Paolo Guiotto: Until a point at the left of 0. Then, you grow linearly at 1, symmetrically.
52:18:220Paolo Guiotto: and then you stay constant equal. So this is the function FN. The idea is that, the larger is n.
52:27:190Paolo Guiotto: the step is this straight piece. So let's say that, for example, this is 1 over N, and this is minus 1 over n. If you want, we can write the analytical formula for this, but we don't need, so it's completely useless, yeah.
52:44:600Paolo Guiotto: Okay, so let this FN.
52:49:160Paolo Guiotto: D…
52:52:940Paolo Guiotto: Fn, that you see here, are, of course, continuous function in minus 1, so they are good functions for this space.
53:03:880Paolo Guiotto: Okay? We checked that.
53:11:610Paolo Guiotto: That. Number one, the sequence FN is a Cauchy.
53:20:510Paolo Guiotto: sequence.
53:22:720Paolo Guiotto: with respect to the L1 nore.
53:28:320Paolo Guiotto: And this won't be difficult, this will be easy. Number two, that's a little bit more delicate. The sequence F and
53:37:470Paolo Guiotto: cannot… be convergent, In one normal.
53:48:70Paolo Guiotto: Okay, so these are the two…
53:51:390Paolo Guiotto: steps. So, step one is, is relatively easy, because we have now to compute
53:59:560Paolo Guiotto: FN minus FM in one norm, and show that for NM large, this can be made a bit really small, okay? The Cauchy property says you can make
54:12:760Paolo Guiotto: This distance between Sn and SM are literally small, a bitterly because absolute is, for every absolute positive.
54:21:890Paolo Guiotto: Provided N and L should indexes are big enough, larger than this initial capital L.
54:29:710Paolo Guiotto: Now, to compute the one norm, we don't need to write formulas integrals, because here the situation is relatively easy. So, to fix ideas, unless the two indexes are the same.
54:43:280Paolo Guiotto: In that case, we get zero, no? FN minus Fn is 0. So, assume that N is different from M.
54:50:50Paolo Guiotto: Okay? And, for example, that N is greater than N. Otherwise, you, you…
54:57:540Paolo Guiotto: to, flip and with that. So, what is the situation?
55:03:880Paolo Guiotto: Now, since n is larger than M, everything, of course, depends on that 1 over N, 1 over M, no? The 1 over M will be greater than the 1 over n, so we have this situation. So let's say that here we have the 1 over N, and here we have the 1 over M.
55:22:970Paolo Guiotto: So here we will have the minus 1 over N, and here the minus 1 over M. So let's use the two colors. Let's say the green one is for FM. FM starts at value minus 1 here.
55:39:60Paolo Guiotto: Then, at this point, at point minus 1 over M, it grows linearly.
55:47:370Paolo Guiotto: Up to this, where it takes value constantly equal to 1.
55:55:790Paolo Guiotto: Well, let's say… What color is this? I don't know what…
56:01:820Paolo Guiotto: Maybe, let's use the red. For the FN, it's similar. It is equal to minus 1 here, but it continues a little bit until this point.
56:13:580Paolo Guiotto: And then it grows linearly.
56:16:780Paolo Guiotto: Up to this point, then it is equal to 1.
56:21:410Paolo Guiotto: So now, I want to compute the…
56:25:600Paolo Guiotto: By definition, integral from minus 1, here, the interval is minus 1, of the absolute value FN minus Fn.
56:35:860Paolo Guiotto: Now, what you see here is that,
56:39:340Paolo Guiotto: The two, the green and the red, are the same.
56:43:590Paolo Guiotto: out of this interval, minus 1 over n to 1 over n, because you see, from minus 1 to this, they are the same, and from 1 over n to 1, they are the same, so the difference would be 0. So it means that I can just cut off the integral.
56:58:280Paolo Guiotto: to the interval from minus 1 over M to 1 over M of the same thing.
57:05:730Paolo Guiotto: modulus FN minus Fn.
57:12:150Paolo Guiotto: But now, you don't need to write exactly what is that difference, and make complicated discussion, because you should split the integral now from minus 1 of n to minus 1 over n, then from minus 1 over n to 1 over n, so it would be a mess.
57:28:780Paolo Guiotto: What we are doing is what? We are assessing, basically this area, no?
57:36:160Paolo Guiotto: Because, the absolute value is,
57:41:590Paolo Guiotto: is non-zero there, and we are computing that area.
57:47:160Paolo Guiotto: Actually, even more simple, we just need to know that since these two functions are, as you can see in the figure, the difference will be always between minus 1 and 1.
57:59:450Paolo Guiotto: So in absolute value, this is less than 1. So I could say, okay, let's throw away with just 1 this, and what remains is less or equal than integral of 1 between minus 1 over M to 1 over M, and this is 2 over N.
58:17:970Paolo Guiotto: Okay, so what I get is this bound, the distance between Fn, and FM is controlled by 2 over M,
58:28:70Paolo Guiotto: for every n greater… I cannot add equal, because for n equal to N, I get 0, so greater or equal than M.
58:38:400Paolo Guiotto: Now, you want to make this less than epsilon, provided you choose… well, you see that this becomes less than epsilon when M is greater than 2 over epsilon.
58:53:520Paolo Guiotto: Okay? Now, you take the first natural greater than that 2 over epsilon. That natural is the integer part of 2 over epsilon plus 1.
59:06:730Paolo Guiotto: So that's the first natural, greater than 2 over epsilon, and then you have that for every n greater or equal than M, greater or equal than this capital N,
59:21:500Paolo Guiotto: it is verified that M is greater than 2 over epsilon, because you took M greater than the first integer after 2 over epsilon. And therefore, you will have that the distance between Fn and FM will be less or equal than epsilon.
59:36:920Paolo Guiotto: And that's the Cauchy property.
59:39:150Paolo Guiotto: Okay?
59:40:670Paolo Guiotto: So this is the check of the cushy property.
59:45:10Paolo Guiotto: Now, let's discuss the convergence.
59:51:910Paolo Guiotto: Here, the discussion is a bit, delicate, let's say.
59:56:140Paolo Guiotto: Because if we look at this sequence, this sequence has been introduced, imagine that the limit is the function which is minus 1, from minus 1 to 0, and plus 1 from 0 to 1. So let's call that function F. So let F…
00:13:690Paolo Guiotto: be the function defined as minus 1 when x is from minus 1 and 0,
00:21:530Paolo Guiotto: well, at zero, let's put a zero, the value, it doesn't matter, it will be never be continuous, this one, okay? And plus…
00:30:810Paolo Guiotto: when x is greater than 0, x to equal 1. That's for X equals 0. It doesn't matter what is the value I give to this function at 0, because there is no value that makes this function continuous.
00:42:860Paolo Guiotto: Okay, now… I claim that my FM should convert to this guy, no?
00:50:480Paolo Guiotto: By, by the, graphical, Argument, FN.
00:58:730Paolo Guiotto: is this.
01:01:830Paolo Guiotto: When N is bigger, This will be very close to this.
01:08:250Paolo Guiotto: which is F. This is F, and the green is FN.
01:14:80Paolo Guiotto: So, I would say the natural limit is that F.
01:18:840Paolo Guiotto: But that F is not in V, so I cannot say FN converges in one norm to F in V.
01:29:110Paolo Guiotto: However, that F is in L1, right? So I can say that…
01:34:400Paolo Guiotto: the distance between F, N, and F in one norm, this makes sense if I imagine that that F is not continuous, but it is L1. So I can compute the L1 norm, no? That F is in L1 minus 1,
01:50:10Paolo Guiotto: So, I can complete the one norm. And you see that this distance is what? It's the integral from minus 1 to 1 of models FN minus F.
01:59:990Paolo Guiotto: As above, as you can see, the two, the red and the green, are the same out of this minus 1 over n to 1 over n, so the integral reduces to the integral from minus 1 over n to 1 over n of the difference.
02:14:670Paolo Guiotto: modulus Fn minus F. Again, I don't need to compute that quantity.
02:21:470Paolo Guiotto: If you want, it's the area here, so in this case, I have an even simpler formula, it's the area sum of two triangles, so it's a rectangle, basically. In any case, this is less or equal than, same argument as above, this is no larger than 1, it is less or equal than 2 over N.
02:40:350Paolo Guiotto: So, it goes to zero.
02:42:570Paolo Guiotto: So what I'm gonna say is that, so my FN converges in one normal
02:51:470Paolo Guiotto: to F, that belongs to L1. So if I look as space, the space at 1,
02:59:560Paolo Guiotto: In N1, this request is really converted to a vector, which is in N1, not a continuous function, not in the space we produce, okay?
03:12:150Paolo Guiotto: Now, I want to show… I want to convince you that DFN cannot converge to anything in V. Let's see why, and this is the delicate point. This remark will be… will be a clear remark,
03:27:200Paolo Guiotto: So… Since, F is not UV, from this,
03:39:190Paolo Guiotto: I cannot, did use… Anything, for the moment.
03:47:670Paolo Guiotto: No? I proved that my sequence is convergent, but the limit is out of my space, so in my space, there is no limit, there is not that limit, okay? But this…
03:59:420Paolo Guiotto: is not yet sufficient to conclude, because I've not yet proved that FN is not convergent. So, to show that this is the case, we will do a contradiction argument.
04:10:540Paolo Guiotto: So, assume that…
04:17:680Paolo Guiotto: my FN converges in one normal to some G, which is in V, so space of continuous function 0, sorry, minus 1, 1.
04:33:620Paolo Guiotto: Now, look how I deduce that this is impossible.
04:38:220Paolo Guiotto: Now, since my G, the supposed limit of DFN, is continuous, it is in L1, right?
04:50:480Paolo Guiotto: Okay?
04:52:20Paolo Guiotto: So look at now what I have. I know that in L1, my sequence FN converges to, according to the L1 norm, to sum F.
05:01:600Paolo Guiotto: And at the same time, FN converges to a G, which is continuous, but still in the space L1.
05:10:190Paolo Guiotto: Now, can the sequence have two limits?
05:14:40Paolo Guiotto: Well, if L1 norm were a true norm, I could say, no, that's not possible. I could deduce F equals G, and then I have the contradiction, because G should be F, which is not continuous, that's a contradiction. But unfortunately, I cannot say that.
05:31:420Paolo Guiotto: Because the one norm does not fulfill the vanishing in the strong form we have seen at the beginning of this class, no? We have seen that what happens when
05:43:910Paolo Guiotto: when I have convergence, like in LP norm, no? What if I have two limits? What is the conclusion? I cannot say F equals G, but F equals Z almost everywhere. Okay, let's take this.
05:59:430Paolo Guiotto: So what I can say here is that, from this, since… We… tab.
06:11:420Paolo Guiotto: Only.
06:13:620Paolo Guiotto: week.
06:17:780Paolo Guiotto: vanishing.
06:22:380Paolo Guiotto: So we can reduce that only apparently we can result that F is equal to G almost everywhere.
06:34:790Paolo Guiotto: Okay.
06:37:530Paolo Guiotto: So… Imagine how… how… what the situation. This is my F.
06:45:120Paolo Guiotto: Minus 1, 1. MyF is done like this, let's say.
06:50:810Paolo Guiotto: that this is the F.
06:56:400Paolo Guiotto: I don't know what is this gym.
06:58:790Paolo Guiotto: And I don't have even a characterization of measures of sets, so I don't know what that means, I don't go anywhere. It needs ABBA for a measures of set a point, but who knows what these measures of the point?
07:14:480Paolo Guiotto: But, I know that F is equal to G almost everywhere.
07:20:130Paolo Guiotto: So, what could you think it should happen to this gene? How this gene should be made?
07:32:590Paolo Guiotto: We have seen something like this. Okay.
07:36:100Paolo Guiotto: Do you remind what happens?
07:38:780Paolo Guiotto: I have that G continuous is equal to zero almost everywhere.
07:45:500Paolo Guiotto: What happens in this case?
07:49:410Paolo Guiotto: Can a function…
07:51:820Paolo Guiotto: Yeah, it forces the continuity forces that G must be constantly equal to 0. Why? Because if I have a point where G is a positive.
08:02:20Paolo Guiotto: In a neighborhood of that point, G will be positive, so there is a positive measure zero set, where G is non-zero, and that would contradict the almost everywhere. So, from this, we then use that G is constant equal to zero.
08:16:60Paolo Guiotto: So, you see that here we are not a constant function, it's not constantly 0, but it is constant here and here. So what would you expect about that G? That G must be constant equal to 1, almost everywhere, from 01,
08:33:399Paolo Guiotto: No? So from this, you have that, huh?
08:37:410Paolo Guiotto: G is equal to 1, almost every X in 01.
08:44:479Paolo Guiotto: So, what would you expect about that gene? It is continual, can be different from one somewhere.
08:52:240Paolo Guiotto: No, because it would be different in an neighborhood. So there would be a positive measure 0, a positive measure set. So imagine that this is my G. At this point, this is the value of G.
09:04:840Paolo Guiotto: But G is continuous, so you would have that around that point, G would be different from 1. So there would be a set of points down here with positive measure where G is not equal to 1.
09:20:470Paolo Guiotto: And that's impossible. So the conclusion is that, from this, it follows that G is identically equal to 1,
09:27:510Paolo Guiotto: On… 0… 1.
09:31:840Paolo Guiotto: I exclude 0 because 0 is a problem.
09:35:609Paolo Guiotto: For the same argument, I have that, also, G is equal to minus 1 minus 1.
09:46:630Paolo Guiotto: Almost everywhere, this time on minus 1, 0.
09:50:950Paolo Guiotto: But for the same argument, I have, that G is constant equal to minus 1 on minus 1, 0.
09:59:280Paolo Guiotto: So, if this is the case, my G must be like that. Here, it is minus 1. Here it is plus 1. I don't know, and I don't care what is here at 0, because this function cannot be continuous.
10:14:370Paolo Guiotto: So… G… should be…
10:23:330Paolo Guiotto: equal to minus 1 for X between minus 1 and 0, excluded, and plus 1 for X
10:32:240Paolo Guiotto: Greater than 0, less or equal than 1.
10:35:630Paolo Guiotto: And if you have a function like that, you see that, in this case.
10:45:950Paolo Guiotto: Gee wouldn't…
10:51:80Paolo Guiotto: B… Continos.
10:54:270Paolo Guiotto: at zero.
10:56:460Paolo Guiotto: Because if you compute the right limit, limit for x going to 0 positive of G , these values are constantly equal to 1, so you get 1.
11:08:350Paolo Guiotto: While, if you compute the limit when x goes to G0 negative of G , these values are constantly equal to minus 1, so the limit is minus 1. Left limit, different from right limit, the function is not continuous. So the conclusion would be, finally, what we desired.
11:27:890Paolo Guiotto: shouldn't be in V.
11:30:60Paolo Guiotto: And that's a contradiction, because we started saying, assume that FL converges in one norm to that… to that G in V, okay?
11:41:270Paolo Guiotto: And the conclusion is that G cannot be V, so that's a contradiction.
11:46:510Paolo Guiotto: Okay?
11:49:240Paolo Guiotto: So this is an example showing that, in general, the Cauchy property is not a characteristic property of convergence. It's necessary, but not sufficient.
12:01:70Paolo Guiotto: However, we will see that… It becomes a sufficient condition for all the spaces we have seen.
12:09:420Paolo Guiotto: Okay? But this will take a little bit of time.
12:14:320Paolo Guiotto: Anything.
12:16:490Paolo Guiotto: So, let's say, conclusion… D.
12:23:290Paolo Guiotto: Oh, shit.
12:26:420Paolo Guiotto: property… is a necessity.
12:32:760Paolo Guiotto: But… not sufficient.
12:39:550Paolo Guiotto: In general.
12:42:980Paolo Guiotto: condition.
12:45:770Paolo Guiotto: to have… convergence.
12:49:920Paolo Guiotto: So, if I stop here, it is, like, more or less a useless condition, because I would like a test.
12:56:370Paolo Guiotto: We have a test to check if the sequence verifies this test, it is convergent. A test that does not involve any limit.
13:05:370Paolo Guiotto: Well, the good news is that, However…
13:13:520Paolo Guiotto: for… most… Come on.
13:21:310Paolo Guiotto: spaces.
13:24:300Paolo Guiotto: of functions.
13:30:720Paolo Guiotto: Notably, for example, LP Spaces.
13:33:820Paolo Guiotto: The spaces of continuous functions with the uniform norm, the LP with the P norm. We will see a list later.
13:42:280Paolo Guiotto: Koshi property, is… Also, sufficient.
13:52:850Paolo Guiotto: It's not just a necessary condition. So that's why we introduced this definition.
13:59:270Paolo Guiotto: space, normed… space.
14:06:250Paolo Guiotto: V with a certain nore.
14:09:520Paolo Guiotto: For which?
14:13:830Paolo Guiotto: all.
14:17:250Paolo Guiotto: Cauchy sequences.
14:22:300Paolo Guiotto: Convergent.
14:29:190Paolo Guiotto: is.
14:30:650Paolo Guiotto: cold.
14:34:680Paolo Guiotto: complete.
14:38:410Paolo Guiotto: Or… in honor of a guy, a Polish mathematician.
14:44:530Paolo Guiotto: for the… among the first to study this,
14:49:250Paolo Guiotto: these, structures, or panacch, space.
14:58:400Paolo Guiotto: So a Banach space is a normal space for which the Cauchy sequences are convergent. So whenever you have verified the Cauchy sequence… the Cauchy property, this sequence is convergent.
15:12:320Paolo Guiotto: Now, to understand that most of the spaces or functions are actually Bannock spaces, complete or complete spaces, we need to do now to understand better the relation between convergence in norm and convergence
15:32:320Paolo Guiotto: what happens for convergence… how to characterize convergency norms for those specific norms. So let's now focus, start, at least we start, focusing on…
15:45:700Paolo Guiotto: So, we now… Focus.
15:53:410Paolo Guiotto: On… a bit more.
16:00:150Paolo Guiotto: on specific, normal.
16:09:830Paolo Guiotto: Nonset.
16:11:630Paolo Guiotto: in function… spaces.
16:18:190Paolo Guiotto: So let's start with the first example we actually introduced that was the case of bounded functions.
16:26:690Paolo Guiotto: So… let's say… space B of X.
16:33:640Paolo Guiotto: Equipped with the infinity normal.
16:37:260Paolo Guiotto: I recall you that.
16:39:450Paolo Guiotto: B of X.
16:41:710Paolo Guiotto: stands for functions.
16:44:810Paolo Guiotto: defined on a set, generic set X, real or complex valued.
16:51:950Paolo Guiotto: Such that they are bounded. So, away is supremum of modulus f of x, when x is in capital X, is finite.
17:04:770Paolo Guiotto: And this quantity, this supremum, is what we call the infinity norm of F. We already checked that this B of X is a vector space, and that quantity is a well-defined norm.
17:19:420Paolo Guiotto: Now, what does it mean that,
17:25:110Paolo Guiotto: A sequence here is convergent according to this norm.
17:30:70Paolo Guiotto: Now, if we have a sequence FN, which is convergent.
17:36:730Paolo Guiotto: to sum F of this space.
17:40:490Paolo Guiotto: Well, this means that, the,
17:44:970Paolo Guiotto: Let's write down the definition, and adapting the definition to this case. For every epsilon positive.
17:53:550Paolo Guiotto: there exists an N initial index such that the infinity norm of Fn minus F is less or equal than epsilon for every n larger than capital F.
18:06:160Paolo Guiotto: Now, that quantity here is the infinity norm. This means that the supremum
18:12:260Paolo Guiotto: when X is in capital X of modulus FNX, minus FX.
18:21:850Paolo Guiotto: This quantity must be less or equal than epsilon.
18:26:210Paolo Guiotto: Well, since the supremum is less or equal than epsilon, we can say that, in particular, all these numbers, Fn of x minus F of X,
18:38:250Paolo Guiotto: is less or equal than epsilon for every X.
18:44:10Paolo Guiotto: In capital X.
18:45:630Paolo Guiotto: So I'm just exploding, if you want, this condition here. So this condition here, which is written in terms of a norm of Fn minus F, means that the suprem of these quantities must be less equal than to the other.
19:01:790Paolo Guiotto: And, in particular, this is actually an if and only if, because if the supremum… let's talk roughly. If the maximum is less than x0, each of the elements must be less than x0. But vice versa, if each of the elements is less than x0, then the maximum is less than 0, so it's an if and only if.
19:21:670Paolo Guiotto: So we can say that,
19:23:790Paolo Guiotto: Writing Literally, this condition, we have this a little bit involved formulation for every epsilon positive.
19:33:370Paolo Guiotto: There exists an initial index such that distance between F and X
19:38:780Paolo Guiotto: minus FX, less or equal epsilon.
19:43:160Paolo Guiotto: this for every X in capital X, and for every n greater or equal, then, capital N. So this is, let's say, the exposed formulation of convergence.
19:57:550Paolo Guiotto: From this property, we can deduce a first consequence.
20:02:280Paolo Guiotto: Because, if you forget of this, or if you freeze and choose an answer, this is saying that,
20:09:570Paolo Guiotto: For every action, there existed initially X such that distance between FNX and FX is less than axon for every n larger than that.
20:18:450Paolo Guiotto: So this means that for that X, this sequence of numbers
20:22:840Paolo Guiotto: converges to this number, and that's the point of ice convergence.
20:28:10Paolo Guiotto: So, it means, in particular, from this, huh?
20:37:790Paolo Guiotto: In particular, we have data.
20:46:200Paolo Guiotto: of Cabo.
20:48:570Paolo Guiotto: if you, for, well, let's write that the conclusion, that, FN of X
20:59:200Paolo Guiotto: converges to F of X,
21:02:170Paolo Guiotto: for every X in the set capital X, the domain of these functions.
21:09:700Paolo Guiotto: And this is the point-wise convergence, what we call the point… Why is that?
21:17:170Paolo Guiotto: convergence.
21:19:130Paolo Guiotto: Which is, let's say when I have a sequence of functions, so a sequence means I have functions, depending on the parameter, discrete parameter, 1, 2, 3, 4, 5, etc.
21:34:860Paolo Guiotto: But they are functions, sir.
21:37:550Paolo Guiotto: The easiest way to check what happens when n goes to infinity is it frees and X. You add a sequence of numbers, do the limit in n, and you see what happens, no? If this limit exists X by X, you can call it F of X, you add the point by the limit.
21:54:890Paolo Guiotto: It's the weakest, simplest way of defining the converges or systems of functions.
22:03:500Paolo Guiotto: Actually, as we see here, we basically proved Effected, so simple proposition.
22:11:330Paolo Guiotto: So let's say it's not needed, and you proof. We proved
22:18:390Paolo Guiotto: this, that if I have the sequencer fan.
22:22:300Paolo Guiotto: converges in infinity norm to F, then the sequence FN converges to F in this point-wise way. I will use this abbreviation, it's not universal agreement, but this PW means point F.
22:40:810Paolo Guiotto: wise.
22:42:790Paolo Guiotto: So, in other words, it means that Fn of X
22:47:790Paolo Guiotto: We go to F of X,
22:50:670Paolo Guiotto: when n goes to plus infinity for every X in capital X.
22:57:950Paolo Guiotto: Okay.
23:00:730Paolo Guiotto: And this is interesting, because you want to study the convergence of a sequence at N in this normal, and you don't know yet what is the limit. This theory.
23:13:390Paolo Guiotto: Gives you, this proposition gives you a way to determine a candidate.
23:18:790Paolo Guiotto: How? You put it to the point where I say, which is something that's say, easier, because it's an ordinary in one value, no? You take that F and your X computer, delimit X by X, 9 goes to infinity.
23:33:720Paolo Guiotto: Now, the point is, is that sufficient?
23:36:770Paolo Guiotto: As you may understand, this is a one-way error, it's not an infin-related, no? So, the vice versa is false, so… so…
23:45:930Paolo Guiotto: We cannot say that uniform convergence coincide with the pointwise convergence. And we can have a… we have already seen the example of this, just quickly.
23:59:60Paolo Guiotto: Example… The hour below of the sequence, so the space is C01.
24:07:590Paolo Guiotto: the sequence FN of X is the sequence made of functions X to the n, no? The powers. In this case, you can see
24:18:560Paolo Guiotto: that, the sequence FN of X,
24:24:480Paolo Guiotto: So, if you look at the point-wise limit, you freeze an X, and you compute the limit in N. So, where X is a parameter, is alive, but it is freezed.
24:37:320Paolo Guiotto: It is not variable. Now, this means that you have to compute the limit in n going to plus infinity of the quantity X to the n.
24:47:00Paolo Guiotto: But, if we said if X is equal to 1, this is the limit of 1 to the n, so it is equal to 1, and the limit will be 1. If X is less than 1, greater or equal than 0,
25:01:690Paolo Guiotto: We have basically an exponential, no? Because base is fixed, you are moving the exponent, base is less than 1, this goes down to 0.
25:12:160Paolo Guiotto: Okay, so the point-wise limit exists in this case, and we can baptize this F of X.
25:18:870Paolo Guiotto: So F of X is the function which is 1 at x equal 1 and 0 elsewhere.
25:25:270Paolo Guiotto: figure.
25:28:950Paolo Guiotto: We have already done many times this figure. So that's DF.
25:34:330Paolo Guiotto: So, at this point, it's equal to 1.
25:38:120Paolo Guiotto: And these are the FN.
25:42:20Paolo Guiotto: So FINR, the green F is the red one.
25:45:480Paolo Guiotto: So we have the pointwise convergence.
25:48:850Paolo Guiotto: So… This Fn converges pointwise to that F, But…
25:58:520Paolo Guiotto: As I already proved, your reminder, it was an example we discussed today, I think, huh?
26:06:360Paolo Guiotto: We already, proved that this sequence is… it was here, right? This sequence is not convergent in infinity naught.
26:18:170Paolo Guiotto: Okay.
26:21:440Paolo Guiotto: So, but FM is not convergent in infinity norm to anything, not to that F. That F is not in the space, so cannot be convergent to that F, definitely.
26:36:660Paolo Guiotto: Okay? So this means that point-wise convergence is weaker than uniform convergence. Now, let's see why this is called uniform convergence. We understand from this.
26:50:770Paolo Guiotto: So, remark.
26:56:00Paolo Guiotto: why… Infinity Norma is called… uniform, convergence.
27:11:90Paolo Guiotto: Because if you go back,
27:13:940Paolo Guiotto: to this, red box we have written here, no? Let me just copy down here. So, FN…
27:23:860Paolo Guiotto: Goes in infinite norm.
27:27:190Paolo Guiotto: to F. If and only if…
27:30:230Paolo Guiotto: we have this property. For every epsilon positive, there exists an initial N such that distance FN minus X minus FX is less or equal than epsilon. That's for every X in capital X, and for every n larger or equal than capital X.
27:49:930Paolo Guiotto: That's the explosion of this, norm convergence.
27:56:440Paolo Guiotto: Now, if you have only point-wise convergence, what could you write?
28:02:260Paolo Guiotto: If FN converges point-wise to F,
28:07:970Paolo Guiotto: This would mean that the limit in n of the quantity Fn of x
28:14:920Paolo Guiotto: is equal to F of X for every X in capital X.
28:19:610Paolo Guiotto: And now let's use the same language to compare these two conditions. So what should we write?
28:26:660Paolo Guiotto: Now, to write this, I should say, For every X fixed.
28:35:830Paolo Guiotto: it happens that the limit of FNX is F of X. So, what does it mean that the sequence has a limit? That's the epsilon property. For every epsilon positive, there would be an initial n, such that distance between Fn x and limit fx.
28:51:950Paolo Guiotto: is less or equal than epsilon for every n, larger or equal than capital N. That looks the same.
28:59:160Paolo Guiotto: No? I just inverted the order of this for every laps, you know, prevalence from this place to that place. In fact, there is a big difference.
29:08:980Paolo Guiotto: Because in this second property, you said, pick an X, then you can say that for every episode, there exists an X. The data initial index depends on what? Of course, on X, how much you want to get close to the lead.
29:23:570Paolo Guiotto: Backup also, it depends on the sequence. It's not limited as a quantity.
29:28:430Paolo Guiotto: You change the sequence, you know, I think you must wait a bit more. Here, why you change the sequence? You change the sequence, and the fit one changes the X.
29:38:290Paolo Guiotto: So, what is missing here is that, in general, this N depends on X. So, more properly, I should write N that is a function of epsilon, of course, but also on X. While this N here, that you see here.
29:58:800Paolo Guiotto: If that's a universal N, that depends only on epsilon. The same N falls for every answer. So that N is an N for epsilon.
30:12:610Paolo Guiotto: So the difference is that when you have infinity convergence, there is a uniform, a uniform initial index.
30:21:550Paolo Guiotto: after winter, you have pursued the business between FN and AFA, whatever is the point, last election. So you are uniformly close to the ribbon.
30:33:450Paolo Guiotto: Why? For this X, I have to wait, maybe, initially for 1 million. For the other X, I have to wait maybe initially equal to 1 billion. For another, it's 1 trillion, and so on.
30:49:660Paolo Guiotto: But if there are infinitely many points, as it happens, that X is in the domain of these functions, so if x is not finding, it is not sure that you can find an initial X that works for every X.
31:05:90Paolo Guiotto: Okay? So that's why we say that's a uniform convergence, because the initial, the start to the point after which you are sure to be close to the limit, is independent of the point. It's uniform in the point.
31:19:650Paolo Guiotto: Why for pointwise convergence, this will depend in general from point. If you arrive to say that this is independent of the point, you have uniform convergence, so you have the other one, no?
31:32:590Paolo Guiotto: Okay.
31:50:750Paolo Guiotto: Okay, I would say, we will… today is Wednesday, so we have class on Friday.
31:58:30Paolo Guiotto: So, do… Exercises… Tan.
32:06:150Paolo Guiotto: 3… One… The number 2?
32:15:590Paolo Guiotto: Well, I… I will do… I… I'm… I… I will, definitely do the number 2, so…
32:23:270Paolo Guiotto: You can also do number 3.
32:27:580Paolo Guiotto: Maybe I will do also this one.
32:30:990Paolo Guiotto: These are exercises, let's say, concrete exercises. Well, number 3 is already on the L2 convergence, but you have the L2 norm, so in principle, you have everything to do. It's just a matter of flying the definitions. So, also, you could do the number 4.
32:49:870Paolo Guiotto: Let's see… I'm not sure I will do all the… at least 2 out of 3.
33:02:310Paolo Guiotto: Yeah, basically… All the exercises can be made, but let's say… Also 5, 6,
33:13:30Paolo Guiotto: Just to give you, maybe, something… The number 7…
33:22:310Paolo Guiotto: And number 8, which is a more XM-like,
33:26:890Paolo Guiotto: Exam-like, because there are several questions, normally. So, do these exercises by Friday.
33:33:570Paolo Guiotto: Okay.
33:35:240Paolo Guiotto: Let's stop here, have a nice day.