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Transcript
00:12:690Paolo Guiotto: Okay. Good morning.
00:17:740Paolo Guiotto: So, today we start with some of the exercises left on convergence.
00:24:390Paolo Guiotto: And, in particular, we start with Exercise 10.
00:29:730Paolo Guiotto: 3… 2… Here we have the spaces, the space of continuous functions on 0, 1,
00:39:590Paolo Guiotto: Equipped with the uniform normal.
00:44:960Paolo Guiotto: the sequ… we have two sequences. One is Fn of x equal to X to the n minus X to the n plus 1.
00:54:210Paolo Guiotto: And the second sequence is Gn , pretty similar, x to n minus X to 2N.
01:03:670Paolo Guiotto: Well, clearly these sequences are sequences in D because they are made of continuous functions.
01:13:170Paolo Guiotto: So, number one, discuss com… discuss… convergence…
01:22:420Paolo Guiotto: of these two sequences, F and GN…
01:28:80Paolo Guiotto: in V, with respect to the uniform norm.
01:32:870Paolo Guiotto: Number two, We have to discuss… well, it says what happens, what happens?
01:44:40Paolo Guiotto: if… We consider… the one norm on V.
01:52:110Paolo Guiotto: We know that this norm is well-defined on V, it is a true norm.
01:57:250Paolo Guiotto: So what… what about the convergence of these sequences in V? Okay.
02:02:820Paolo Guiotto: It's a pretty standard exercise, but let's start with the question mark.
02:11:120Paolo Guiotto: Now, last time, we said that
02:16:530Paolo Guiotto: uniform convergence is stronger than pointwise convergence. So, if, for example, FN
02:24:360Paolo Guiotto: is convergent in uniform norm to F, then FN must be convergent, in pointwise sense, to F.
02:34:970Paolo Guiotto: So let's use this to identify the possible limit. We know that this is not an equivalence, so we cannot say that once we have the pointwise convergence, we have automatically the uniform convergence, but at least this will be helpful to identify the limit F.
02:52:20Paolo Guiotto: Now, F of X is for X fixed in 01,
02:58:880Paolo Guiotto: Is, if it exists, of course, the limit
03:03:50Paolo Guiotto: for n going to plus infinity over Fn , so in this case, limit for n going to infinity of X to the n minus X to the n plus 1.
03:13:850Paolo Guiotto: So what can be said about this?
03:16:210Paolo Guiotto: Now, since our X is between 0 and 1,
03:21:220Paolo Guiotto: X to the n will be between 0 and 1. Basically, we have to distinguish two cases, because there is a slightly different behavior. When x is 1, and when X is less than 1, greater than or equal than 0. Because if X is 1, here we have 1 to N,
03:40:30Paolo Guiotto: minus 1 to n plus 1, which are both equal to 1, so 1 minus 1 equals 0.
03:47:260Paolo Guiotto: While 1X is less than 1 positive, these two quantities, X to N,
03:55:550Paolo Guiotto: and X ton plus 1, they both go to 0, no? So, the limit in this case is 0. In any case, we did use that. This limit is 0 for every X in 0, 1.
04:12:370Paolo Guiotto: So now we have a candidate, so F equal… identical equals 0, which is an element of V.
04:21:510Paolo Guiotto: is… D… unique.
04:26:920Paolo Guiotto: possible… limit.
04:30:800Paolo Guiotto: To check that it is a limit, we have to compute the norm of Fn minus F, so the distance in uniform norm.
04:40:910Paolo Guiotto: So, to check… If… FN.
04:49:790Paolo Guiotto: goes in uniform norm to F, which is identically equals 0, we have… to… show… that,
05:04:860Paolo Guiotto: the universal norm of Fn minus F, goes to zero.
05:12:520Paolo Guiotto: Now, let's compute this quantity. The uniform norm of Fn minus F, the infinorm, in this case, so F is 0, so this is the norm of Fn.
05:26:560Paolo Guiotto: In this case, since we have continuous functions, this is the maximum.
05:32:30Paolo Guiotto: of the absolute value of FNX.
05:37:990Paolo Guiotto: for X in 01.
05:41:850Paolo Guiotto: This is, X to the n minus X to the N plus 1.
05:47:980Paolo Guiotto: We may notice here, incidentally, that for X between 0, 1,
05:54:300Paolo Guiotto: the quantity X to N plus 1 is…
05:58:150Paolo Guiotto: respect to X to the N is…
06:06:780Paolo Guiotto: smaller. So in particular, this means that this is positive. So we can, just write the maximum of X to the n minus X to the n plus 1.
06:21:600Paolo Guiotto: for X in 01, which is our FN.
06:28:320Paolo Guiotto: So now, to determine this mass maximum.
06:31:450Paolo Guiotto: So this function is positive. We are studying on 0, 1. It is equal to 0 at the two endpoints. It is positive, so there is no part of…
06:43:310Paolo Guiotto: the graph of F and here, so we do not care what happens here, also, and for X negative.
06:51:580Paolo Guiotto: So, basically, what we do is a little, study of function to determine the maximum. Actually, we just compute the derivative. Fn prime of x is equal to n times x to the n minus 1 minus n plus 1 times X to the n.
07:11:780Paolo Guiotto: So we can factorize x to the n minus 1, then it comes n minus
07:17:500Paolo Guiotto: N plus 1 times XM.
07:21:730Paolo Guiotto: Because from this, we see that F prime and FN is greater or equal than zero, if and only if, since X for us, it is greater or equal than zero, so this quantity here is positive or zero, if and only if n minus n plus 1
07:40:190Paolo Guiotto: X is greater or equal than 0. That means that X must be less or equal than n over n plus 1.
07:50:10Paolo Guiotto: So it means that the function FN from 0 to 1, there is this point n over n plus 1, that 4N large, for example, will be close to 1.
08:03:970Paolo Guiotto: So we have that the sine of derivative is positive here, negative here, function is increasing here, decreasing here.
08:11:250Paolo Guiotto: Here, so that point is the maximum for the function. So we have here the point n over m plus 1,
08:20:610Paolo Guiotto: I… I draw close to… to 1, because for N big, it will be there.
08:28:470Paolo Guiotto: And what is the maximum, then, of the function? It is the value of the function at that point. So we can continue this calculation here, saying that…
08:40:220Paolo Guiotto: so the infinity norm of Fn…
08:43:780Paolo Guiotto: is equal to the value of Fn at this point, n over n plus 1.
08:51:410Paolo Guiotto: So when we replace this into the definition of Fn, this is n over n plus 1 to power n minus n over n plus 1 to power n plus 1.
09:06:230Paolo Guiotto: Now, we don't know what is this value, but let's say that more or less the function, it is irrelevant for the solution, is made like that, and this is the value FN at point n over n plus 1.
09:20:100Paolo Guiotto: Now, since this quantity represents the norm of Fn, which is the norm of Fn minus 0,
09:27:460Paolo Guiotto: We have to understand if this goes to zero or not. Now, this is a numerical secret, there is no more X, so it's a sort of first-year calculus exercise.
09:38:340Paolo Guiotto: So what can be said about these quantities?
09:42:80Paolo Guiotto: Now, as you can see, when n goes to infinity, this goes to 1, this goes to infinity. 1 to infinity is an indeterminate format, so we cannot just solve easily. And the same for this one, this is another 1 to infinity.
09:56:940Paolo Guiotto: However, You may recognize, perhaps, that this seems to be similar to an exponential limit.
10:05:410Paolo Guiotto: For example, if I write as 1 over, so I put this in denominator, inverting numerator with denominator, this becomes n plus 1 over n to the n.
10:18:70Paolo Guiotto: This quantity is 1 plus 1 over n to the n, which is the fundamental limit of the exponential.
10:26:140Paolo Guiotto: And more or less similarly here, because we have 1 over n plus 1 over n now to the n plus 1.
10:35:350Paolo Guiotto: So we can write this as 1 over 1 plus 1 over n to the N.
10:42:900Paolo Guiotto: minus 1 over. We can split this exponent here, and we have, again, 1, sorry, 1 plus
10:52:340Paolo Guiotto: 1 over n to the n times another coefficient, 1 plus 1 over n.
11:00:710Paolo Guiotto: So in any case, we see that this goes to E, this goes to E, this goes to 1, so everything goes to 1 over E minus 1 over E, which is equal to 0.
11:13:830Paolo Guiotto: And this is, what, the desired conclusion. So this says that FN converges in uniform norm to zero.
11:27:350Paolo Guiotto: Now, let's repeat this kind of work for GN, okay?
11:33:910Paolo Guiotto: 4.
11:35:210Paolo Guiotto: Jen… of X, we have.
11:41:870Paolo Guiotto: So first, as we understood, since if we want to have a uniform limit, we must have a point-wise limit. Let's compute the pointwise limit.
11:51:400Paolo Guiotto: So… D.
11:54:720Paolo Guiotto: pointwise, limit.
11:58:790Paolo Guiotto: is limited.
12:01:410Paolo Guiotto: for N going to plus infinity, GNX, so it is equal to the limit
12:09:40Paolo Guiotto: For n going to plus infinity of X to DN minus X to 2N.
12:16:680Paolo Guiotto: Which is exactly the same of…
12:19:200Paolo Guiotto: before, not for X equals 1, you get 0. For X less than 1, you have that these two, X to the n and x to 2N, they both go to 0. So this limit is constantly equal to 0 for every X.
12:34:580Paolo Guiotto: Okay, so the unique possibility… the unique.
12:42:640Paolo Guiotto: possibility… is that this GN goes in infinity norm, to zero.
12:53:80Paolo Guiotto: Which happens if and only if the norm of GN minus zero infinity norm, which is the norm of Gn.
13:03:340Paolo Guiotto: This goes to zero.
13:05:730Paolo Guiotto: Okay, now let's compute the norm to see if… whether this is true or not. So we have infinity norm of Gn is, again, the maximum.
13:17:760Paolo Guiotto: of the modulus of GNX.
13:21:930Paolo Guiotto: for axi 01.
13:25:710Paolo Guiotto: Now, we replace the function. Gn is X to the n minus X to 2N. Exactly as above, we have that X to 2N is smaller than X.
13:39:10Paolo Guiotto: to the n. Now, because the exponent is higher, so this is less or equal than x to the n, so this is greater or equal than zero, so we can say that this is, at the end, the maximum of the function X ton minus X to x to 2n.
14:00:820Paolo Guiotto: X to 2N for X in 01. So now, which is our GN, of course, in this case?
14:08:610Paolo Guiotto: So, let's see now what is the situation. It's pretty similar to the previous one.
14:13:810Paolo Guiotto: So we are in interval 0, 1, our function is positive, so we do not care what happens down here. We are looking between 0 and 1. The function at 0 is 0, at 1 is 0 as well. The derivative of this n
14:30:940Paolo Guiotto: is equal to NX to the n minus 1 minus 2NX to 2N minus 1.
14:41:430Paolo Guiotto: So, We can factorize an N here, an x to the n minus 1,
14:48:660Paolo Guiotto: So, we get 1 minus 2…
14:56:210Paolo Guiotto: Then we have X to 2N minus 1 minus n minus 1, so it is X to the n.
15:03:970Paolo Guiotto: Of course, I'm not strong.
15:05:790Paolo Guiotto: Now, this is greater or equal than 0 if and only if, remind that our x is between 0, 1, so the quantity x to the n plus minus 1 is positive, this is positive and is an integer, so if only 1 minus 2x to the n is greater or equal than 0.
15:23:840Paolo Guiotto: So if, and only if X to the n is less or equal than 1 half.
15:30:320Paolo Guiotto: And since, our X is positive, this means X less or equal than, 1 of,
15:39:730Paolo Guiotto: 2 to 1 over N.
15:42:140Paolo Guiotto: Okay? So the nth root of 2. Now, at this point, when n is large, 1 over n is close to 0, so 2 to the 1 over n will be close to 2.
15:54:850Paolo Guiotto: 2 to 0, so 1. So again, it's a pointer somewhere here, so this is 1 over 2 to 1 over n.
16:05:180Paolo Guiotto: And, therefore, the infinorm of GN,
16:12:100Paolo Guiotto: which I remind is also, in this case, the distance between Gn and 0, which is the unique possible distance to be checked. It is equal to the value of this Gn at this point, 1 over 2 to 1 over N.
16:28:310Paolo Guiotto: So let's see what happens when we plug this into the function.
16:31:940Paolo Guiotto: So it is 1 over, if you want, 1 over 2 exponents, 1 over n, so this is DX to power n.
16:42:290Paolo Guiotto: minus the same 1 over 2 to 1 over n to power 2N.
16:51:380Paolo Guiotto: So here, it happens that the two powers, because of the rules of multiplication of exponents, not power over power, you multiply the exponents, so 1 over n times n is 1, so you get 1 half here.
17:05:329Paolo Guiotto: Minus… also here, 1 over n times 2n is 2, so 1 half square is 1 fourth.
17:12:930Paolo Guiotto: So it is equal, you see, even constantly to 1 fourth.
17:17:750Paolo Guiotto: So that norm is constantly equal to 1 fourth, it doesn't go to 0.
17:23:660Paolo Guiotto: So we concluded that this sequence, GN, Conclusion.
17:32:560Paolo Guiotto: Giann… Cannot.
17:37:800Paolo Guiotto: be uniformly… convergent.
17:42:680Paolo Guiotto: on 01.
17:45:620Paolo Guiotto: Which is a way to say it's not convergent in the uniform norm.
17:49:610Paolo Guiotto: On the interval 0, 1. You may say, maybe it converges to someone else. No, because we know that if you have uniform convergence, you have pointwise convergence. So the unique possible limit is the point-wise limit.
18:03:310Paolo Guiotto: So there cannot be any other limit. Even if we check just one possibility, that's the unique one for this convergence.
18:12:650Paolo Guiotto: Okay, so you can see that point-wise convergence, so this is a concrete example, the example that says, even if the point-wise convergence is weaker than uniform convergence, so it does not show anything about the uniform convergence, it is very helpful
18:31:860Paolo Guiotto: Because it is useful to identify
18:35:550Paolo Guiotto: the candidate to be delivered, okay? So, if possible, we can first look for the point where it's convergent, and then check with that candidate if it is the right one.
18:49:510Paolo Guiotto: Now, the second question asks to basically repeat the discussion, but with a different norm.
18:57:580Paolo Guiotto: So, in principle, the norm is different. Well, we know that if the norm are equivalent, convergence in one norm is the same of convergence in any other norm, no? But we know that this is not the case. The infinity norm and one norm are not equivalent. This was an example showing that in infinite dimensions, we normally have that norms are not equivalent.
19:20:910Paolo Guiotto: But we may remind that the infinity norm is stronger than one norm. And that's helpful because, for example, I can say.
19:29:150Paolo Guiotto: Okay, since, infinity norm.
19:35:390Paolo Guiotto: is stronger.
19:40:520Paolo Guiotto: Then… One Norma.
19:45:970Paolo Guiotto: So, and we know that DFN is convergent, In the infinite norma.
19:56:220Paolo Guiotto: to zero.
19:57:650Paolo Guiotto: then we can conclude that it will converge also in the L1 norm and to the same limit.
20:06:220Paolo Guiotto: So this is a… you see, it's a quick, faster way to… otherwise, I should repeat the discussion, no? And this would become more complicated.
20:20:410Paolo Guiotto: However, since we have to do 4GN this discussion, let's see 4GN. So, 4GN, we do not have the uniform convergence, so for example, we cannot invoke this fact, okay? So we have to restart from scratch, basically, okay?
20:35:600Paolo Guiotto: So, what about the…
20:43:190Paolo Guiotto: GM.
20:44:780Paolo Guiotto: And now, here, we should restart from scratch, saying, what should be the limit?
20:51:140Paolo Guiotto: We have not yet seen what is the relation between one normal convergent And…
20:58:800Paolo Guiotto: pointwise convergence. That will be somehow complicated, as you will see, okay?
21:05:120Paolo Guiotto: Why I'm saying this? Because we used the fact that here.
21:10:610Paolo Guiotto: The point-wise convergence is weaker, is implied by the uniform. So, I usually start to identify the possibility, no? That's the strategy.
21:22:830Paolo Guiotto: But since 4D1 we have not yet seen if this implication falls, I'm 3.v2, in fact.
21:31:600Paolo Guiotto: But there will be a weaker version of this, we will see the star. So, for the moment, we don't know what is the relation between the 1 normal, L1 normal convergence, and the pointwise convergence.
21:42:910Paolo Guiotto: So, at this stage, I… I don't know what is the relation between this point-wise unit that, in any case, G has, you know, because we know that G is pointwise convergent to zero.
21:57:630Paolo Guiotto: Okay, we know that, no? But we don't know if this is the unique possibility.
22:04:220Paolo Guiotto: However, why don't you try to check if this is, is the limit, no?
22:12:120Paolo Guiotto: we have a possibility, let's try that one, no? So…
22:18:500Paolo Guiotto: Is JN perhaps converging in 1 Norma to 0?
22:25:360Paolo Guiotto: It's a sort of one-shot we have, let's try with this one. Now, if we want to try this.
22:31:780Paolo Guiotto: to check this, again, I have to compute the one norm of the difference. To check this, We,
22:44:730Paolo Guiotto: to show… If the distance between Gn and 0, now in one normal, goes to 0.
22:55:400Paolo Guiotto: Now, this is, of course, the one norm.
22:59:150Paolo Guiotto: of GN…
23:00:810Paolo Guiotto: And so we have to prove, we have to say if this goes to zero or not.
23:07:650Paolo Guiotto: So, let's compute this one node.
23:10:690Paolo Guiotto: No.
23:12:460Paolo Guiotto: The one normal of Gianna
23:15:930Paolo Guiotto: is, by definition, the integral from 0 to 1 of modulus of Gn x dx.
23:24:60Paolo Guiotto: Again, we plug into this formula DGN, which is X ton minus X to 2N, and we remind that this quantity is positive, so we can just say that this is the integer form, 0 to 1 of X to the n minus X to 2N.
23:42:900Paolo Guiotto: Fortunately, for this case, this quantity can be easily computed.
23:47:300Paolo Guiotto: In fact, the first one is the evaluation of X to the n plus 1 divided n plus 1 between 0 and 1, no? Because this function here, in brackets, has derivative X to the n.
24:03:320Paolo Guiotto: And similarly, the other one is the evaluation of X to 2N plus 1 divided 2N plus 1 between 0 and 1.
24:14:370Paolo Guiotto: So, of course, if we can compute explicitly, it's always…
24:18:430Paolo Guiotto: 99% of times, the best thing to do, because we have the exact evaluation, you see?
24:27:450Paolo Guiotto: Now, what happens here? This is 1 over n plus 1 minus 0 minus… this is 1 over 2n plus 1 minus 0, so 1 over 2n plus 1.
24:39:180Paolo Guiotto: So this is the exact calculation of the one norm of GN, and as you can see, when you send N to plus infinity, this goes to zero.
24:49:450Paolo Guiotto: So, we have… we have been lucky, because this means that GN goes to 0 in 1 norel.
25:01:200Paolo Guiotto: There is no contradiction, no? Because,
25:05:170Paolo Guiotto: you can well have that a sequence is convergent in a weaker norm, but not in a stronger norm. That's perfectly, fine.
25:15:530Paolo Guiotto: Okay, so let's do another example, like the exercise 10… 3… 4…
25:29:70Paolo Guiotto: So here we have this Fn of X,
25:32:900Paolo Guiotto: equal 1 over the root of X plus 1 over n for X into 0, 1.
25:45:660Paolo Guiotto: Well, plot the graph, of, Graph.
25:53:100Paolo Guiotto: of FN… Then it asks, is the sequence of functions FN contained in L1, 01,
26:07:710Paolo Guiotto: Is it contained in L201?
26:13:960Paolo Guiotto: Then, question 2… is FN convergent?
26:24:240Paolo Guiotto: in L1… Or in L2.
26:29:240Paolo Guiotto: And of course, in the case, in affirmative case, to what is convergent dysfunction?
26:35:850Paolo Guiotto: This sequence, sorry.
26:39:620Paolo Guiotto: So, you know, the graph is possible only in few circumstances, but if… policy, but… Why… why… why…
26:51:790Paolo Guiotto: it is better to do. So we are on 01… 01.
26:57:600Paolo Guiotto: These functions are 1 over the root plus something, you know? So they are basically the root, but with the translation, no? They would be like if the origin is at point minus 1 over n, so we imagine that
27:11:890Paolo Guiotto: for example, for X equals 0, these fn at 0 are equals to 1 over the root of 1 over n, so basically they are root of n, they are bigger for N bigger.
27:25:930Paolo Guiotto: root of n, and then, for, n equal 1, FN of 1,
27:32:860Paolo Guiotto: is 1 over root of 1 plus 1 over n, which is about 1, no? A little bit bigger than 1, so it will be a little bit smaller than 1, but very close to 1. So if this is 1,
27:46:710Paolo Guiotto: We may imagine something like this, no?
27:50:740Paolo Guiotto: This is the plot of the airplanes.
27:53:500Paolo Guiotto: Now, in particular, as you can see, these FN are, of course, continuous, well-defined continuous functions on 0, 1. Here, you are looking within freeze, you are not doing yet the limit, okay?
28:06:690Paolo Guiotto: So for n freezed, positive, no? You don't have the problem of 0, because the argument can never be 0. So you have a nice continuous function on 0, 1, and therefore it will be contained in any LP space.
28:23:980Paolo Guiotto: So it will be measurable, integral, with any power, because the power will be continuous. You see the argument, or should I write?
28:34:110Paolo Guiotto: I mean, to stay in IP, you must have… the integral of modulus F to power P is finite, no? But since F is continuous, modulus of F is continuous. To the power p, it is continuous, so the integral is final. So there is basically nothing to say. This for every P, but let's say we are here with P equals 1, 2, so…
28:53:670Paolo Guiotto: So, basically, we responded to the first question.
28:57:920Paolo Guiotto: Now, let's pass the questions.
29:01:250Paolo Guiotto: To question two, we have to discuss convergence in L1 and in L2 of this sequence.
29:09:340Paolo Guiotto: And here again, we need to understand, yes, but if there is convergence, convergence to what?
29:17:620Paolo Guiotto: What should be the bet? Well, intuitively, you see here there is, again, a pointwise limit, no? So, even if we don't have now, at this point, we don't have the relation between LP convergence and pointwise convergence, but we have understood that somehow point-wise convergence is the weakest form, no? It's not connected to a norm. It can be even proved that there is not a norm.
29:42:550Paolo Guiotto: that gives us convergence, the pointwise convergence. But however, let's consider the point-wise convergence to get some idea. So we may notice
29:56:440Paolo Guiotto: That… the limit when n goes to plus infinity.
30:06:450Paolo Guiotto: So it is the limit for n going to plus infinity of 1 over the root of X plus
30:14:830Paolo Guiotto: 1 over n. Now, this is easy, because when n goes to infinity, this goes to 0. The argument to the root of the root goes to X,
30:23:960Paolo Guiotto: So the root is continuous, it will go to root of X. So 1 over root of X. So this, at least for x positive, so for x between 0 and 1 excluding 0, where we should say it is equal to plus infinity.
30:39:440Paolo Guiotto: But however, for all X except one, so we may say for almost every X here, of course, the underlying measure will be the Lebec measure. So this is the pointwise limit function.
30:54:430Paolo Guiotto: No.
30:55:980Paolo Guiotto: So the question is, could be this the candidate?
31:01:440Paolo Guiotto: So, to be the candidate in L1, at least this should be an L1 function, otherwise you are going out of the space. Now, so, the gas is… well, the gas.
31:17:140Paolo Guiotto: Is FN now converging in L1 to this F?
31:26:510Paolo Guiotto: Well, first, let's see… if this F is in L1, otherwise we cannot say that.
31:33:980Paolo Guiotto: No? F is in L1, and that's clear, because, Yes.
31:43:160Paolo Guiotto: Because, now, this function is not 100% continuous, because there is one dead point at zero. However, we can say that the function f is continuous.
31:54:50Paolo Guiotto: everywhere, except at zero, so almost everywhere, and this, as we know, makes the function measurable. So, we can say that
32:05:330Paolo Guiotto: that the F… It's measurable on 01.
32:11:400Paolo Guiotto: And moreover.
32:13:450Paolo Guiotto: The integral 01 of the modulus of F, well, F is positive, is the integral between 01 of 1 over root of x, which is 5.
32:25:370Paolo Guiotto: Okay?
32:26:550Paolo Guiotto: You can even compute, and you get value… to… That helps, okay?
32:33:670Paolo Guiotto: So this is a 2 root of X evaluation between 0, 1.
32:38:690Paolo Guiotto: equal to.
32:41:280Paolo Guiotto: Okay, so we know now that this is in L1, so it could be a potential limit, and so to check this.
32:50:600Paolo Guiotto: We have to, now, to discuss,
32:54:790Paolo Guiotto: the behavior of norm of Fn minus F in 1.
33:00:40Paolo Guiotto: L1. So, the integral between 01 of absolute value 1 over root of X plus 1 over n minus 1 over root of X.
33:13:230Paolo Guiotto: the X.
33:14:870Paolo Guiotto: Now, let's see, and we have to send this to infinity.
33:20:380Paolo Guiotto: Now, we may notice here something, first of all, because in this case, we can do the calculation exactly, so why not? Otherwise, we may use, for example, some limit theorem, like dominated convergence, monoton convergence. Let's see what happens here.
33:38:960Paolo Guiotto: Here I may notice that, first of all, I don't need that absolute value, because this denominator, root of X plus 1 over n, is
33:48:960Paolo Guiotto: is with respect to this one? What is the relation to denominators?
33:59:250Paolo Guiotto: Yeah, the first is greater, so the fraction will be smaller. And therefore, when you have the absolute value, the absolute value is exactly 1 over root of X minus 1 over root of X plus 1 over n.
34:16:469Paolo Guiotto: Where this is now positive.
34:20:940Paolo Guiotto: Okay, so if you want, right here, you could, use, to, to, you could compute this.
34:29:739Paolo Guiotto: This can be done, exactly. If you want to use the NEVI, let's say, you could say that here you can apply, I think, even both theoren-dominated or monotone convergence.
34:42:980Paolo Guiotto: Because, for example, this is positive and less than… since I am subtracting, to, 1 over root of X, this 1 over positive, so this is clearly less than 1 over root of X, right?
34:59:300Paolo Guiotto: You see what I'm saying? This is the function g of x, which belongs in L10 is the integral dominant, no, of the sequence.
35:10:970Paolo Guiotto: Then, I have the integral dominant. I can also notice that when I send inside the integral n to infinity, that's the pointwise limit I checked above, no? So this goes to zero, so by dominated convergence.
35:27:800Paolo Guiotto: Otherwise, in this case, you can't compute the value and resolve the question. You get that this goes to the integral 01 of 0, which is equal to 0, and so the conclusion is, yes, our FN goes to F
35:43:600Paolo Guiotto: So, 1 over root of X, in one norm.
35:50:130Paolo Guiotto: And that's fine, no?
35:52:460Paolo Guiotto: So we… we discussed the convergence in L1. What about L2?
36:07:210Paolo Guiotto: Now, can we say that,
36:12:820Paolo Guiotto: We have a pointwise limit, no? So why don't we try again to use that one? Can we say that FN converges also in L2 sensor
36:24:100Paolo Guiotto: to, F.
36:26:720Paolo Guiotto: We have not yet seen
36:29:770Paolo Guiotto: possible relations between NP norms. There are relations under second.
36:34:960Paolo Guiotto: So, let's say one is stronger than the other. This is important, because we know that if, for example, one number is stronger than two normal, we would say, since it is converted here, we would have come to this also here. But in fact, what turns out if they are normal?
36:50:770Paolo Guiotto: So what is knowing that the attitude normal for this gate is stronger than anyone who will stay there.
36:57:80Paolo Guiotto: For the moment, we do not have such a relation, so we restart the discussion from scratch. The first remark is that, to be true, this F must be in L2. So, first, is F
37:12:710Paolo Guiotto: in L2… Because if this is false, L201.
37:21:140Paolo Guiotto: If this is false, Doesn't mean that the sequence FN is not convergent in L2.
37:27:510Paolo Guiotto: But it means that it cannot be convergent to that F, no? Maybe it converges to someone else.
37:34:110Paolo Guiotto: Well, let's check. F is, again, a measurable function on 01.
37:40:850Paolo Guiotto: And to be in L2, I have to check that the integral of F squared is fine.
37:47:900Paolo Guiotto: Right? But what is this? Is the integral 01 of the square of root… 1 over root of X? That's 1 over X.
37:57:690Paolo Guiotto: And that integral, we know, is infinum. It is the log between 1 and 0, and therefore it is infinite. So the conclusion is that, no, our F is not in L2, so, F…
38:11:360Paolo Guiotto: can… FN cannot be convergent in L2 sensor to this F, because this is simply something, a statement, without any meaning in L2, no? You are converging to someone which is not in the space. What does it mean?
38:28:450Paolo Guiotto: So… What this says… this says for the moment that FN is not convergent to F.
38:36:710Paolo Guiotto: But it does not tell us yet if FN is convergent to anything else or not, okay?
38:46:10Paolo Guiotto: However, now here we may suspect something, you know? You see that the altonorm of the limit is infinite, you know, because we did the square of the root.
38:57:420Paolo Guiotto: What happens if we do the norm of Fn?
39:01:460Paolo Guiotto: of Fn in L2 sense. So we do the square of Fn, you see that you get 1 over X plus 1 over N, that probably will go to infinity. We define it, because the function is certainly in L2, but the norm goes to infinity. So let's compute that.
39:21:250Paolo Guiotto: let's compute…
39:29:320Paolo Guiotto: the true norm of Fn, because this might be suspect. Now, when you compute an LP norm, apart for P equals 1, it is always better to compute to the power p.
39:40:810Paolo Guiotto: Because you take out the root, okay? So this becomes the integral between 01 of modules F and
39:49:420Paolo Guiotto: X square.
39:52:160Paolo Guiotto: Now, because of the definition, FN is positive, so the models
39:57:220Paolo Guiotto: Can be a mirror. There is also the square. This is 1 over X plus 1 over N dx.
40:05:770Paolo Guiotto: A quantity that we can compute, because this is a logarithm of X plus 1 over n, to be evaluated between 0 and 1. So we get log of 1 plus 1 over n minus log of 1 over n.
40:26:630Paolo Guiotto: Which is plus log of n, because of the properties of logarithma.
40:33:690Paolo Guiotto: And now you can see that this quantity goes to plus infinity.
40:39:470Paolo Guiotto: And you may be mind that,
40:42:330Paolo Guiotto: To be convergent, a sequence must be…
40:47:930Paolo Guiotto: bounded, okay? If it is unbounded, it cannot be convergent.
40:53:460Paolo Guiotto: So, the sequence of N Ease.
40:59:840Paolo Guiotto: Unbounded?
41:03:60Paolo Guiotto: And therefore, FN.
41:06:720Paolo Guiotto: cannot…
41:08:150Paolo Guiotto: Of course, this should be always referred to the norm, no? Because we have seen that in L1 norm, sequence is fine. So, in L2,
41:19:620Paolo Guiotto: So, cannot be convergent.
41:24:310Paolo Guiotto: in L2. And that's all for the exercise.
41:31:580Paolo Guiotto: Okay, do you have any questions?
41:39:150Paolo Guiotto: Okay, I would say, so, if you have not yet done, do, the 10-3-3, which is pretty similar to these ones.
41:51:860Paolo Guiotto: Then, the 5… Well, this… Yeah, this seeks to… the 7…
42:02:40Paolo Guiotto: Well, D8, I don't know why it is here. It's an exercise on norm space. There is… yeah, there is a sequence, but it's, in fact, is an exercise on norm, but you can, of course, do…
42:14:370Paolo Guiotto: Well, the last one, it's a theoretical exercise, so… Hmm.
42:20:690Paolo Guiotto: Try to do.
42:24:40Paolo Guiotto: Okay.
42:25:330Paolo Guiotto: So, let's go back to the discussion we started yesterday and tried to finish. So, yesterday, we started to explore
42:37:470Paolo Guiotto: relations between, the, uniform convergence and pointwise convergence. We discovered that
42:46:760Paolo Guiotto: On the space of bounded functions, uniform convergence imply the pointwise convergence, but vice versa is not true.
42:57:190Paolo Guiotto: Okay, well, yes. So, I want to close, this, discussion about the space P of X with two important, facts. So, let's…
43:15:360Paolo Guiotto: Recall… that, On.
43:23:300Paolo Guiotto: B of X.
43:26:60Paolo Guiotto: Equipped with the uniform. Norma.
43:30:540Paolo Guiotto: We proved that.
43:35:70Paolo Guiotto: that if a sequence FN converges in uniform normal.
43:40:300Paolo Guiotto: to F, then it converges also in pointwise sense.
43:45:890Paolo Guiotto: But not vice versa.
43:48:680Paolo Guiotto: Okay.
43:50:230Paolo Guiotto: The two facts I want to show you here… Are the following. First.
43:57:170Paolo Guiotto: We said that, yesterday also, we introduced the definition of what is a Cauchy sequence, because Cauchy sequence, the Cauchy property is, must be necessarily fulfilled by convergent sequences, no?
44:16:910Paolo Guiotto: And the nice thing with this condition is that it is an intrinsic condition that does not depend on the knowledge of any limit.
44:25:690Paolo Guiotto: And this says, if a sequence is convergent, it must verify this condition. Unfortunately, this condition is not characteristic for convergence, because we have seen an example that shows you may have a Cauchy sequence which is not convergent in that space.
44:43:290Paolo Guiotto: However, this is rather exceptional. As I told you yesterday, most of the common spaces with the common concerns of convergence are, verify these conditions, so that each sequence
44:58:870Paolo Guiotto: each sequence verified equit property is convergent. We call them panic or complete spaces. What I want to show you now, in this case, because it is simple, is that this space with that norm is complete, is a Bannock space. So, proposition…
45:19:420Paolo Guiotto: the space BX.
45:22:240Paolo Guiotto: They keep the weekday uniform normal.
45:25:900Paolo Guiotto: Is. E. Bana.
45:31:30Paolo Guiotto: space.
45:32:350Paolo Guiotto: So, for this case, this means that… a sequence
45:37:840Paolo Guiotto: That is… let me refresh here… that is a sequencer FN.
45:44:40Paolo Guiotto: is convergent.
45:47:310Paolo Guiotto: if, and only if, FN is a cushy.
45:55:420Paolo Guiotto: sequence.
45:57:940Paolo Guiotto: It might seem pretty abstract, but it is not, because I told you, when you will do modeling, usually you will have to solve a problem, so you will have a solution, which is the solution of some equation.
46:13:580Paolo Guiotto: That could be a differential equation, integral equation, whatever.
46:17:480Paolo Guiotto: Now, normally, you are not able to solve explicitly that equation, because the number of equations we are able to solve explicitly is a very limited number, particular cases, etc, okay? So, how do we do in general to solve an equation? We try to solve an approximation of that equation, where we are able to solve the equation for
46:38:960Paolo Guiotto: different reasons, no? Because we have something that ensures that it's just a solution. And then, we try to show that this sequence of approximation converges to the right solution.
46:52:530Paolo Guiotto: And that's where you have to know these facts, because normally, you don't know what is the limit. Even… you have to prove that a limit exists.
47:02:240Paolo Guiotto: So, you have your sequence, and you want to establish if a limit exists. And the unique, reasonable test you can do is the Cauchy property. So, you are able to assess the quantity Fn minus FM in norma, ensure that it gets small when M are big. That's the point. You need to know that, however, that norma makes… is a norma on which the space is complete.
47:27:10Paolo Guiotto: So, on which, when the Cauchy property is verified, the sequence is converged. So that's why it's so important to know that the underlying space is a Bannock space.
47:37:260Paolo Guiotto: Now, let's see the proof. The proof is more or less always the same for all these cases. So, I start with the Cauchy sequence, so let…
47:48:870Paolo Guiotto: F, N, B, cushy sequence.
47:55:630Paolo Guiotto: In infinity norm, that is, for every epsilon positive, there exists an initial index n, such that the uniform norm FN minus FM is smaller than epsilon for every indexes n and m larger than capital N.
48:13:900Paolo Guiotto: And now, the goal is to show that, of course, this sequence is convergent. But to show that it is convergent, we need to identify the limit. So, the goal is the following.
48:26:720Paolo Guiotto: goal is, find…
48:30:550Paolo Guiotto: NF, that must be a vector of the space, so in this case a bounded function, such that this FN converges in uniform norm to F.
48:43:130Paolo Guiotto: Now, how do I find this F?
48:46:980Paolo Guiotto: Well, we know that if we have the uniform converters, we have, obviously, the bin class converter.
48:53:350Paolo Guiotto: So that must be the point by Z of DFN. So the first step should be, let's show that DFN has a point by Z, okay? So…
49:06:640Paolo Guiotto: Step 1…
49:14:270Paolo Guiotto: Let's… a shoe.
49:19:740Paolo Guiotto: that, the sequence FN has… a point-wise limit.
49:29:200Paolo Guiotto: Okay? And so how do we do this? Well, let's just rewrite this. This says that for every epsilon positive, there exists in index n, such that
49:39:40Paolo Guiotto: The uniform norm is the supremum, right? Supremum for X in capital X, of absolute value FNX minus FM.
49:50:300Paolo Guiotto: X, less or equal than epsilon for every N and M larger than this capital N.
49:58:380Paolo Guiotto: So, since the supremum must be less or equal than epsilon, it's similar what we said yesterday to deduce from uniform convergence, pointwise convergence, then this means that the modulus FNX
50:12:680Paolo Guiotto: minus FMX less or equal than epsilon, this for every X in X, and for every N and M larger than capital N.
50:26:670Paolo Guiotto: But then, since, see if you freeze X, so… 4NX.
50:36:870Paolo Guiotto: generic X. In capital X, huh?
50:40:480Paolo Guiotto: say, freezed.
50:44:530Paolo Guiotto: So you choose an X, that one.
50:47:820Paolo Guiotto: This is saying that for every epsilon positive the existing initial index n, such that distance FNX minus FMX is less or equal than epsilon for every n and m larger than capital N.
51:05:50Paolo Guiotto: So this means that the sequence of numbers, Fn ,
51:12:150Paolo Guiotto: This is a sequence of real numbers, no? Because we now evaluated all functions at this point, X. We have a sequence of numbers, so we have a numerical sequence. This is a Cauchy sequence.
51:30:740Paolo Guiotto: But R fulfills the completeness property, so it's a Bannock space with respect to models. And since R
51:45:610Paolo Guiotto: is a banner.
51:50:850Paolo Guiotto: space. This implies also that there exists the limit in n, when n goes to plus infinity, of this sequence of numbers, fn of X.
52:03:270Paolo Guiotto: Now, this happens X by X.
52:07:360Paolo Guiotto: You change X, you change the sequence, but still you have the limit.
52:11:480Paolo Guiotto: So this automatically defines a function. We baptize this limit, f of x. It will depend on X, because if you change X, you change the length here. Now, look at the examples we have seen above, you know? Take whatever is your sequence.
52:30:390Paolo Guiotto: You see here, FN of X is this quantum f of x. You change X, you change that quantum. But when x is 3, that is one fixed quantum, that's a numerous.
52:42:410Paolo Guiotto: We have a unit that really depends on x, so we use a function on X, we call it F of X.
52:48:520Paolo Guiotto: Okay, now we have the point-wise limit.
52:52:500Paolo Guiotto: Are we done? Of course not, because pointwise limit is not equivalent to uniform limit. So, the step two is to show that this F is also the uniform limiter. So, step two…
53:08:890Paolo Guiotto: We now claim that Fn goes to that F we built just here.
53:16:970Paolo Guiotto: And how can we prove this?
53:19:280Paolo Guiotto: No.
53:20:780Paolo Guiotto: We look at this property again.
53:27:280Paolo Guiotto: Okay.
53:28:550Paolo Guiotto: Take this property and send one of the two, N or M, for example, M to plus infinity.
53:35:730Paolo Guiotto: So… letting… M to plus infinity, in star.
53:46:720Paolo Guiotto: We can do that, because this property holds for every N larger than some carbon n.
53:54:490Paolo Guiotto: Okay? So I have my little woman back there, who blocks his feet, huh?
53:59:540Paolo Guiotto: Right?
54:03:790Paolo Guiotto: So, it, it should be… supposed to, to, to charge.
54:11:40Paolo Guiotto: The wire is not charging? Very good.
54:16:500Paolo Guiotto: Technology search. Okay.
54:20:20Paolo Guiotto: So, when we let M2 plus infinity, this guy will go to, because of the point was, limit to f of x.
54:29:500Paolo Guiotto: And the inequality will be conserved because of the permanence of sine, you know? You preserve bounds in limit. So, we gather that for every epsilon positive.
54:41:510Paolo Guiotto: there exists a capital N, such that distance between F and X
54:46:340Paolo Guiotto: And now FX, this is less or equal than epsilon, for every n larger than capital N.
54:54:350Paolo Guiotto: And remind that that n is uniform in X. That's because it comes from this property, okay?
55:02:410Paolo Guiotto: So I can cite this also for every X in capital X.
55:07:530Paolo Guiotto: Then, if these quantities, if these quantities are bounded by epsilon, their maximum will be bounded by epsilon as well, so the supremum
55:17:440Paolo Guiotto: for X in capital X of this modulus FNX minus F of X.
55:24:700Paolo Guiotto: Also, this one will be bounded by epsilon.
55:28:160Paolo Guiotto: And this is exactly the uniform norm, FN minus F infinity norm. So what we have is that
55:38:550Paolo Guiotto: For every epsilon positive, we find an initial N such that distance between F, N, and F in uniform norm is less or equal than epsilon for every n larger than capital N. And that's the conclusion, because it means exactly by definition.
55:56:00Paolo Guiotto: that FN goes to F, okay?
55:59:810Paolo Guiotto: So, this is the, let's say, the unique proof I will do for completeness for these spaces, but they all are based on this kind of strategies, no? You first find a pointwise limit to identify the limit, and then second, you show that that point-wise limit is actually
56:18:750Paolo Guiotto: a limit in norm, okay?
56:22:310Paolo Guiotto: That is not because pointwise limit is a limited number, but because of the situation, because of the Cauchy problem.
56:29:560Paolo Guiotto: Okay, now, this space, the space of banded function on an abstract set with the uniform norm is,
56:39:930Paolo Guiotto: interesting, but not particularly concrete example. An interesting example is, for example, still with the uniform normal.
56:50:460Paolo Guiotto: a particular case of this. A particular case… It's particularly, in part, case off.
57:01:510Paolo Guiotto: bounded… functions, is, decays…
57:12:830Paolo Guiotto: of space, you remind this is important, of continuous function on a compact set.
57:21:570Paolo Guiotto: where K is contained in RD, is compact.
57:29:380Paolo Guiotto: So, closed and bound.
57:34:550Paolo Guiotto: It is important, because being compact domain, we automatically know, by such theorem, that the modulus is continuous and it is bounded. There is a maximum for the modulus. So in this case, in this case, the infinity norm, as we know.
57:57:980Paolo Guiotto: So the infinity norm for NF, which should be the supremum of modules F of X for X, belonging to this compact K, is actually a maximum. This is because of the biostass
58:18:390Paolo Guiotto: theorem. It is a maximum for x in k of modulus F of X.
58:26:310Paolo Guiotto: Now, what turns out is the following fact.
58:30:220Paolo Guiotto: proposition.
58:32:240Paolo Guiotto: That if we consider the space of continuous function on a compact set equipped with the uniform normal, also this is a Banach space.
58:50:260Paolo Guiotto: Well, exit-proof, because, you may think,
58:54:360Paolo Guiotto: It seems to be obvious. Why?
58:57:520Paolo Guiotto: We may notice that if you have a continuous function on K, because of… by Sust theorem, they are bounded, right? So this is contained in the space of bounded function of K. So by…
59:13:710Paolo Guiotto: via SAS TRM.
59:19:240Paolo Guiotto: models of F, pass.
59:23:420Paolo Guiotto: Max, huh?
59:25:730Paolo Guiotto: on cave… So, in particular, it is bound.
59:35:740Paolo Guiotto: So, this is a part of a larger space. Now, so if you want to have a figure, we cannot, of course, do a figure to represent these spaces, because they are infinitely dimensional, but let's say that this is the big space of a bounded function on K,
59:52:570Paolo Guiotto: Inside, we have a smaller set, which is the set of continuous functions on K.
59:59:590Paolo Guiotto: We may have bounded function which are not continuous, and so they are not in that set.
00:06:110Paolo Guiotto: So, what can be said then? So, if…
00:10:410Paolo Guiotto: We take a sequence F, N.
00:13:300Paolo Guiotto: in C of K, is a Cauchy sequence.
00:23:110Paolo Guiotto: So, since this is a subset of B of K,
00:27:460Paolo Guiotto: And we just come to see that BLK is complete. Whenever you have a Cauchy sequence, that sequence is convergent, right? So we know that…
00:39:90Paolo Guiotto: This is the previous… TRM, proposition.
00:45:40Paolo Guiotto: this sequence F and will converge in uniform norm to sum F.
00:50:660Paolo Guiotto: So the proof is over.
00:54:160Paolo Guiotto: This is not completely over. Why?
00:57:360Paolo Guiotto: He goes, that deaf is where?
01:02:980Paolo Guiotto: We said that the space B of K is complete, so if you have a Cauchy sequence in B of K, it will converge in B of K, so this is an element of B of K.
01:16:940Paolo Guiotto: And this is not yet what I want, because to conclude that my space is a bank space, I want that a Cauchy sequence converges to a point of my space. So, to a continuous function.
01:31:470Paolo Guiotto: So, in other words, I'm saying, suppose that you have a pushy sequence here, this is the sequence FN.
01:38:430Paolo Guiotto: So you look at this sequence as a space that is pending in the pin space. In that case, the sequence is convergent to something
01:48:850Paolo Guiotto: But the point is, can be this… well, of course, the right points cannot be outside of the space of continuous functions, but can I have a limit F here?
02:03:210Paolo Guiotto: If the limit is out of C of K, the sequence is not convergent in C of K, because it has not an image in C of K. You see? Now, what I want to show is that actually, in this case, that limit must be inside, as you may expect.
02:19:870Paolo Guiotto: So… The key point…
02:27:560Paolo Guiotto: is that limit f belongs to the space of continuous function on K.
02:35:800Paolo Guiotto: So, this is a remarkable fact.
02:39:510Paolo Guiotto: Let's underline this. This is a… Remarkable.
02:47:230Paolo Guiotto: Fact.
02:49:400Paolo Guiotto: Because it says that, huh?
02:52:160Paolo Guiotto: If you have a sequence of continuous function which is uniformly convergent to something, that something must be continuous.
03:02:520Paolo Guiotto: So, the continuity is preserved, In the uniform limit.
03:10:580Paolo Guiotto: Now, you're in mind that, for example, measurability is such a weak property which is preserved by pointwise limit.
03:19:70Paolo Guiotto: You do not preserve a continuity by pointwise limit. We have seen the example of the powers x to the n. The limit is 0, and with the jump at 1, no? So, the pointwise limit is too weak.
03:31:90Paolo Guiotto: And, measurability is a weak property which is preserved by that kind of limit.
03:37:220Paolo Guiotto: The convergence we need to preserve continuity is uniform convergence, a very strong form of convergence, okay? So, uniform…
03:51:550Paolo Guiotto: limit.
03:54:610Paolo Guiotto: off.
03:56:170Paolo Guiotto: continuous.
03:58:440Paolo Guiotto: functions.
04:01:720Paolo Guiotto: is… Continuos.
04:06:90Paolo Guiotto: function.
04:08:180Paolo Guiotto: So this factor is saying this, that continuity is preserved when we pass to the limit with the uniform norm.
04:18:420Paolo Guiotto: So, to show the Eastern.
04:20:960Paolo Guiotto: to show that F is continuous, F is continuous at, on C of K, if and or if F is continuous at every point. So, if I am able to prove that the limit when
04:36:730Paolo Guiotto: I don't know, Y goes to X of F of Y is equal to F of X,
04:45:20Paolo Guiotto: for every X in K, no?
04:49:20Paolo Guiotto: Continuity at each point means that the lead of the function is the value of the function at that point. So that's what I want to prove. To prove this.
05:01:870Paolo Guiotto: I do this. Let's take the difference between the two quantities, FY and FX. I want to convince you that this difference goes to zero.
05:12:150Paolo Guiotto: So let's take the absolute value.
05:15:80Paolo Guiotto: And we do an assessment of this distance.
05:18:350Paolo Guiotto: And this is made by what is a standard type argument, which is called the 3 epsilon 3…
05:29:150Paolo Guiotto: Epsilon argument.
05:35:370Paolo Guiotto: So the idea is that, let's introduce Fn into this. There is no FN, so to introduce Fn, I have to add and subtract. So let's do this. F of Y, I subtract FNY, and I have to add FNY.
05:53:100Paolo Guiotto: And they do the same for f of x. So I add and subtract F of X, Fn . So let's write this, minus FNX.
06:04:590Paolo Guiotto: plus FNX minus FX.
06:09:500Paolo Guiotto: So this is still F over Y minus F of X.
06:13:320Paolo Guiotto: Now, let's group this sum into three blocks. This, this, and this.
06:19:70Paolo Guiotto: And applied the triangular inequality.
06:23:70Paolo Guiotto: triangular inequality for the absolute value. You get modulus of F of Y minus F and Y.
06:33:10Paolo Guiotto: plus modulus F and Y.
06:37:130Paolo Guiotto: minus F and X.
06:40:710Paolo Guiotto: And plus… F, M, X.
06:45:780Paolo Guiotto: minus FX.
06:48:510Paolo Guiotto: Now, we have to convince you that these three quantities are small, and therefore, this one will be small.
06:57:190Paolo Guiotto: Now, the first and the last one are similar.
07:02:250Paolo Guiotto: They are multiple, they are distances at point Y between a fan in there, you see.
07:10:500Paolo Guiotto: So, both these two can be controlled by the same kind of entity. So, modulus of F of Y minus F, N of Y
07:23:790Paolo Guiotto: as well as modulus F and X minus FX,
07:30:390Paolo Guiotto: They are both controlled by the uniform norm, because the uniform norm is what? It's the maximum between modules of these differences. So, by the supremum, if you want to use a different letter.
07:43:970Paolo Guiotto: when Z is in K of modulus FNZ minus FZ.
07:51:330Paolo Guiotto: But that's the uniform norm of Fn minus F.
07:56:290Paolo Guiotto: And we know that Fn is going to F in uniform norms, so this quantity for N large, is going to be small. So let's say that here, continuing with the inequality, I have less or equal
08:09:130Paolo Guiotto: These two guys, the first and the third term, are both bounded by this one. So I will have a 2 norm of FN minus F, infinite norm. Plus, the other term.
08:28:439Paolo Guiotto: is not of the same type, because this is the difference between the distance between F and Y and that represents. So here, n is the same for the two functions.
08:42:200Paolo Guiotto: So here we will have the distance between the value of N at point X and point Y.
08:47:560Paolo Guiotto: What will you know about Wi-Fi?
08:50:649Paolo Guiotto: My friend, easy.
08:53:109Paolo Guiotto: Now, a context FN is taken where?
08:57:790Paolo Guiotto: Our sequence is a sequence of continuous function.
09:01:210Paolo Guiotto: So for Fn, this property, the limit of Y going to X Fn of Y equal FN of X is true.
09:09:890Paolo Guiotto: So what can we say? And now I will do a little bit informally.
09:13:939Paolo Guiotto: So, if you look from the mathematician point of view, this is not 100% correct, as of right, but can be made correct.
09:25:10Paolo Guiotto: So, let's rewrite our bound, first of all. So, we got this modulus FY minus FX,
09:34:750Paolo Guiotto: can be controlled by 2, the norm of Fn minus F infinity plus modulus Fn, Y minus FNX. This is what we obtain. Now, let's compute the limit.
09:51:910Paolo Guiotto: So, the limit… when Y goes to X of modulus F of Y minus F of X,
10:02:900Paolo Guiotto: will be less or equal than the right-hand side. Well, you see that this quantity here is a constant for X and Y.
10:10:660Paolo Guiotto: So when I do the limit, it remains itself. It's 2, infinity norm of Fn minus F,
10:17:520Paolo Guiotto: plus limit when Y goes to X,
10:21:730Paolo Guiotto: of this, modulus FNX minus FNY, okay? You agree?
10:30:630Paolo Guiotto: Now, this limit here is…
10:37:710Paolo Guiotto: is 0, because the function fn is supposed to be continuous.
10:44:370Paolo Guiotto: I don't know for F yet, but I know for FN this.
10:48:80Paolo Guiotto: So, I have that, my limit.
10:51:500Paolo Guiotto: when Y goes to, X,
10:56:190Paolo Guiotto: of, F of Y minus F of F, F of X is less or equal than to this quantity norm of Fn minus F infinity norm.
11:08:760Paolo Guiotto: But now, notice that the left-hand side does not depend on A, see?
11:14:990Paolo Guiotto: It's a constant, right? Amen. You don't see anywhere, there is no end.
11:21:410Paolo Guiotto: So if I take now the leading dengan, this will remain, etc.
11:26:340Paolo Guiotto: So, I will have that this quantity Here, the limit
11:32:990Paolo Guiotto: when Y goes to X of modulus F of Y minus F of X,
11:40:360Paolo Guiotto: will be less or equal than limit when n goes to plus infinity.
11:46:290Paolo Guiotto: of 2 norm of Fn minus F infinity. But that limit is zero, because we are assuming
11:54:700Paolo Guiotto: That we are assuming. We know, no, we are not assuming. We know it comes from this point, from the completeness of B of K, that that sequence goes uniformly to F. So…
12:08:790Paolo Guiotto: This limit is zero, because…
12:13:660Paolo Guiotto: FN goes in infinite norm to F. And therefore, we have the conclusion, because this must be greater or equal than zero, so the unique possibility is that the limit
12:30:340Paolo Guiotto: when y goes to X of absolute value F of Y minus f of x, it is equal to 0. And that's the conclusion, because this means that the quantity inside the models must go to 0.
12:44:320Paolo Guiotto: So, limit.
12:47:10Paolo Guiotto: when Y goes to X, of F of Y.
12:52:560Paolo Guiotto: minus F of X is 0.
12:56:110Paolo Guiotto: And that explains… of course, this solves for every X in K, and this explains the continuum, okay? So now this is a first example of completeness for a space
13:12:00Paolo Guiotto: Let's say, a complete real space of functions, continuous functions.
13:18:410Paolo Guiotto: on… On, on, a combat set.
13:23:990Paolo Guiotto: Okay, so…
13:26:310Paolo Guiotto: Still, in this class, so uniform norm, there is the completeness of the other important case. I will just limit to the statement, proposition. What if we now consider this space NL infinity space? So, X, F,
13:45:940Paolo Guiotto: mule… Measure… space, huh?
13:51:780Paolo Guiotto: Equip de.
13:55:310Paolo Guiotto: We did.
13:56:720Paolo Guiotto: what is the natural norm for this space is the L infinity norm, which is not the soup, you remind, is that quantity the essential soup, no? With the…
14:12:670Paolo Guiotto: Infinity.
14:14:550Paolo Guiotto: Norma.
14:17:250Paolo Guiotto: So, just to refresh your ideas.
14:21:410Paolo Guiotto: It was here. We introduced the L infinity space, defining the infinity norm as the smallest of the essential upper bounds, okay? That's not the stream moving, okay?
14:40:940Paolo Guiotto: Okay? But the function here is a load to me.
14:44:930Paolo Guiotto: I'm bomb, nigga!
14:46:880Paolo Guiotto: The unique point is that it must be unbounded over the mutual zero set, so you don't care what happens around there, no? So, on the zero set, this is bound, no? We take as normal the best of these essential upper bounds.
15:04:340Paolo Guiotto: And we have seen that this is a norm with weaker form for the benching.
15:10:870Paolo Guiotto: And that's the canonical norm for that space. However, it's in fact very similar to the… to the…
15:20:220Paolo Guiotto: to the infinity norm because of this characteristic, which is, as you can see, this quantity, the norm, bounds modulus f of x, not for every X, as it would be for the supremum.
15:35:870Paolo Guiotto: But for almost every act. So, basically, it plays the same role of the supremum. It works, so if you want to intuitively
15:45:120Paolo Guiotto: you can still think to the, supremum. Well, it turns out that this is a Banach space.
15:53:630Paolo Guiotto: So, let's eliminate the dot.
15:56:660Paolo Guiotto: Is. E. Bana.
16:02:740Paolo Guiotto: space.
16:05:650Paolo Guiotto: Well, you have the proof in the notes, if you are curious. Basically, the proof is very similar to the proof of the set of bounded functions. So, step one, you have a Cauchy sequence in this lymphang genome, and you prove that
16:21:430Paolo Guiotto: the sequence Fn of x is pointwise convergent almost everywhere in that case, okay? So you have a pointwise limit, you call it F. Since it is point-wise limit of measurable function is measurable.
16:35:280Paolo Guiotto: And then you prove that it is actually a limit in the L-infinity norm, okay? So it's similar, but a little bit more tricky, because the definition of the norm is more complicated, and so on.
16:48:620Paolo Guiotto: Now, let's come to the important point, which is the LP norm for P less than infinity, so for the traditional LP spaces.
17:00:590Paolo Guiotto: So, now… Let's… from… to… the… K's… Ugh.
17:21:980Paolo Guiotto: Well, I…
17:25:90Paolo Guiotto: I forgot to mention. It is implicitly… it is implicitly true, clear, but it's better if I specify… cancel a second this one.
17:36:420Paolo Guiotto: about this convergence, the Linfinity, we have to mention, let's say, moreover.
17:47:140Paolo Guiotto: Which is important. If FN converges in L infinity norm to some F, also this one, exactly as the uniform norm.
18:00:690Paolo Guiotto: implies point-wise convergence, but with the unique difference that this will be almost everywhere, because here we are dealing with measurable functions. So, you have always that measures your set of points where
18:13:200Paolo Guiotto: The property, the conclusion is false. So FN converges to F, pointwise almost everywhere.
18:22:360Paolo Guiotto: Okay?
18:23:510Paolo Guiotto: So it means that Fn of X goes to F of X for all X, except for a measure 0 set.
18:30:180Paolo Guiotto: Okay?
18:32:270Paolo Guiotto: Good. Now, let's come to…
18:38:490Paolo Guiotto: Now… to the case.
18:44:280Paolo Guiotto: off.
18:45:580Paolo Guiotto: LPX.
18:48:200Paolo Guiotto: with the… be between 1 and strictly less than plus infinity.
18:56:590Paolo Guiotto: Now, a first important pointer
19:00:860Paolo Guiotto: is that in general, LP convergence does not imply point-wise convergence, and that's different from all previous examples. In all previous examples, that basically were variations on the infinity normal. The infinity norma
19:18:540Paolo Guiotto: in the sense of the supremum norm, or in the sense of the essential supremum Norm implies
19:25:180Paolo Guiotto: The pointwise convergent.
19:27:430Paolo Guiotto: For the infinity norm, we have the pointwise convergence for all points. For the L infinity, we have the pointwise convergence for almost every point, okay? But basically, the message is the same.
19:39:520Paolo Guiotto: Infinity normal convergence implies pointwise convergence.
19:44:350Paolo Guiotto: Here, it is no more true.
19:47:430Paolo Guiotto: In this case.
19:53:40Paolo Guiotto: We have that FN convergence in P normal to F, does not imply
20:01:500Paolo Guiotto: that FN converges pointwise to F, even in the sense we can sense that almost everywhere sense.
20:11:890Paolo Guiotto: Now, there is a nice example.
20:18:870Paolo Guiotto: that instead of writing functions, we can do… they are basically indicator functions of intervals, so they are relatively easy to be written. I will do the figure of the graphs of this sequence, so you will see concretely what happens. So, let's take L1,
20:37:400Paolo Guiotto: fake.
20:40:780Paolo Guiotto: L1 on the interval 01.
20:45:920Paolo Guiotto: So, with usual a back measure, blah blah blah, and this sequence that we build in this way. So, I will draw the graphs of these functions.
20:56:910Paolo Guiotto: The first function is the function constantly equal to
21:01:750Paolo Guiotto: Or maybe it is better if we discard the first, we start from the… we consider as first this function. It is equal to 1…
21:10:720Paolo Guiotto: On 0, 1 half.
21:13:560Paolo Guiotto: and 0 from 1 half 1. So this is our… well, let's put the indexes later, okay? This is the first function.
21:22:410Paolo Guiotto: Then the second function.
21:26:850Paolo Guiotto: It's similar, only that this is 0 from 0 to 1 half, and 1 from, 1 half to 1.
21:45:100Paolo Guiotto: Why, let's call that this F0.
21:48:980Paolo Guiotto: And it's F1. Let's hope that the choice of indexes is correct. Now, what we do, we build an X group of functions that will be four functions.
21:59:540Paolo Guiotto: So, what we do, we divide the interval 0, 1 in 4 equal parts. So, we have 0, 1 fourth, 1 half, 3 fourths, 1.
22:10:780Paolo Guiotto: And the first function of this group is 1 on the first quarter of interval, 0, 1, and 0 in the other 3 quarters. So this is F2.
22:24:820Paolo Guiotto: Now, F3…
22:27:870Paolo Guiotto: is similar. We're still dividing the unitary interval in four equal parts. The function is 0 in the first quarter, then it is 1 in the second, and it is 0 in the last two, okay?
22:42:850Paolo Guiotto: That's F3.
22:45:230Paolo Guiotto: So you imagine there's in the F4,
22:49:520Paolo Guiotto: Similar, that will be something like this.
22:53:460Paolo Guiotto: Zero here, one here, 0 here.
22:57:150Paolo Guiotto: Okay.
22:58:540Paolo Guiotto: And the last one is the…
23:01:210Paolo Guiotto: F… you see, that's… it was wrong with the choice of indexes, because this is a F4 and this is F5.
23:09:960Paolo Guiotto: So probably I should start from…
23:15:540Paolo Guiotto: Okay, it doesn't matter, the index is,
23:22:910Paolo Guiotto: So this one is like this.
23:25:550Paolo Guiotto: Look, the next group will be a group made of 8 functions. We divide the interval 0, 1 in 8 equal parts, and we have 8 functions made in this way. Each function is 1 on one of these intervals, and 0 in all others, okay?
23:44:250Paolo Guiotto: So, you basically understand that they are something like an indicator of an interval.
23:54:450Paolo Guiotto: That interval, you see, I can do the division in equal parts.
24:00:910Paolo Guiotto: By using, something like powers of 2, no?
24:04:500Paolo Guiotto: So, here we have intervals of length 1 half.
24:07:790Paolo Guiotto: These are the intervals of length 1 fourth, next step will be intervals of length 1 over 8, and so on. So 1 over 2 to the n will be the length. So I have… these points are over 2 to the N, 2 to the n, and this is something like, I will have indexes K, K plus 1, where k range from 0 to 2 to the N,
24:31:510Paolo Guiotto: minus 1.
24:32:950Paolo Guiotto: Now, this is… as you see, this depends on two indexes, no? So, you start with n equal to 1, and you have two functions. Then you have an n equal 2, you have four functions, so on, no? However, it doesn't matter, this sequence is this one.
24:48:120Paolo Guiotto: What you can see here is that…
24:51:620Paolo Guiotto: The following happens. Here, we have…
24:56:190Paolo Guiotto: Number one, this sequence, FN, goes to 0 in L1.
25:02:440Paolo Guiotto: But, number two, and this is really disturbing, F-anovexa
25:10:90Paolo Guiotto: if I fix an X and I look at this sequence, is never convergent, so there is no X for which this is convergent.
25:19:250Paolo Guiotto: You see? So it is, exactly the opposite of this thing.
25:25:10Paolo Guiotto: It's not true that for some X converges, maybe, but for some other, no. No! Here, it is never convergent, so is never…
25:35:890Paolo Guiotto: convergent.
25:38:590Paolo Guiotto: for every X in 01.
25:43:270Paolo Guiotto: So, as you can see, here we have an example of a sequence which is converging in L1 norm. Sorry, let's write the D norm, yeah?
25:53:230Paolo Guiotto: But it is not convergent point-wise at any point.
25:58:270Paolo Guiotto: And why this? Because if you compute the one norm of an F,
26:04:00Paolo Guiotto: well, instead of using indexes 1, 2, 3, let's use indexes K and N, no? So that we can refer to this… let's call this F, K, N, this one, okay? If you want, we can write a unique index with a formula, but I don't want to,
26:23:250Paolo Guiotto: to insist on this, because it's completely useless. You see that for this, the integral, this would be the integral of modulus of the function FKN.
26:38:760Paolo Guiotto: on 01.
26:41:140Paolo Guiotto: But if you look at FKN, it's an indicator, no? It's a function which is 1 on that interval, and therefore, the integral of the absolute value, they are positive, so it is the integral of the function.
26:54:570Paolo Guiotto: FKN.
26:57:120Paolo Guiotto: This function is equals 0 everywhere, except that little interval, so this will become the integral of 1 on the interval from k over 2 to the n, k plus 1 over 2 to the n.
27:14:110Paolo Guiotto: So, it is the length of that interval. It is 1 over 2 to the n.
27:20:270Paolo Guiotto: So when you send N to infinity, this quantity goes to zero.
27:25:200Paolo Guiotto: Now, as it is reasonably, if you… if you look at the areas, you see the area here is this.
27:32:450Paolo Guiotto: this, the area of this green region is the value of the norm, no? This is 1 half, one half, this is 1 fourth, one fourth.
27:45:190Paolo Guiotto: 1 fourth, and 1 fourth. The next sequence will be 1 over 8, 1 over 8, 8 times, then 1 over 16, 16 times. In any case, they are going to zero.
27:56:710Paolo Guiotto: But!
28:00:660Paolo Guiotto: So this was the fact one.
28:02:660Paolo Guiotto: 2. What happens to the sequence FN if you pick an X?
28:09:00Paolo Guiotto: Look, let's look to the block of the four functions here.
28:15:250Paolo Guiotto: Take an X wherever you like. For example, here.
28:19:840Paolo Guiotto: The first sequence, the first function of this block has value 0.
28:25:690Paolo Guiotto: The second function of this block has value 0. Then the third function has value 1, and the last function has value 0.
28:33:960Paolo Guiotto: So, what do you see?
28:35:680Paolo Guiotto: is that in this block of 4 functions, 3 are 0 and 1 is 1. Exactly that one where your point lies in the interval, in the corresponding interval.
28:46:470Paolo Guiotto: So, when you have the next block of 8 functions, what will happen?
28:50:900Paolo Guiotto: That you will have that for 7 of this function, the value will be 0, and for 1 will be 1.
28:57:500Paolo Guiotto: So, in other words, the sequence Fn , it's made of 0 and 1, because these functions take on these two values.
29:06:170Paolo Guiotto: And you have both 0 and 1 in the sequence repeating infinitely many times.
29:11:240Paolo Guiotto: You see the point? So, Fn of X is a sequence, Oh.
29:20:780Paolo Guiotto: Zero.
29:22:480Paolo Guiotto: And… 1… that.
29:26:960Paolo Guiotto: R… Repeated.
29:33:370Paolo Guiotto: infinitely many.
29:37:150Paolo Guiotto: times.
29:39:800Paolo Guiotto: So such a sequence cannot have a limit, no?
29:43:740Paolo Guiotto: Because what should be derived? It should be 0, no, because there are… there are… the value 1 is taken infinitely many times, so you cannot say that for n large, you are close to 0, because for n large, you are also close to 1.
29:57:580Paolo Guiotto: And so this means that the limit, never exists, whatever is the point X, okay?
30:05:40Paolo Guiotto: So this is a sort of big problem, because this convergence does not imply, the LP convergence does not imply the pointwise convergence. However, it can be proved that it is a bit tricky. I just mentioned here the statement.
30:23:140Paolo Guiotto: that even if the full sequence, FN is not convergent point-wise.
30:30:360Paolo Guiotto: We can always extract a subsequence that we converge point-wise. So, we can say that
30:38:10Paolo Guiotto: If we have the sequence FN converging to some F in P norm, then it is not true that the sequence FN itself converges point-wise, we just proved it with this example, but there exists a subsequence, FNK,
30:56:180Paolo Guiotto: in the FN.
30:58:480Paolo Guiotto: Such that this one, the F and K, converges point-wise to F.
31:08:290Paolo Guiotto: If you want, we can see the example of this sequence in the previous example.
31:14:290Paolo Guiotto: Because look at this example. What is the sub-sequence, for example, that converts to Z?
31:21:700Paolo Guiotto: But even today, just be swamped.
31:26:80Paolo Guiotto: He's got the beast. That big piece.
31:29:520Paolo Guiotto: Then it's got the minus 3. They are the group of 8, let me take the first one, okay? So, in this case, the subsequence is the following.
31:41:170Paolo Guiotto: Example… In D.
31:47:990Paolo Guiotto: Case. Off.
31:50:620Paolo Guiotto: previous.
31:52:610Paolo Guiotto: Example… The subsequence… F and K is…
32:03:180Paolo Guiotto: The sequence made by… well, here we can write is the sequence made by the indicators of 0, 1 over 2 to the n.
32:14:760Paolo Guiotto: So, it is the sequence made of, this is one, between 0 and 1 half, and 0, between 1 half and one. Then you take… discard the second, and take the third. So, it is this and zero here.
32:31:120Paolo Guiotto: Then you discard the other 3, and you take the next one, which is 1 in half of that interval, and 0 in the remaining part.
32:40:760Paolo Guiotto: You see? In this case, you have that this sequence, FNK, actually goes to 0 pointwise, but here you have an even stronger fact. This also actually for every X. FNKX goes to zero.
32:58:640Paolo Guiotto: for every X in 01, except for X equals 0, where all functions are equal 1, f and k of 0 are constant equal to 1, it goes to 1. But in any case, here you have pointwise convergence at every point x.
33:17:920Paolo Guiotto: Okay? Now, why this weaker result is important?
33:24:150Paolo Guiotto: is crucial to discuss convergence for, for NLP norm. Because, again, it says that if nothing defined, you can compute the point by Z.
33:38:00Paolo Guiotto: Now, if the sequence is pointwise converted to actor, that must be the really fast converted.
33:47:470Paolo Guiotto: You see the volume?
33:49:220Paolo Guiotto: So this is exciting. We cannot do that alpha will converge pointwise to alpha.
33:55:130Paolo Guiotto: But, at least there's some sequence master version of that.
34:00:320Paolo Guiotto: If you know, which is more, that the full sequence converted 5 times better to find something, that something must be the F we're looking for.
34:10:730Paolo Guiotto: So, again, you can use the point where it's not going to identify the liquid.
34:15:650Paolo Guiotto: So if I go back to the example, the exercise we have done this morning, where we, we say, we do not have connection between pointwise convergence and LP convergence. So how do we identify that limit? So we say, well, since it converges point-wise from zero, let's try with zero.
34:35:00Paolo Guiotto: But that would be the unique tribe, because now we know that since it converges pointwise to zero.
34:41:570Paolo Guiotto: That can be the unique possibility, because if T converges in IP to F, something should converge pointwise, a subsequence to F, but we know that the full sequence goes to 0, F equals 0. You see the logic.
34:56:640Paolo Guiotto: Okay, last, just, one second to finish.
35:01:670Paolo Guiotto: in theorem, the LP space is a Banach space. So, LP… X is A.
35:11:690Paolo Guiotto: Bonac.
35:13:140Paolo Guiotto: space, for every P.
35:16:610Paolo Guiotto: between 1 and plus infinity. So, also, these spaces are good spaces.
35:22:460Paolo Guiotto: Okay, so as you can see at the end, all important function spaces, and not only these ones, you will see in differential equations, soluble spaces, and so on, all these spaces are Barnab spaces, so that's why it's so important to know this, because we normally use these spaces to solve
35:41:400Paolo Guiotto: Applied problems.
35:43:100Paolo Guiotto: Okay, I'm 5 minutes over, however, thank you, and have a nice day.
35:49:990Paolo Guiotto: Okay, let's stop doing court.