WEBVTT 00:00:00.000 --> 00:00:02.898 align:middle line:90% [SQUEAKING] 00:00:02.898 --> 00:00:04.347 align:middle line:90% [RUSTLING] 00:00:04.347 --> 00:00:05.796 align:middle line:90% [CLICKING] 00:00:05.796 --> 00:00:11.420 align:middle line:90% 00:00:11.420 --> 00:00:16.850 align:middle line:84% PROFESSOR: So we're going to prove 00:00:16.850 --> 00:00:19.620 align:middle line:90% the Picard-Lindelof theorem. 00:00:19.620 --> 00:00:22.530 align:middle line:90% And we'll see. 00:00:22.530 --> 00:00:25.920 align:middle line:90% It may take us-- 00:00:25.920 --> 00:00:30.360 align:middle line:84% we may need to continue on it also on Tuesday. 00:00:30.360 --> 00:00:32.119 align:middle line:84% And then when we're done with it, 00:00:32.119 --> 00:00:35.010 align:middle line:84% we will just do review on Tuesday. 00:00:35.010 --> 00:00:36.710 align:middle line:90% But let's see how far we get. 00:00:36.710 --> 00:00:49.970 align:middle line:84% So the Picard-Lindelof theorem is the following theorem. 00:00:49.970 --> 00:00:56.060 align:middle line:84% And it says that it's an ODE, so it's 00:00:56.060 --> 00:00:58.405 align:middle line:84% what's called an ordinary differential equation. 00:00:58.405 --> 00:01:09.300 align:middle line:90% 00:01:09.300 --> 00:01:11.670 align:middle line:84% And so again, a differential equation 00:01:11.670 --> 00:01:16.660 align:middle line:84% is something about an unknown function and its derivative. 00:01:16.660 --> 00:01:20.130 align:middle line:84% So a differential equation just means it's an equation. 00:01:20.130 --> 00:01:37.380 align:middle line:84% So this here means that it's an equation in an unknown function 00:01:37.380 --> 00:01:38.980 align:middle line:90% and its derivative. 00:01:38.980 --> 00:01:47.190 align:middle line:90% 00:01:47.190 --> 00:01:49.990 align:middle line:84% Ordinary, just as we talked about last time, 00:01:49.990 --> 00:01:54.550 align:middle line:84% ordinary refers to that is just of one variable. 00:01:54.550 --> 00:01:59.680 align:middle line:84% And so the ODE that the Picard-Lindelof theorem is about 00:01:59.680 --> 00:02:00.530 align:middle line:90% is-- 00:02:00.530 --> 00:02:05.200 align:middle line:84% so the unknown function is y, and the ODE 00:02:05.200 --> 00:02:09.880 align:middle line:84% is this that we want the derivative 00:02:09.880 --> 00:02:13.570 align:middle line:84% of this unknown function to be equal to a function 00:02:13.570 --> 00:02:18.130 align:middle line:84% of the unknown function, plus another function that 00:02:18.130 --> 00:02:19.550 align:middle line:90% just depends on x. 00:02:19.550 --> 00:02:22.420 align:middle line:84% So y here is the unknown function, 00:02:22.420 --> 00:02:26.131 align:middle line:90% and y is a function of x. 00:02:26.131 --> 00:02:30.790 align:middle line:90% x is a real number. 00:02:30.790 --> 00:02:33.130 align:middle line:90% And it's just a-- yeah. 00:02:33.130 --> 00:02:35.720 align:middle line:84% So this is the differential equation, 00:02:35.720 --> 00:02:39.160 align:middle line:84% and then we have some initial condition. 00:02:39.160 --> 00:02:45.110 align:middle line:84% So at zero, we want it to take a value "a." 00:02:45.110 --> 00:02:47.740 align:middle line:84% So this is the differential equation. 00:02:47.740 --> 00:02:56.720 align:middle line:84% And what we assume here is-- so f and g are function on R. 00:02:56.720 --> 00:03:03.810 align:middle line:84% And you should think about those as being known, and they are-- 00:03:03.810 --> 00:03:07.110 align:middle line:90% and f here is differentiable. 00:03:07.110 --> 00:03:11.960 align:middle line:90% 00:03:11.960 --> 00:03:21.050 align:middle line:84% And the derivative is continuous, 00:03:21.050 --> 00:03:24.290 align:middle line:90% and g is just continuous. 00:03:24.290 --> 00:03:26.040 align:middle line:84% Doesn't need to be differentiable. 00:03:26.040 --> 00:03:32.450 align:middle line:90% 00:03:32.450 --> 00:03:35.870 align:middle line:84% And the Picard-Lindelof theorem is-- 00:03:35.870 --> 00:03:49.040 align:middle line:84% so again, the Picard-Lindelof theorem is the following, 00:03:49.040 --> 00:04:00.200 align:middle line:90% that given f of g as above and-- 00:04:00.200 --> 00:04:00.700 align:middle line:90% sorry. 00:04:00.700 --> 00:04:03.670 align:middle line:90% And a, a was this constant. 00:04:03.670 --> 00:04:05.930 align:middle line:84% It's just a real number, as above. 00:04:05.930 --> 00:04:09.690 align:middle line:90% 00:04:09.690 --> 00:04:22.980 align:middle line:84% There exists a delta greater than zero, such that-- 00:04:22.980 --> 00:04:24.513 align:middle line:90% and a solution. 00:04:24.513 --> 00:04:30.960 align:middle line:90% 00:04:30.960 --> 00:04:41.940 align:middle line:84% y defined on the integral from minus delta to delta 00:04:41.940 --> 00:04:49.770 align:middle line:90% and a solution y to the ODE. 00:04:49.770 --> 00:04:54.650 align:middle line:84% And so again, meaning that the solution-- 00:04:54.650 --> 00:04:59.320 align:middle line:84% that means that at zero, it takes value a, 00:04:59.320 --> 00:05:01.620 align:middle line:84% and then is satisfying that equation. 00:05:01.620 --> 00:05:04.250 align:middle line:90% 00:05:04.250 --> 00:05:09.760 align:middle line:84% And moreover, and I'll get back to that, but we won't. 00:05:09.760 --> 00:05:12.710 align:middle line:84% It's quite possible I won't get to that today. 00:05:12.710 --> 00:05:20.790 align:middle line:90% Moreover, solutions are unique. 00:05:20.790 --> 00:05:30.430 align:middle line:90% 00:05:30.430 --> 00:05:35.200 align:middle line:84% And so this is like a standard problem, 00:05:35.200 --> 00:05:38.450 align:middle line:84% in when you're looking at differential equations, 00:05:38.450 --> 00:05:42.400 align:middle line:84% you have a differential equation that may describe something 00:05:42.400 --> 00:05:43.250 align:middle line:90% in nature. 00:05:43.250 --> 00:05:47.980 align:middle line:84% And you wonder, does this equation have a solution? 00:05:47.980 --> 00:05:54.460 align:middle line:84% And if it has a solution, are solutions unique? 00:05:54.460 --> 00:05:58.740 align:middle line:84% So this is what this answer in this case. 00:05:58.740 --> 00:06:01.990 align:middle line:84% And the interesting thing, or part of the interesting thing, 00:06:01.990 --> 00:06:04.620 align:middle line:84% so this is a fundamental question 00:06:04.620 --> 00:06:07.060 align:middle line:84% used all over science and engineering. 00:06:07.060 --> 00:06:08.760 align:middle line:84% And one of the interesting things 00:06:08.760 --> 00:06:14.650 align:middle line:84% is that it uses the proof of this uses metric spaces. 00:06:14.650 --> 00:06:18.480 align:middle line:90% And so the proof has been used-- 00:06:18.480 --> 00:06:20.370 align:middle line:84% various versions of The proof have 00:06:20.370 --> 00:06:24.600 align:middle line:84% been used to prove existence and uniqueness 00:06:24.600 --> 00:06:27.540 align:middle line:84% of various differential equations. 00:06:27.540 --> 00:06:42.750 align:middle line:84% So the proof uses the theory of metric spaces, which 00:06:42.750 --> 00:06:45.150 align:middle line:84% is sort of surprising because it seems to have nothing 00:06:45.150 --> 00:06:46.295 align:middle line:90% to do with metric spaces. 00:06:46.295 --> 00:06:49.260 align:middle line:90% 00:06:49.260 --> 00:06:54.160 align:middle line:84% And so it uses, in particular, the metric space that it's using 00:06:54.160 --> 00:06:59.640 align:middle line:84% is the continuous functions on some interval here. 00:06:59.640 --> 00:07:02.650 align:middle line:90% 00:07:02.650 --> 00:07:05.410 align:middle line:84% And on the space of continuous function, 00:07:05.410 --> 00:07:10.930 align:middle line:84% we have the metric that we talked about, the metric. 00:07:10.930 --> 00:07:13.570 align:middle line:90% So if f and g-- 00:07:13.570 --> 00:07:17.050 align:middle line:90% so f is bad because it's the n. 00:07:17.050 --> 00:07:18.070 align:middle line:90% g also. 00:07:18.070 --> 00:07:21.010 align:middle line:90% Let's call it h1 and h2. 00:07:21.010 --> 00:07:24.010 align:middle line:84% Suppose that these are continuous functions. 00:07:24.010 --> 00:07:29.110 align:middle line:84% h1 and h2 are continuous functions 00:07:29.110 --> 00:07:34.270 align:middle line:90% on an interval like this. 00:07:34.270 --> 00:07:36.670 align:middle line:84% Then the distance is this that we 00:07:36.670 --> 00:07:39.020 align:middle line:90% have considered various times. 00:07:39.020 --> 00:07:46.840 align:middle line:84% This is just the maximum over x in this interval here 00:07:46.840 --> 00:07:49.940 align:middle line:90% of f of x, of h1-- 00:07:49.940 --> 00:07:50.570 align:middle line:90% sorry. 00:07:50.570 --> 00:07:58.430 align:middle line:84% h1 of x minus h2 of x, and then the absolute value of that. 00:07:58.430 --> 00:08:02.660 align:middle line:84% So this is the distance on the space of continuous functions 00:08:02.660 --> 00:08:04.235 align:middle line:90% that we have looked at a lot. 00:08:04.235 --> 00:08:06.800 align:middle line:90% 00:08:06.800 --> 00:08:12.450 align:middle line:84% And it's actually not quite this space, as I mentioned last time. 00:08:12.450 --> 00:08:14.995 align:middle line:84% It's a ball in this space that it's using. 00:08:14.995 --> 00:08:19.910 align:middle line:90% 00:08:19.910 --> 00:08:24.390 align:middle line:84% So what do we need about this metric space? 00:08:24.390 --> 00:08:28.610 align:middle line:84% So one thing we've proven a couple of lectures ago 00:08:28.610 --> 00:08:39.566 align:middle line:84% is that the space here of continuous functions 00:08:39.566 --> 00:08:43.660 align:middle line:84% with this metric here is Cauchy complete. 00:08:43.660 --> 00:08:51.870 align:middle line:90% 00:08:51.870 --> 00:08:53.285 align:middle line:90% So we're going to use that. 00:08:53.285 --> 00:08:56.146 align:middle line:90% 00:08:56.146 --> 00:08:57.990 align:middle line:84% Another thing we're going to use is 00:08:57.990 --> 00:09:00.460 align:middle line:90% this notion of contracting map. 00:09:00.460 --> 00:09:13.340 align:middle line:84% So another tool will be that of a contracting map. 00:09:13.340 --> 00:09:20.190 align:middle line:90% 00:09:20.190 --> 00:09:26.590 align:middle line:84% And remember, this is a number of lectures ago. 00:09:26.590 --> 00:09:28.710 align:middle line:84% We talked about a contracting map, 00:09:28.710 --> 00:09:31.300 align:middle line:84% but we need a more general notion of contracting map, 00:09:31.300 --> 00:09:35.160 align:middle line:84% because when we talked about it, we talked about a map from R 00:09:35.160 --> 00:09:40.800 align:middle line:84% into R. And then we used this notion of contracting map 00:09:40.800 --> 00:09:47.300 align:middle line:84% to finding zeros using Newton's method. 00:09:47.300 --> 00:09:51.170 align:middle line:84% But here we're going to use it for something different, 00:09:51.170 --> 00:09:53.480 align:middle line:84% and so we need a more general notion. 00:09:53.480 --> 00:10:02.860 align:middle line:84% So suppose so a contracting map uses the general version as 00:10:02.860 --> 00:10:06.190 align:middle line:84% opposed to the version that we talked 00:10:06.190 --> 00:10:13.170 align:middle line:84% about already that was for the line for R. So contracting map. 00:10:13.170 --> 00:10:13.670 align:middle line:90% Sorry. 00:10:13.670 --> 00:10:16.900 align:middle line:90% 00:10:16.900 --> 00:10:21.355 align:middle line:84% And so you have a metric space x, d. 00:10:21.355 --> 00:10:24.160 align:middle line:84% It could be like this, but it also just 00:10:24.160 --> 00:10:27.040 align:middle line:90% could be any metric space. 00:10:27.040 --> 00:10:33.880 align:middle line:84% And then you have a map from this metric space into the self. 00:10:33.880 --> 00:10:36.220 align:middle line:90% And T here is said to be-- 00:10:36.220 --> 00:10:38.906 align:middle line:90% T is said to be contracting. 00:10:38.906 --> 00:10:47.420 align:middle line:90% 00:10:47.420 --> 00:10:57.560 align:middle line:84% If there exists a constant c less than 1, 00:10:57.560 --> 00:11:04.850 align:middle line:84% such that if you're looking at the distance between the images, 00:11:04.850 --> 00:11:07.560 align:middle line:90% then they are squeezed together. 00:11:07.560 --> 00:11:11.180 align:middle line:84% So this distance here between the images under the map 00:11:11.180 --> 00:11:15.860 align:middle line:84% is less than or equal to this constant 00:11:15.860 --> 00:11:19.100 align:middle line:84% that is less than 1 times the distance 00:11:19.100 --> 00:11:21.650 align:middle line:90% before you apply the map. 00:11:21.650 --> 00:11:26.960 align:middle line:84% So this means that under the map, things get closer together. 00:11:26.960 --> 00:11:30.110 align:middle line:90% So that's a contracting map. 00:11:30.110 --> 00:11:33.510 align:middle line:84% And so now, we prove the theorem. 00:11:33.510 --> 00:11:38.130 align:middle line:84% When we did the Newton's method, we proved a theorem, 00:11:38.130 --> 00:11:41.400 align:middle line:84% but we needed about fixed points. 00:11:41.400 --> 00:11:44.560 align:middle line:84% And we need a generalized version of that. 00:11:44.560 --> 00:11:51.360 align:middle line:84% So if you take a map just in general from a space 00:11:51.360 --> 00:12:00.611 align:middle line:84% to a space, then a fixed point, a fixed point 00:12:00.611 --> 00:12:09.600 align:middle line:90% is a point x in the space. 00:12:09.600 --> 00:12:12.840 align:middle line:90% So that T of x. 00:12:12.840 --> 00:12:15.460 align:middle line:90% So the T leaves x invariant. 00:12:15.460 --> 00:12:16.960 align:middle line:90% So T of x is equal to x. 00:12:16.960 --> 00:12:17.880 align:middle line:90% That's a fixed point. 00:12:17.880 --> 00:12:21.930 align:middle line:90% 00:12:21.930 --> 00:12:24.610 align:middle line:84% And so now, if you have a contracting map, 00:12:24.610 --> 00:12:36.750 align:middle line:84% so if T here is contracting, then T 00:12:36.750 --> 00:12:38.915 align:middle line:84% can have at most one fixed point. 00:12:38.915 --> 00:12:53.650 align:middle line:90% 00:12:53.650 --> 00:12:54.730 align:middle line:90% Why? 00:12:54.730 --> 00:13:01.150 align:middle line:84% Because suppose that x-- so if x and y are both fixed 00:13:01.150 --> 00:13:15.620 align:middle line:84% points, well, then you have that T of x is equal to T of y. 00:13:15.620 --> 00:13:16.120 align:middle line:90% Sorry. 00:13:16.120 --> 00:13:23.395 align:middle line:84% T of x is equal to x, and T of y is equal to y. 00:13:23.395 --> 00:13:28.140 align:middle line:84% And so now you have that the distance between T of x. 00:13:28.140 --> 00:13:32.080 align:middle line:90% 00:13:32.080 --> 00:13:35.110 align:middle line:90% Let me make a little more space. 00:13:35.110 --> 00:13:37.270 align:middle line:90% Then you have the-- 00:13:37.270 --> 00:13:39.230 align:middle line:84% actually, we'll just continue down here. 00:13:39.230 --> 00:13:47.120 align:middle line:84% So that the distance between T of x, comma, T of y-- 00:13:47.120 --> 00:13:50.030 align:middle line:84% the map is contracting, so this is 00:13:50.030 --> 00:13:55.230 align:middle line:84% less than or equal to this constant times this year. 00:13:55.230 --> 00:13:57.540 align:middle line:84% On the other hand, they're both fixed points. 00:13:57.540 --> 00:13:59.010 align:middle line:90% So T of x is x. 00:13:59.010 --> 00:14:01.230 align:middle line:90% This has x, and this has y. 00:14:01.230 --> 00:14:05.700 align:middle line:84% So if we write this out, so this is x comma y. 00:14:05.700 --> 00:14:07.730 align:middle line:84% And so you see that this thing here 00:14:07.730 --> 00:14:11.240 align:middle line:84% is some constant less than 1, strictly less than 1, 00:14:11.240 --> 00:14:12.750 align:middle line:90% times itself. 00:14:12.750 --> 00:14:16.220 align:middle line:84% So this means, of course, that this number here, 00:14:16.220 --> 00:14:19.100 align:middle line:84% this non-negative number must be zero. 00:14:19.100 --> 00:14:25.970 align:middle line:84% And so this, of course, implies by the actions of a metric space 00:14:25.970 --> 00:14:28.760 align:middle line:90% that x is equal to y. 00:14:28.760 --> 00:14:35.120 align:middle line:84% So a contracting map has at most one fixed point. 00:14:35.120 --> 00:14:42.390 align:middle line:84% Now, the other thing is that if you take a metric space-- 00:14:42.390 --> 00:14:51.370 align:middle line:84% if you take a metric space, so this here is Cauchy complete. 00:14:51.370 --> 00:14:57.190 align:middle line:84% So now I'm going to take a Cauchy complete metric space. 00:14:57.190 --> 00:15:00.210 align:middle line:84% So it could be continuous function on some closed 00:15:00.210 --> 00:15:02.920 align:middle line:84% interval, and that's how we will apply it, 00:15:02.920 --> 00:15:06.090 align:middle line:90% but it works more generally. 00:15:06.090 --> 00:15:09.360 align:middle line:90% So we have this fine theorem. 00:15:09.360 --> 00:15:29.670 align:middle line:84% So if this here a Cauchy complete metric space, 00:15:29.670 --> 00:15:49.960 align:middle line:84% and T from x into x is a contracting map, 00:15:49.960 --> 00:15:56.800 align:middle line:90% then T has a unique fixed point. 00:15:56.800 --> 00:16:01.720 align:middle line:84% We know it only has at most one, so this statement really 00:16:01.720 --> 00:16:05.390 align:middle line:90% is that it has one right. 00:16:05.390 --> 00:16:07.520 align:middle line:90% So it has a unique fixed point. 00:16:07.520 --> 00:16:13.064 align:middle line:90% 00:16:13.064 --> 00:16:16.695 align:middle line:90% So let's try to prove this. 00:16:16.695 --> 00:16:19.600 align:middle line:90% 00:16:19.600 --> 00:16:22.420 align:middle line:90% And you might remember. 00:16:22.420 --> 00:16:24.590 align:middle line:90% And if you don't, that's fine. 00:16:24.590 --> 00:16:28.700 align:middle line:84% But you might remember how we used it in the Newton method 00:16:28.700 --> 00:16:30.880 align:middle line:84% because the Newton method was like-- 00:16:30.880 --> 00:16:37.130 align:middle line:84% it was a version of this on R. So you had a map from R 00:16:37.130 --> 00:16:39.900 align:middle line:84% to R, and to find the fixed point. 00:16:39.900 --> 00:16:41.870 align:middle line:84% And so you wondered what you were there, 00:16:41.870 --> 00:16:43.610 align:middle line:84% and your Newton method was something 00:16:43.610 --> 00:16:47.160 align:middle line:84% about finding a point where a function vanishes. 00:16:47.160 --> 00:16:50.960 align:middle line:84% And that was set up as a fixed point theorem. 00:16:50.960 --> 00:16:54.530 align:middle line:84% And so what you did is you made a good choice, 00:16:54.530 --> 00:16:57.650 align:middle line:84% I mean, a choice that looked reasonable. 00:16:57.650 --> 00:17:01.050 align:middle line:84% This is not exactly where the function vanishes, 00:17:01.050 --> 00:17:03.870 align:middle line:84% but something that looks close to where it vanishes. 00:17:03.870 --> 00:17:06.210 align:middle line:90% And then you iterate. 00:17:06.210 --> 00:17:10.170 align:middle line:84% You took the image under the map and you iterated this. 00:17:10.170 --> 00:17:13.399 align:middle line:84% And so it's the same we're going to do here. 00:17:13.399 --> 00:17:18.349 align:middle line:84% And this time, we don't even need to make a good choice. 00:17:18.349 --> 00:17:20.720 align:middle line:90% So we just take a-- 00:17:20.720 --> 00:17:27.720 align:middle line:84% we just take-- let x here be any point in the metric space. 00:17:27.720 --> 00:17:31.280 align:middle line:84% So you just fixing some point in the metric space. 00:17:31.280 --> 00:17:33.360 align:middle line:90% And now you're looking at-- 00:17:33.360 --> 00:17:36.970 align:middle line:84% so you're looking at the sequence x, T of x, 00:17:36.970 --> 00:17:41.120 align:middle line:90% and then you apply T to T of x. 00:17:41.120 --> 00:17:43.250 align:middle line:90% So I write this as this. 00:17:43.250 --> 00:17:45.790 align:middle line:84% And then you apply T to T squared 00:17:45.790 --> 00:17:48.790 align:middle line:90% of x, so that's like this. 00:17:48.790 --> 00:17:53.890 align:middle line:84% And so you keep applying T more and more times, 00:17:53.890 --> 00:17:57.710 align:middle line:84% and so this gives you a sequence, and so on. 00:17:57.710 --> 00:18:00.160 align:middle line:90% So this gives you a sequence. 00:18:00.160 --> 00:18:03.500 align:middle line:84% And now the claim is that this sequence here-- 00:18:03.500 --> 00:18:15.440 align:middle line:84% so I claim that this sequence here converge to-- 00:18:15.440 --> 00:18:19.000 align:middle line:84% so it converge, and the limit here, 00:18:19.000 --> 00:18:30.100 align:middle line:84% this here, is a fixed point for T. So that's the claim. 00:18:30.100 --> 00:18:34.280 align:middle line:90% 00:18:34.280 --> 00:18:41.196 align:middle line:84% Now, so we wanted to prove first that is converging. 00:18:41.196 --> 00:18:50.380 align:middle line:90% 00:18:50.380 --> 00:18:52.770 align:middle line:90% So we're going to prove first-- 00:18:52.770 --> 00:19:07.765 align:middle line:84% so we'll first show that Tn is a Cauchy sequence. 00:19:07.765 --> 00:19:15.230 align:middle line:90% 00:19:15.230 --> 00:19:17.960 align:middle line:90% So now let's look at-- 00:19:17.960 --> 00:19:20.900 align:middle line:84% so we're supposed to look-- if you're looking at a Cauchy-- 00:19:20.900 --> 00:19:24.060 align:middle line:84% if you try to prove that something is a Cauchy sequence, 00:19:24.060 --> 00:19:28.880 align:middle line:84% then you're looking at the distance between elements 00:19:28.880 --> 00:19:33.330 align:middle line:84% in the sequence where both of these elements 00:19:33.330 --> 00:19:36.120 align:middle line:90% is sufficiently far. 00:19:36.120 --> 00:19:39.060 align:middle line:84% That's what you're supposed to look at. 00:19:39.060 --> 00:19:42.130 align:middle line:90% And so let me write the m here. 00:19:42.130 --> 00:19:44.700 align:middle line:84% So I can think about one of them as being 00:19:44.700 --> 00:19:46.390 align:middle line:90% further out than the other one. 00:19:46.390 --> 00:19:48.330 align:middle line:84% So I'm going to think about m here 00:19:48.330 --> 00:19:55.020 align:middle line:84% as being n plus some number k, some integer k. 00:19:55.020 --> 00:19:57.870 align:middle line:84% And so I'm interested in estimating 00:19:57.870 --> 00:20:07.745 align:middle line:84% the distance between Tn of x and Tn plus k of x. 00:20:07.745 --> 00:20:12.420 align:middle line:90% 00:20:12.420 --> 00:20:18.330 align:middle line:84% Now, in order to do this, let me start 00:20:18.330 --> 00:20:23.340 align:middle line:90% with something simpler first. 00:20:23.340 --> 00:20:28.520 align:middle line:84% Let me look at the distance between T of x. 00:20:28.520 --> 00:20:31.180 align:middle line:90% 00:20:31.180 --> 00:20:42.370 align:middle line:84% Let me look at the distance between Tl of x and Tl 00:20:42.370 --> 00:20:46.420 align:middle line:90% plus 1 of x. 00:20:46.420 --> 00:20:50.570 align:middle line:84% So two consecutive elements in the sequence. 00:20:50.570 --> 00:20:56.170 align:middle line:84% Now, I can think that this thing here, well, this 00:20:56.170 --> 00:21:07.780 align:middle line:84% is equal to the distance between T of Tl minus 1 of x. 00:21:07.780 --> 00:21:16.115 align:middle line:84% And then T of Tl of x, like that. 00:21:16.115 --> 00:21:17.670 align:middle line:90% I can write it like that. 00:21:17.670 --> 00:21:21.340 align:middle line:90% 00:21:21.340 --> 00:21:24.130 align:middle line:84% But now, of course, so you see here there's 00:21:24.130 --> 00:21:28.310 align:middle line:84% some elements in x in the space x. 00:21:28.310 --> 00:21:31.130 align:middle line:90% Here's has another element. 00:21:31.130 --> 00:21:33.500 align:middle line:90% They both have T applied to it. 00:21:33.500 --> 00:21:44.620 align:middle line:84% T is contracting, so you gets c, Tl minus 1, x, Tl of x. 00:21:44.620 --> 00:21:47.690 align:middle line:90% 00:21:47.690 --> 00:21:52.430 align:middle line:84% So you see that if you take two consecutive elements 00:21:52.430 --> 00:21:55.010 align:middle line:84% and you're taking the distance, then 00:21:55.010 --> 00:22:02.880 align:middle line:84% it's bounded by this constant c times where you have 1 less-- 00:22:02.880 --> 00:22:06.840 align:middle line:84% where you have applied T 1 less time on both. 00:22:06.840 --> 00:22:09.770 align:middle line:84% These are, again, consecutive elements, 00:22:09.770 --> 00:22:12.185 align:middle line:90% but the highest one is 1 less. 00:22:12.185 --> 00:22:14.810 align:middle line:90% 00:22:14.810 --> 00:22:29.010 align:middle line:84% So now, if you iterate this, so iterating this, so let's see. 00:22:29.010 --> 00:22:32.500 align:middle line:84% Maybe let me wait with iterating. 00:22:32.500 --> 00:22:35.590 align:middle line:84% Let me just look at what we need. 00:22:35.590 --> 00:22:39.090 align:middle line:90% So we have this thing here. 00:22:39.090 --> 00:22:43.770 align:middle line:90% And this here I can write as-- 00:22:43.770 --> 00:22:50.610 align:middle line:84% so here you see, here I can write this as T, the distance T, 00:22:50.610 --> 00:23:03.030 align:middle line:84% And then Tn minus 1x, and then T, Tn minus 1 plus kx, 00:23:03.030 --> 00:23:05.916 align:middle line:90% like this. 00:23:05.916 --> 00:23:12.520 align:middle line:84% And so here I can use the same trick that you guys see, 00:23:12.520 --> 00:23:23.160 align:middle line:84% and then d, Tn minus 1x comma Tn minus 1 plus k of x. 00:23:23.160 --> 00:23:27.400 align:middle line:90% 00:23:27.400 --> 00:23:28.730 align:middle line:90% And this one here. 00:23:28.730 --> 00:23:31.450 align:middle line:90% I can keep going like this. 00:23:31.450 --> 00:23:36.830 align:middle line:84% I can keep going until there is the-- 00:23:36.830 --> 00:23:40.540 align:middle line:90% I've used up all my ends. 00:23:40.540 --> 00:23:42.410 align:middle line:90% And so I can keep going. 00:23:42.410 --> 00:23:45.710 align:middle line:84% And so this is less or equal to c to the n. 00:23:45.710 --> 00:23:54.510 align:middle line:84% And then the last one I get is, I get x, and then I get Tk of x. 00:23:54.510 --> 00:24:01.200 align:middle line:90% 00:24:01.200 --> 00:24:06.070 align:middle line:84% I want to show that this thing here becomes small 00:24:06.070 --> 00:24:11.080 align:middle line:84% as long as n-- remember, k is a positive integer as long as n 00:24:11.080 --> 00:24:12.130 align:middle line:90% is sufficiently large. 00:24:12.130 --> 00:24:14.900 align:middle line:90% 00:24:14.900 --> 00:24:18.490 align:middle line:84% So this is a hopeful sign, because when n gets large, 00:24:18.490 --> 00:24:20.560 align:middle line:84% this here gets really, really small. 00:24:20.560 --> 00:24:23.800 align:middle line:84% But I still need to estimate this here. 00:24:23.800 --> 00:24:25.580 align:middle line:90% Now, so how do I estimate it? 00:24:25.580 --> 00:24:28.280 align:middle line:90% By using what I did over here. 00:24:28.280 --> 00:24:35.780 align:middle line:90% So you're looking at-- 00:24:35.780 --> 00:24:45.800 align:middle line:84% so you have the distance between x and Tk of x. 00:24:45.800 --> 00:24:49.410 align:middle line:84% This thing here, I can use this triangle inequality. 00:24:49.410 --> 00:24:55.640 align:middle line:84% So I can say that this is less than dx T of x 00:24:55.640 --> 00:25:07.940 align:middle line:84% plus dT of x, T squared of x plus dT 00:25:07.940 --> 00:25:13.080 align:middle line:90% squared of x, cubed of x. 00:25:13.080 --> 00:25:25.320 align:middle line:84% And then all the way up to d, Tk minus 1 of x, Tk of x. 00:25:25.320 --> 00:25:27.480 align:middle line:84% So this is just the triangle inequality. 00:25:27.480 --> 00:25:30.360 align:middle line:90% 00:25:30.360 --> 00:25:33.150 align:middle line:84% Now, this may seem rather bad estimate, 00:25:33.150 --> 00:25:35.890 align:middle line:90% but let's see how bad it is. 00:25:35.890 --> 00:25:41.280 align:middle line:84% So now, this first thing here, I'm just leaving it. 00:25:41.280 --> 00:25:42.970 align:middle line:90% I can't do anything about that. 00:25:42.970 --> 00:25:46.650 align:middle line:90% So I need to just take that. 00:25:46.650 --> 00:25:52.060 align:middle line:84% This one here, so I'm using this with between two consecutive. 00:25:52.060 --> 00:25:59.064 align:middle line:84% So in this case, it's just bounded by c times dx, Tx. 00:25:59.064 --> 00:26:02.130 align:middle line:90% 00:26:02.130 --> 00:26:05.550 align:middle line:84% And the next one here, this one here 00:26:05.550 --> 00:26:12.090 align:middle line:90% is bounded by c squared, dx, Tx. 00:26:12.090 --> 00:26:18.960 align:middle line:84% And the next, and so on, up to this one here 00:26:18.960 --> 00:26:28.560 align:middle line:84% is bounded by a c, k minus 1, dx, Tx. 00:26:28.560 --> 00:26:31.940 align:middle line:90% 00:26:31.940 --> 00:26:36.290 align:middle line:84% And so you see that now, this here factor out. 00:26:36.290 --> 00:26:38.865 align:middle line:84% So this here is this factored out. 00:26:38.865 --> 00:26:41.380 align:middle line:90% 00:26:41.380 --> 00:26:44.320 align:middle line:84% And then, in this bracket, you get 00:26:44.320 --> 00:26:51.830 align:middle line:84% 1 plus c plus c squared, plus all the way up to ck minus 1. 00:26:51.830 --> 00:26:54.980 align:middle line:90% But this is a geometric series. 00:26:54.980 --> 00:26:57.160 align:middle line:90% This is a geometric series. 00:26:57.160 --> 00:27:01.670 align:middle line:84% And this is certainly less than when I take-- 00:27:01.670 --> 00:27:04.040 align:middle line:84% remember c is strictly less than 1. 00:27:04.040 --> 00:27:05.600 align:middle line:90% So it's a geometric series. 00:27:05.600 --> 00:27:07.760 align:middle line:84% It's a convergent geometric series. 00:27:07.760 --> 00:27:12.610 align:middle line:84% So this here is less than dx, T of x. 00:27:12.610 --> 00:27:16.120 align:middle line:84% And then this sum, well, it's less 00:27:16.120 --> 00:27:19.100 align:middle line:84% than when you're summing it all the way to infinity. 00:27:19.100 --> 00:27:21.620 align:middle line:90% So this is less than 1. 00:27:21.620 --> 00:27:24.150 align:middle line:84% This is the sum of the geometric series. 00:27:24.150 --> 00:27:27.840 align:middle line:90% C is strictly less than 1. 00:27:27.840 --> 00:27:30.085 align:middle line:84% C is positive, and it's strictly less than 1. 00:27:30.085 --> 00:27:33.596 align:middle line:90% 00:27:33.596 --> 00:27:37.880 align:middle line:90% So/ you see now that we have-- 00:27:37.880 --> 00:27:40.440 align:middle line:90% so we have that. 00:27:40.440 --> 00:27:42.350 align:middle line:84% So what we were interested in is, 00:27:42.350 --> 00:27:44.810 align:middle line:84% we were interested in saying that, as long 00:27:44.810 --> 00:27:47.340 align:middle line:84% as I choose little n sufficiently large, 00:27:47.340 --> 00:27:51.590 align:middle line:84% then this thing here must be very small. 00:27:51.590 --> 00:27:54.540 align:middle line:84% And we know already that it's bounded like this. 00:27:54.540 --> 00:27:57.990 align:middle line:84% And now I fill in this, and you see this here. 00:27:57.990 --> 00:28:01.110 align:middle line:84% So this means that this is less than cn. 00:28:01.110 --> 00:28:08.120 align:middle line:84% And then what you get here is 1 minus c, and then 00:28:08.120 --> 00:28:13.680 align:middle line:90% times dx T of x. 00:28:13.680 --> 00:28:18.540 align:middle line:84% x is fixed, and so you see that when n, little n, 00:28:18.540 --> 00:28:21.690 align:middle line:84% is sufficiently large, then this whole thing here 00:28:21.690 --> 00:28:24.060 align:middle line:90% is as small as you want. 00:28:24.060 --> 00:28:27.000 align:middle line:84% So this means that what we have proven here 00:28:27.000 --> 00:28:32.465 align:middle line:84% is that this sequence here is a Cauchy sequence. 00:28:32.465 --> 00:28:39.570 align:middle line:90% 00:28:39.570 --> 00:28:40.565 align:middle line:90% So we have proven that. 00:28:40.565 --> 00:28:43.695 align:middle line:90% 00:28:43.695 --> 00:28:46.440 align:middle line:84% Now, of course, this is where the assumption comes in 00:28:46.440 --> 00:28:51.490 align:middle line:84% that this metric space is Cauchy complete. 00:28:51.490 --> 00:29:10.560 align:middle line:84% So since x, this metric space is Cauchy complete, 00:29:10.560 --> 00:29:25.010 align:middle line:84% and this sequence here is a Cauchy sequence, 00:29:25.010 --> 00:29:27.206 align:middle line:90% it is convergent. 00:29:27.206 --> 00:29:31.884 align:middle line:90% 00:29:31.884 --> 00:29:32.658 align:middle line:90% Yeah. 00:29:32.658 --> 00:29:33.450 align:middle line:90% STUDENT: I'm sorry. 00:29:33.450 --> 00:29:34.950 align:middle line:84% Just back to the end of that repeat. 00:29:34.950 --> 00:29:35.840 align:middle line:90% PROFESSOR: Yeah. 00:29:35.840 --> 00:29:39.290 align:middle line:84% STUDENT: Are you able to explain again how we got that feature, 00:29:39.290 --> 00:29:40.730 align:middle line:90% or how we were trying to-- 00:29:40.730 --> 00:29:46.190 align:middle line:84% did we find a bound for the dx, Tx, or we're just saying that-- 00:29:46.190 --> 00:29:48.750 align:middle line:84% PROFESSOR: So I'm saying that x is fixed. 00:29:48.750 --> 00:29:55.001 align:middle line:84% You pick any point you want, and I claim that if I-- 00:29:55.001 --> 00:29:58.260 align:middle line:84% it may be a very bad candidate for the fixed point, 00:29:58.260 --> 00:29:59.370 align:middle line:90% but it doesn't matter. 00:29:59.370 --> 00:30:00.500 align:middle line:90% It's OK. 00:30:00.500 --> 00:30:03.270 align:middle line:84% It may just take you longer to get to the fixed point. 00:30:03.270 --> 00:30:08.450 align:middle line:84% But you fix x, and then you're looking at the sequence. 00:30:08.450 --> 00:30:11.180 align:middle line:84% And then I claim that this sequence 00:30:11.180 --> 00:30:14.185 align:middle line:90% will be a Cauchy sequence. 00:30:14.185 --> 00:30:20.020 align:middle line:84% And so x is fixed, and so we have this inequality here. 00:30:20.020 --> 00:30:26.010 align:middle line:84% And what we need is that when little n is sufficiently far 00:30:26.010 --> 00:30:31.120 align:middle line:84% large, then for all k, k is a positive integer, 00:30:31.120 --> 00:30:33.960 align:middle line:90% This is further out than in. 00:30:33.960 --> 00:30:35.500 align:middle line:90% Then for all k-- 00:30:35.500 --> 00:30:37.690 align:middle line:84% and this should be independent of k-- 00:30:37.690 --> 00:30:39.070 align:middle line:90% this thing here small. 00:30:39.070 --> 00:30:40.080 align:middle line:90% But this is just-- 00:30:40.080 --> 00:30:42.163 align:middle line:84% STUDENT: Doesn't that get rid of-- the whole thing 00:30:42.163 --> 00:30:45.420 align:middle line:84% from the previous step was now it's not a Tk of x. 00:30:45.420 --> 00:30:52.230 align:middle line:84% PROFESSOR: Yeah, so since x is fixed, x is fixed. 00:30:52.230 --> 00:30:53.950 align:middle line:90% Tx is fixed. 00:30:53.950 --> 00:30:55.275 align:middle line:90% So this is just a constant. 00:30:55.275 --> 00:31:00.150 align:middle line:90% 00:31:00.150 --> 00:31:01.900 align:middle line:90% So right. 00:31:01.900 --> 00:31:03.780 align:middle line:90% So what we have proven is that-- 00:31:03.780 --> 00:31:08.560 align:middle line:84% so we have so we have this Cauchy complete metric space. 00:31:08.560 --> 00:31:10.740 align:middle line:90% And we have this sequence. 00:31:10.740 --> 00:31:12.210 align:middle line:90% X was fixed again. 00:31:12.210 --> 00:31:13.960 align:middle line:90% That is a Cauchy sequence. 00:31:13.960 --> 00:31:17.330 align:middle line:84% And so since this metric space is Cauchy complete, 00:31:17.330 --> 00:31:23.020 align:middle line:84% this is convergent, and I can denote the limit. 00:31:23.020 --> 00:31:26.590 align:middle line:84% So I just called the limit x infinity. 00:31:26.590 --> 00:31:27.555 align:middle line:90% So we have this. 00:31:27.555 --> 00:31:30.100 align:middle line:90% 00:31:30.100 --> 00:31:32.141 align:middle line:90% Now, I want to show that-- 00:31:32.141 --> 00:31:34.690 align:middle line:90% 00:31:34.690 --> 00:31:55.684 align:middle line:84% so the next claim is that x infinity is a fixed point for T. 00:31:55.684 --> 00:31:58.750 align:middle line:84% So if I'm looking at this sequence here, 00:31:58.750 --> 00:32:00.770 align:middle line:90% suppose I take the sequence. 00:32:00.770 --> 00:32:03.443 align:middle line:84% And suppose I take T of this sequence. 00:32:03.443 --> 00:32:08.290 align:middle line:90% 00:32:08.290 --> 00:32:11.890 align:middle line:84% So if I take T of this sequence, this here 00:32:11.890 --> 00:32:17.000 align:middle line:84% is nothing but the next element in the sequence. 00:32:17.000 --> 00:32:23.660 align:middle line:84% So this one here will still converge to x infinity, 00:32:23.660 --> 00:32:27.290 align:middle line:90% because it's just shifted by 1. 00:32:27.290 --> 00:32:30.845 align:middle line:84% But then on the other hand, so now, if you're looking at-- 00:32:30.845 --> 00:32:33.620 align:middle line:90% 00:32:33.620 --> 00:32:35.990 align:middle line:90% so you have that. 00:32:35.990 --> 00:32:41.023 align:middle line:84% If you're looking at T, so we want to prove that-- 00:32:41.023 --> 00:32:44.060 align:middle line:90% 00:32:44.060 --> 00:32:51.050 align:middle line:84% so now, if we're looking at T of x infinity, 00:32:51.050 --> 00:33:05.400 align:middle line:84% this distance between T of x infinity and Tn plus 1 of x, 00:33:05.400 --> 00:33:08.945 align:middle line:84% suppose I'm looking at this distance here. 00:33:08.945 --> 00:33:13.800 align:middle line:84% Well, this here I can think about as the distance between T 00:33:13.800 --> 00:33:20.400 align:middle line:90% of x infinity and T of Tn of x. 00:33:20.400 --> 00:33:24.010 align:middle line:84% But here there are now both of these elements. 00:33:24.010 --> 00:33:26.460 align:middle line:84% You can think about this as just some elements. 00:33:26.460 --> 00:33:27.870 align:middle line:90% Apply T to it. 00:33:27.870 --> 00:33:29.920 align:middle line:90% Here there's Ts applied to that. 00:33:29.920 --> 00:33:35.490 align:middle line:84% So this is less than the distance x infinity 00:33:35.490 --> 00:33:41.400 align:middle line:84% to Tn as a constant c less than 1 in front. 00:33:41.400 --> 00:33:49.440 align:middle line:84% So like this by the contracting map property. 00:33:49.440 --> 00:33:53.910 align:middle line:84% But now you see that as n goes to infinity, 00:33:53.910 --> 00:33:57.280 align:middle line:90% this thing here goes to zero. 00:33:57.280 --> 00:34:03.300 align:middle line:84% So this means that this thing here converges to this point. 00:34:03.300 --> 00:34:05.613 align:middle line:90% So you have that-- 00:34:05.613 --> 00:34:09.000 align:middle line:90% 00:34:09.000 --> 00:34:10.210 align:middle line:90% so you have that-- 00:34:10.210 --> 00:34:20.920 align:middle line:90% 00:34:20.920 --> 00:34:25.100 align:middle line:84% you have that if you're looking at-- 00:34:25.100 --> 00:34:28.100 align:middle line:90% if you're looking at-- 00:34:28.100 --> 00:34:33.850 align:middle line:90% again, as n go to-- 00:34:33.850 --> 00:34:46.120 align:middle line:84% as n goes to infinity, since Tn of x converges to x infinity, 00:34:46.120 --> 00:34:48.310 align:middle line:84% then we have that the distance-- this 00:34:48.310 --> 00:34:55.780 align:middle line:84% is the equivalent to that the distance here goes to zero. 00:34:55.780 --> 00:35:02.320 align:middle line:84% And then by this inequality, it means that the distance here 00:35:02.320 --> 00:35:12.960 align:middle line:84% Tx infinity and Tn plus 1 of x goes to zero. 00:35:12.960 --> 00:35:16.320 align:middle line:84% And so this means that this is just some fixed point. 00:35:16.320 --> 00:35:18.240 align:middle line:84% It means that this sequence here-- 00:35:18.240 --> 00:35:22.820 align:middle line:84% so this implies that this sequence here 00:35:22.820 --> 00:35:27.980 align:middle line:90% converts to T of x infinity. 00:35:27.980 --> 00:35:30.590 align:middle line:84% But on the other hand, we already 00:35:30.590 --> 00:35:36.450 align:middle line:84% observed that this sequence also converges to x infinity. 00:35:36.450 --> 00:35:40.830 align:middle line:90% 00:35:40.830 --> 00:35:43.950 align:middle line:84% So it converges to this and it converges to that, 00:35:43.950 --> 00:35:47.220 align:middle line:84% but it can only have one limit, so those two things are equal. 00:35:47.220 --> 00:35:52.970 align:middle line:84% So you conclude that x infinity is equal to T of x infinity. 00:35:52.970 --> 00:35:57.100 align:middle line:84% And so x infinity is a fixed point. 00:35:57.100 --> 00:36:01.460 align:middle line:90% 00:36:01.460 --> 00:36:04.040 align:middle line:84% So this proves this theorem right. 00:36:04.040 --> 00:36:09.410 align:middle line:84% And again, we already knew that the limit was unique. 00:36:09.410 --> 00:36:13.270 align:middle line:90% 00:36:13.270 --> 00:36:20.460 align:middle line:84% So now, how is this related to the Picard-Lindelof theorem? 00:36:20.460 --> 00:36:23.290 align:middle line:84% And so it's related in the following way. 00:36:23.290 --> 00:36:25.290 align:middle line:84% And again, this is something that 00:36:25.290 --> 00:36:28.226 align:middle line:84% has been used to solve a lot of differential equations. 00:36:28.226 --> 00:36:36.060 align:middle line:90% 00:36:36.060 --> 00:36:41.580 align:middle line:84% And so we need to define a space in a map. 00:36:41.580 --> 00:36:45.910 align:middle line:84% The spaces initially, but we have to restrict it. 00:36:45.910 --> 00:36:54.780 align:middle line:84% But initially, let's think about some continuous functions 00:36:54.780 --> 00:36:57.480 align:middle line:90% on some interval. 00:36:57.480 --> 00:36:59.920 align:middle line:84% Let's start with just continuous function. 00:36:59.920 --> 00:37:02.130 align:middle line:84% But it's not quite going to be the space. 00:37:02.130 --> 00:37:04.480 align:middle line:84% I will have to look at a smaller-- 00:37:04.480 --> 00:37:05.360 align:middle line:90% Yeah. 00:37:05.360 --> 00:37:06.110 align:middle line:90% STUDENT: Question. 00:37:06.110 --> 00:37:09.040 align:middle line:84% Why is it that you didn't need to know 00:37:09.040 --> 00:37:13.453 align:middle line:84% the need before to a good starting point? 00:37:13.453 --> 00:37:14.870 align:middle line:84% PROFESSOR: That's a good question. 00:37:14.870 --> 00:37:16.220 align:middle line:90% So that's right. 00:37:16.220 --> 00:37:19.400 align:middle line:84% So this comes with this Newton method. 00:37:19.400 --> 00:37:22.450 align:middle line:90% So we used the-- 00:37:22.450 --> 00:37:24.940 align:middle line:84% so we used this contracting mapping 00:37:24.940 --> 00:37:30.632 align:middle line:84% for R in the Newton method to find a zero. 00:37:30.632 --> 00:37:34.060 align:middle line:84% And to some extent, there was a great idea. 00:37:34.060 --> 00:37:37.480 align:middle line:84% Newton was a pretty famous guy for a good reason. 00:37:37.480 --> 00:37:40.640 align:middle line:84% But in a way, this is more ingenious, I think, 00:37:40.640 --> 00:37:42.440 align:middle line:90% because it's like metric space. 00:37:42.440 --> 00:37:45.602 align:middle line:84% What the hell does this have to do with differential equation? 00:37:45.602 --> 00:37:49.550 align:middle line:84% And the reason why in the other one, we had to restrict it 00:37:49.550 --> 00:37:54.110 align:middle line:84% was that the map we defined wasn't contracting everywhere. 00:37:54.110 --> 00:37:58.030 align:middle line:84% It needed things to be pretty close initially 00:37:58.030 --> 00:38:02.720 align:middle line:84% for there to exist a c strictly less than 1. 00:38:02.720 --> 00:38:04.620 align:middle line:90% So that's where it came in. 00:38:04.620 --> 00:38:08.130 align:middle line:90% It wasn't really because if-- 00:38:08.130 --> 00:38:08.630 align:middle line:90% yeah. 00:38:08.630 --> 00:38:13.550 align:middle line:90% 00:38:13.550 --> 00:38:16.610 align:middle line:84% So now, again, so we need to relate this. 00:38:16.610 --> 00:38:19.110 align:middle line:84% That seems to have nothing to do with differential equation. 00:38:19.110 --> 00:38:21.720 align:middle line:84% We need to relate it to the differential equation. 00:38:21.720 --> 00:38:25.280 align:middle line:84% So again, what is the space, the metric space 00:38:25.280 --> 00:38:26.880 align:middle line:90% that we're going to look at? 00:38:26.880 --> 00:38:29.040 align:middle line:84% And again, it's not exactly this, 00:38:29.040 --> 00:38:30.750 align:middle line:90% but it's just a subset of it. 00:38:30.750 --> 00:38:37.130 align:middle line:84% So it's this space, and with the usual metric 00:38:37.130 --> 00:38:38.400 align:middle line:90% that we have talked about. 00:38:38.400 --> 00:38:45.710 align:middle line:84% That is going to be the metric space. 00:38:45.710 --> 00:38:49.670 align:middle line:90% Now, what is the map T? 00:38:49.670 --> 00:38:55.280 align:middle line:84% And so the map T is-- so T here is 00:38:55.280 --> 00:39:03.071 align:middle line:84% going to be a map from continuous functions 00:39:03.071 --> 00:39:05.426 align:middle line:84% into the space of continuous functions. 00:39:05.426 --> 00:39:08.700 align:middle line:90% 00:39:08.700 --> 00:39:15.930 align:middle line:84% And is going to be defined as follows. 00:39:15.930 --> 00:39:18.340 align:middle line:84% And I'll come to this in a second. 00:39:18.340 --> 00:39:21.730 align:middle line:90% So T of x, T of y. 00:39:21.730 --> 00:39:25.140 align:middle line:84% So think about y as being a continuous function 00:39:25.140 --> 00:39:27.180 align:middle line:90% on this interval. 00:39:27.180 --> 00:39:35.490 align:middle line:84% And T of y is now equal to a plus, and then the integral, 00:39:35.490 --> 00:39:40.420 align:middle line:90% and then integral. 00:39:40.420 --> 00:39:46.000 align:middle line:84% So this is a-- so T of y, y is of an element here. 00:39:46.000 --> 00:39:48.940 align:middle line:84% This one here is also supposed to be an element here, 00:39:48.940 --> 00:39:50.730 align:middle line:90% so this is a function. 00:39:50.730 --> 00:39:54.780 align:middle line:84% So we want to find out what the value is at x. 00:39:54.780 --> 00:39:59.820 align:middle line:84% And this a, this constant a plus the integral from 00:39:59.820 --> 00:40:12.970 align:middle line:84% zero to x, and then of f of y, plus g of x. 00:40:12.970 --> 00:40:18.250 align:middle line:84% And this here is a function of x, also, so like this. 00:40:18.250 --> 00:40:23.020 align:middle line:84% And I shouldn't write x because I use now x up here. 00:40:23.020 --> 00:40:26.650 align:middle line:90% So I write it like that. d of x. 00:40:26.650 --> 00:40:30.370 align:middle line:84% That's how the map is going to be defined. 00:40:30.370 --> 00:40:33.020 align:middle line:90% So again, if you take-- 00:40:33.020 --> 00:40:38.800 align:middle line:84% so y here, so y is going-- so y is a continuous function 00:40:38.800 --> 00:40:41.320 align:middle line:90% on this interval. 00:40:41.320 --> 00:40:43.780 align:middle line:84% And I will have to restrict it, and this 00:40:43.780 --> 00:40:47.780 align:middle line:84% is exactly when you're asking what in Newton's method, 00:40:47.780 --> 00:40:51.380 align:middle line:84% why did we have to pick this good choice. 00:40:51.380 --> 00:40:56.410 align:middle line:84% It's exactly like this that this map is not contracting 00:40:56.410 --> 00:40:57.770 align:middle line:90% on this whole space. 00:40:57.770 --> 00:41:02.020 align:middle line:84% So we have to-- in a way, we have to do the same stuff. 00:41:02.020 --> 00:41:04.360 align:middle line:84% We have to make a good choice first. 00:41:04.360 --> 00:41:06.610 align:middle line:84% And so that's why we have to restrict the space. 00:41:06.610 --> 00:41:09.990 align:middle line:90% And it actually is-- 00:41:09.990 --> 00:41:13.210 align:middle line:84% part of the beauty is that it's pretty much optimal, 00:41:13.210 --> 00:41:16.920 align:middle line:84% this stuff, this sort of general principle. 00:41:16.920 --> 00:41:18.600 align:middle line:90% We can come back to that. 00:41:18.600 --> 00:41:22.920 align:middle line:84% So this is the map T. And so the first thing 00:41:22.920 --> 00:41:27.600 align:middle line:90% that you observe is that T-- 00:41:27.600 --> 00:41:32.680 align:middle line:84% so this gives you this function here. 00:41:32.680 --> 00:41:39.400 align:middle line:84% Remember that so if y is continuous, g is continuous. 00:41:39.400 --> 00:41:42.190 align:middle line:84% Well, if y is continuous f is continuous. 00:41:42.190 --> 00:41:43.990 align:middle line:90% Everything here is continuous. 00:41:43.990 --> 00:41:47.290 align:middle line:84% And so this means that by the fundamental theorem of calculus, 00:41:47.290 --> 00:41:52.890 align:middle line:84% this function here is actually even differentiable. 00:41:52.890 --> 00:41:55.930 align:middle line:84% And the derivative of this is just a constant. 00:41:55.930 --> 00:41:59.260 align:middle line:84% So this function here is differentiable, 00:41:59.260 --> 00:42:00.710 align:middle line:90% let alone continuous. 00:42:00.710 --> 00:42:06.940 align:middle line:84% So it's definitely in the space and the derivative of this map. 00:42:06.940 --> 00:42:09.440 align:middle line:84% By the fundamental theorem of calculus-- 00:42:09.440 --> 00:42:21.850 align:middle line:90% 00:42:21.850 --> 00:42:23.780 align:middle line:84% so by the fundamental theorem of calculus, 00:42:23.780 --> 00:42:26.660 align:middle line:84% the derivative is just what is inside here. 00:42:26.660 --> 00:42:36.280 align:middle line:84% So this is just f of y plus g of x. 00:42:36.280 --> 00:42:39.445 align:middle line:84% And if you want to write x in, then it should be like this. 00:42:39.445 --> 00:42:47.660 align:middle line:90% 00:42:47.660 --> 00:42:50.330 align:middle line:84% And then one more thing you observe, 00:42:50.330 --> 00:42:55.810 align:middle line:84% which is also immediate, that if you take a continuous function 00:42:55.810 --> 00:42:58.800 align:middle line:84% like this, and now you evaluate this, 00:42:58.800 --> 00:43:01.490 align:middle line:90% now you evaluate this at zero. 00:43:01.490 --> 00:43:05.060 align:middle line:84% But if x is zero within this integral, 00:43:05.060 --> 00:43:07.020 align:middle line:90% it's over just a point. 00:43:07.020 --> 00:43:08.400 align:middle line:90% So this here is zero. 00:43:08.400 --> 00:43:10.760 align:middle line:84% So this means that the value here is a. 00:43:10.760 --> 00:43:15.615 align:middle line:90% 00:43:15.615 --> 00:43:20.553 align:middle line:84% And so now, the next thing you observe is that-- 00:43:20.553 --> 00:43:31.850 align:middle line:90% 00:43:31.850 --> 00:43:40.870 align:middle line:84% so the next thing we observe is that if you have a fixed point. 00:43:40.870 --> 00:43:43.873 align:middle line:90% 00:43:43.873 --> 00:43:45.290 align:middle line:84% I think this is another nice thing 00:43:45.290 --> 00:43:50.840 align:middle line:84% is that the way it's set up is that you just need 00:43:50.840 --> 00:43:54.270 align:middle line:90% to use continuous functions. 00:43:54.270 --> 00:43:57.250 align:middle line:84% But of course, if you want a solution to this, 00:43:57.250 --> 00:44:00.400 align:middle line:84% it means that the function must have a derivative. 00:44:00.400 --> 00:44:00.900 align:middle line:90% Yeah. 00:44:00.900 --> 00:44:04.020 align:middle line:84% STUDENT: So how do you know that [INAUDIBLE]? 00:44:04.020 --> 00:44:06.250 align:middle line:84% PROFESSOR: Yeah we don't know that yet. 00:44:06.250 --> 00:44:07.950 align:middle line:90% That's a major point, actually. 00:44:07.950 --> 00:44:10.270 align:middle line:84% And again, just like in Newton's methods, 00:44:10.270 --> 00:44:12.850 align:middle line:84% we have to restrict where we're looking at it. 00:44:12.850 --> 00:44:14.200 align:middle line:90% Yeah. 00:44:14.200 --> 00:44:16.410 align:middle line:84% But the first thing I want to observe 00:44:16.410 --> 00:44:31.200 align:middle line:84% is that if y is a fixed point, well, then T of y is equal to y. 00:44:31.200 --> 00:44:33.510 align:middle line:90% And so what does this mean? 00:44:33.510 --> 00:44:36.100 align:middle line:90% Well, the derivative. 00:44:36.100 --> 00:44:46.350 align:middle line:84% So it means that if this is the case, then we have that T of y 00:44:46.350 --> 00:44:47.710 align:middle line:90% is of this form. 00:44:47.710 --> 00:44:51.220 align:middle line:84% So it has a derivative, and that derivative is this. 00:44:51.220 --> 00:44:54.590 align:middle line:84% So this means that y would have it, because y is equal to this. 00:44:54.590 --> 00:44:59.625 align:middle line:84% So y is differentiable, and the derivative is equal to this. 00:44:59.625 --> 00:45:02.890 align:middle line:90% 00:45:02.890 --> 00:45:04.930 align:middle line:90% And we always have-- 00:45:04.930 --> 00:45:06.850 align:middle line:84% whether it's a fixed point or not, 00:45:06.850 --> 00:45:14.095 align:middle line:84% we always have that Ty of zero is a. 00:45:14.095 --> 00:45:18.590 align:middle line:84% And since it is a fixed point, this is also this. 00:45:18.590 --> 00:45:22.210 align:middle line:84% So you see that if it is a fixed point, then 00:45:22.210 --> 00:45:24.435 align:middle line:84% it is a solution to the differential equation. 00:45:24.435 --> 00:45:27.490 align:middle line:90% 00:45:27.490 --> 00:45:31.880 align:middle line:90% Now, then the point comes in. 00:45:31.880 --> 00:45:36.430 align:middle line:84% The key point comes in with proving 00:45:36.430 --> 00:45:38.340 align:middle line:90% that the map is contracting. 00:45:38.340 --> 00:46:04.635 align:middle line:90% 00:46:04.635 --> 00:46:10.070 align:middle line:84% And so now, so we need to prove-- 00:46:10.070 --> 00:46:19.090 align:middle line:84% need to show that T is a contracting map. 00:46:19.090 --> 00:46:29.480 align:middle line:90% 00:46:29.480 --> 00:46:34.700 align:middle line:90% So set. 00:46:34.700 --> 00:46:37.160 align:middle line:84% I have to define a number of things, 00:46:37.160 --> 00:46:39.980 align:middle line:90% because again, I have to-- 00:46:39.980 --> 00:46:42.830 align:middle line:84% T is not going to be contracting on the space 00:46:42.830 --> 00:46:44.160 align:middle line:90% of continuous functions. 00:46:44.160 --> 00:46:47.880 align:middle line:84% So I have to look at some subset of continuous function. 00:46:47.880 --> 00:46:50.580 align:middle line:84% And the right subset is just a ball. 00:46:50.580 --> 00:46:52.270 align:middle line:90% It's its a close ball. 00:46:52.270 --> 00:46:56.100 align:middle line:90% So it's going to be-- 00:46:56.100 --> 00:47:00.570 align:middle line:84% so we're going to-- maybe I'll just say first that we're going 00:47:00.570 --> 00:47:07.360 align:middle line:84% to restrict T. So T here will be restricted now to-- 00:47:07.360 --> 00:47:14.910 align:middle line:84% we will think about it as a map on some ball into itself. 00:47:14.910 --> 00:47:18.120 align:middle line:84% But it's not at all clear that this map 00:47:18.120 --> 00:47:21.282 align:middle line:90% takes a ball into itself. 00:47:21.282 --> 00:47:23.350 align:middle line:90% And what do I mean by ball? 00:47:23.350 --> 00:47:25.300 align:middle line:90% So I have this bar up here. 00:47:25.300 --> 00:47:32.220 align:middle line:84% So I just means that this thing here 00:47:32.220 --> 00:47:36.336 align:middle line:84% is going to be the function y that is continuous. 00:47:36.336 --> 00:47:40.230 align:middle line:90% 00:47:40.230 --> 00:47:43.630 align:middle line:84% And then it's not quite going to be this either. 00:47:43.630 --> 00:47:45.480 align:middle line:90% I have to restrict. 00:47:45.480 --> 00:47:47.940 align:middle line:84% But I'll get to this in a second. 00:47:47.940 --> 00:47:50.200 align:middle line:84% So at the moment, you can think about it 00:47:50.200 --> 00:47:53.300 align:middle line:84% as being continuous function on the interval, 00:47:53.300 --> 00:48:00.830 align:middle line:84% so that the norm of y at each point x is less than 00:48:00.830 --> 00:48:04.930 align:middle line:84% or equal to R. And this bar here refers 00:48:04.930 --> 00:48:09.890 align:middle line:90% to that equal is also allowed. 00:48:09.890 --> 00:48:14.290 align:middle line:84% Now, this here is really like the ball. 00:48:14.290 --> 00:48:16.070 align:middle line:90% These are functions. 00:48:16.070 --> 00:48:18.670 align:middle line:84% If you're thinking about the function 00:48:18.670 --> 00:48:23.570 align:middle line:84% in the space of continuous function that is constant zero, 00:48:23.570 --> 00:48:27.310 align:middle line:84% then these are the functions whose distance-- this 00:48:27.310 --> 00:48:32.350 align:middle line:84% is the same as the space of continuous functions 00:48:32.350 --> 00:48:37.780 align:middle line:84% whose distance to the constant function zero is, at most, 00:48:37.780 --> 00:48:43.600 align:middle line:90% R. This is the zero function. 00:48:43.600 --> 00:48:45.350 align:middle line:84% It's less than or equal to R. Sorry. 00:48:45.350 --> 00:48:47.142 align:middle line:84% This is getting a little smaller over here. 00:48:47.142 --> 00:48:53.930 align:middle line:90% 00:48:53.930 --> 00:48:56.510 align:middle line:90% And again, it's going to be-- 00:48:56.510 --> 00:48:59.810 align:middle line:84% I will have to restrict the interval here 00:48:59.810 --> 00:49:02.670 align:middle line:84% to make sure that the map is contracting. 00:49:02.670 --> 00:49:05.825 align:middle line:90% 00:49:05.825 --> 00:49:09.440 align:middle line:84% But first, I want to define this radius here. 00:49:09.440 --> 00:49:12.680 align:middle line:90% And so the radius is-- 00:49:12.680 --> 00:49:14.990 align:middle line:90% I'm setting R here equal-- 00:49:14.990 --> 00:49:16.550 align:middle line:90% a was just a constant. 00:49:16.550 --> 00:49:20.005 align:middle line:84% I take the absolute value of a and I add 2 to it. 00:49:20.005 --> 00:49:22.580 align:middle line:90% 00:49:22.580 --> 00:49:27.650 align:middle line:84% Then I define two other areas, and there's no way around it. 00:49:27.650 --> 00:49:29.510 align:middle line:90% You have to define. 00:49:29.510 --> 00:49:32.995 align:middle line:84% We will require to define several constants first. 00:49:32.995 --> 00:49:35.690 align:middle line:90% 00:49:35.690 --> 00:49:40.490 align:middle line:84% L1, capital L1, is going to be another constant. 00:49:40.490 --> 00:49:43.250 align:middle line:90% It's going to be the maximum. 00:49:43.250 --> 00:49:52.560 align:middle line:84% And then you take off over z, where z here is less than R, 00:49:52.560 --> 00:49:57.815 align:middle line:84% and then you take the value like this. 00:49:57.815 --> 00:50:00.404 align:middle line:90% 00:50:00.404 --> 00:50:03.040 align:middle line:90% Remember, f is continuous. 00:50:03.040 --> 00:50:08.470 align:middle line:84% So looking at where the norm of x is less than 00:50:08.470 --> 00:50:13.560 align:middle line:84% or equal to some constant, then this, the maximum 00:50:13.560 --> 00:50:15.120 align:middle line:90% is achieved on this. 00:50:15.120 --> 00:50:18.630 align:middle line:84% So this is just some real number, 00:50:18.630 --> 00:50:22.980 align:middle line:84% positive real, non-negative real number. 00:50:22.980 --> 00:50:32.760 align:middle line:84% And then I define L2 to be the maximum of where I-- 00:50:32.760 --> 00:50:38.130 align:middle line:84% of maximum on this unit ball here, and where norm of x 00:50:38.130 --> 00:50:43.340 align:middle line:84% is less than 1 of g of x, like that. 00:50:43.340 --> 00:50:48.095 align:middle line:90% 00:50:48.095 --> 00:50:53.320 align:middle line:90% And now, what I want to do is-- 00:50:53.320 --> 00:50:55.990 align:middle line:90% so the first step is-- 00:50:55.990 --> 00:50:58.250 align:middle line:84% so the first step is the following. 00:50:58.250 --> 00:51:09.580 align:middle line:90% 00:51:09.580 --> 00:51:19.660 align:middle line:84% So first step in the proof is to prove the following, 00:51:19.660 --> 00:51:32.560 align:middle line:90% that there exists delta zero. 00:51:32.560 --> 00:51:36.760 align:middle line:84% This is not the delta that I will use eventually. 00:51:36.760 --> 00:51:38.980 align:middle line:84% I would have to restrict it further. 00:51:38.980 --> 00:51:44.670 align:middle line:84% There exists delta zero, so that if you're looking at this space, 00:51:44.670 --> 00:51:50.360 align:middle line:84% if y here is in the space of continuous functions 00:51:50.360 --> 00:52:00.770 align:middle line:84% on this interval here, then T of y-- 00:52:00.770 --> 00:52:01.610 align:middle line:90% sorry. 00:52:01.610 --> 00:52:08.365 align:middle line:84% If y is in this here, and y is in the unit ball-- 00:52:08.365 --> 00:52:11.030 align:middle line:90% 00:52:11.030 --> 00:52:18.240 align:middle line:84% not in the unit ball, then y here is less or equal to R. 00:52:18.240 --> 00:52:20.850 align:middle line:84% So it's a continuous function on this interval, 00:52:20.850 --> 00:52:23.220 align:middle line:84% and the norm is less or equal to R. 00:52:23.220 --> 00:52:26.690 align:middle line:84% So this means that so this is, of course, the same as saying 00:52:26.690 --> 00:52:31.550 align:middle line:84% that y is in this ball here when I'm now 00:52:31.550 --> 00:52:34.290 align:middle line:90% thinking about this as a space. 00:52:34.290 --> 00:52:37.310 align:middle line:84% But you can forget about that if you want. 00:52:37.310 --> 00:52:38.040 align:middle line:90% It's the same. 00:52:38.040 --> 00:52:42.410 align:middle line:90% 00:52:42.410 --> 00:52:46.220 align:middle line:84% And I want to show that T of y is in the same space. 00:52:46.220 --> 00:52:51.130 align:middle line:90% So T of y. 00:52:51.130 --> 00:52:53.300 align:middle line:90% So I want to show-- 00:52:53.300 --> 00:53:07.960 align:middle line:84% so I want to show that T of y at each x here, 00:53:07.960 --> 00:53:10.780 align:middle line:90% the norm is less or equal to 1. 00:53:10.780 --> 00:53:16.450 align:middle line:84% Then if I have that, I already know that this function T of y 00:53:16.450 --> 00:53:17.420 align:middle line:90% is continuous. 00:53:17.420 --> 00:53:20.470 align:middle line:84% It's actually even differentiable. 00:53:20.470 --> 00:53:23.530 align:middle line:84% But it's, in particular, continuous. 00:53:23.530 --> 00:53:27.370 align:middle line:84% And if I have this, then I know that this space 00:53:27.370 --> 00:53:28.470 align:middle line:90% is taken into itself. 00:53:28.470 --> 00:53:33.580 align:middle line:90% 00:53:33.580 --> 00:53:37.150 align:middle line:90% So now, how do I do that? 00:53:37.150 --> 00:53:42.500 align:middle line:84% So that's where I need to restrict the interval that I'm 00:53:42.500 --> 00:53:43.500 align:middle line:90% looking on. 00:53:43.500 --> 00:53:47.130 align:middle line:84% And again, we'll come back to it, 00:53:47.130 --> 00:53:51.180 align:middle line:84% but there's a good reason that we have to restrict. 00:53:51.180 --> 00:53:53.290 align:middle line:84% Otherwise, it couldn't possibly be true. 00:53:53.290 --> 00:53:56.355 align:middle line:90% 00:53:56.355 --> 00:53:56.855 align:middle line:90% Yeah. 00:53:56.855 --> 00:53:58.700 align:middle line:84% STUDENT: To be contracting, do we need 00:53:58.700 --> 00:54:00.710 align:middle line:90% it to be strictly less than 1? 00:54:00.710 --> 00:54:03.020 align:middle line:84% PROFESSOR: So remember, contracting 00:54:03.020 --> 00:54:09.200 align:middle line:84% means that there exists a constant c strictly less than 1. 00:54:09.200 --> 00:54:12.300 align:middle line:84% This here has nothing to do with the constant c. 00:54:12.300 --> 00:54:17.460 align:middle line:84% This here is just the space that we're looking at the map on. 00:54:17.460 --> 00:54:21.120 align:middle line:90% And the reason why I make here-- 00:54:21.120 --> 00:54:24.180 align:middle line:84% so again, so this has nothing to do with that c. 00:54:24.180 --> 00:54:27.380 align:middle line:84% But the reason why I want this less than or equal to 00:54:27.380 --> 00:54:33.850 align:middle line:84% is that I want to have that the space is Cauchy complete. 00:54:33.850 --> 00:54:34.350 align:middle line:90% Sorry. 00:54:34.350 --> 00:54:36.930 align:middle line:84% So let me just try to make that clear. 00:54:36.930 --> 00:54:43.270 align:middle line:84% So to get a fixed point for a contracting map, 00:54:43.270 --> 00:54:46.470 align:middle line:84% you need a space that is Cauchy complete 00:54:46.470 --> 00:54:49.150 align:middle line:84% when we prove this for the metric space. 00:54:49.150 --> 00:54:51.330 align:middle line:90% We have a good candidate. 00:54:51.330 --> 00:54:54.930 align:middle line:84% We have the candidate of continuous functions 00:54:54.930 --> 00:54:57.340 align:middle line:84% on some closed interval like this. 00:54:57.340 --> 00:55:01.440 align:middle line:84% This is Cauchy complete with the usual metric. 00:55:01.440 --> 00:55:03.600 align:middle line:84% But I'm not going to look at that space 00:55:03.600 --> 00:55:07.000 align:middle line:84% because my map is not actually contracting on the space. 00:55:07.000 --> 00:55:09.520 align:middle line:90% It's contracting on a subset. 00:55:09.520 --> 00:55:11.890 align:middle line:90% And so I need that subset. 00:55:11.890 --> 00:55:16.090 align:middle line:84% So in the end, I'm sort of not working on this, 00:55:16.090 --> 00:55:18.622 align:middle line:90% but I'm working on a subset. 00:55:18.622 --> 00:55:22.390 align:middle line:84% And I need that subset to be Cauchy complete. 00:55:22.390 --> 00:55:25.440 align:middle line:84% So basically, what I have is that this here 00:55:25.440 --> 00:55:28.570 align:middle line:84% is the giant metric space I start with. 00:55:28.570 --> 00:55:32.160 align:middle line:84% Then I'm looking at a ball in this. 00:55:32.160 --> 00:55:36.270 align:middle line:84% And if I take the closed ball and it 00:55:36.270 --> 00:55:37.960 align:middle line:84% happened to be at the origin, but it 00:55:37.960 --> 00:55:40.120 align:middle line:90% could be just at any point. 00:55:40.120 --> 00:55:43.300 align:middle line:84% If you take a metric space and you're checking something 00:55:43.300 --> 00:55:47.060 align:middle line:84% like this, if the metric space is Cauchy complete, 00:55:47.060 --> 00:55:49.870 align:middle line:84% then this subset is also Cauchy complete. 00:55:49.870 --> 00:55:55.820 align:middle line:84% But this require I have less than or equal to-- 00:55:55.820 --> 00:55:58.360 align:middle line:84% because I also could kind of converge to something 00:55:58.360 --> 00:56:01.904 align:middle line:90% where you got equality. 00:56:01.904 --> 00:56:06.855 align:middle line:84% So that was why I do it like that. 00:56:06.855 --> 00:56:09.800 align:middle line:90% So let's see. 00:56:09.800 --> 00:56:12.760 align:middle line:90% So we are trying. 00:56:12.760 --> 00:56:18.430 align:middle line:84% We have already defined this radius, capital R. 00:56:18.430 --> 00:56:23.650 align:middle line:84% And then we have these two constants, L1 and L2. 00:56:23.650 --> 00:56:28.390 align:middle line:84% And what I need to prove is-- the first step here is to prove 00:56:28.390 --> 00:56:29.620 align:middle line:90% that if-- 00:56:29.620 --> 00:56:31.670 align:middle line:90% I'm going to look again. 00:56:31.670 --> 00:56:35.870 align:middle line:84% So the first step is that I need to restrict. 00:56:35.870 --> 00:56:38.280 align:middle line:90% So I need to restrict. 00:56:38.280 --> 00:56:44.870 align:middle line:84% So I'm going to look at a subset of these spaces of-- 00:56:44.870 --> 00:56:47.970 align:middle line:84% this is a subset of the continuous functions. 00:56:47.970 --> 00:56:51.410 align:middle line:84% And I need to prove that if y is in here, 00:56:51.410 --> 00:56:54.070 align:middle line:84% then the image is in the same space. 00:56:54.070 --> 00:56:58.590 align:middle line:90% 00:56:58.590 --> 00:57:03.740 align:middle line:84% And so this is the statement that if y is continuous, 00:57:03.740 --> 00:57:09.750 align:middle line:84% the norm of this function is less than or equal to R. 00:57:09.750 --> 00:57:14.840 align:middle line:84% So at each point, each x, the absolute value of y of x 00:57:14.840 --> 00:57:18.170 align:middle line:84% is less or equal to R. Then you have the same 00:57:18.170 --> 00:57:20.010 align:middle line:90% for the image under T. 00:57:20.010 --> 00:57:23.960 align:middle line:84% We already know, again, that the image under the T of y 00:57:23.960 --> 00:57:25.400 align:middle line:90% is continuous. 00:57:25.400 --> 00:57:29.762 align:middle line:90% 00:57:29.762 --> 00:57:36.840 align:middle line:84% And so what I can do is I can restrict my interval. 00:57:36.840 --> 00:57:42.880 align:middle line:84% So I will restrict the interval as follows. 00:57:42.880 --> 00:57:54.990 align:middle line:84% So I'm going to pick delta zero to be the minimum of 1 00:57:54.990 --> 00:57:57.930 align:middle line:90% over L plus 1. 00:57:57.930 --> 00:57:59.650 align:middle line:90% L is non-negative. 00:57:59.650 --> 00:58:01.330 align:middle line:90% In principle, it could be zero. 00:58:01.330 --> 00:58:04.890 align:middle line:84% I don't want to worry about that, so I just add 1. 00:58:04.890 --> 00:58:09.540 align:middle line:84% And L2 could also be zero, although it is pretty unlikely 00:58:09.540 --> 00:58:10.630 align:middle line:90% that any of them are zero. 00:58:10.630 --> 00:58:12.900 align:middle line:84% But just to be safe, I'm just adding 00:58:12.900 --> 00:58:16.050 align:middle line:90% so I don't divide by zero. 00:58:16.050 --> 00:58:17.350 align:middle line:90% So that's why. 00:58:17.350 --> 00:58:21.580 align:middle line:84% And the first thing is that I'm-- 00:58:21.580 --> 00:58:25.300 align:middle line:84% and then I'm going to look at the interval from this. 00:58:25.300 --> 00:58:27.550 align:middle line:84% And this is not going to be my final. 00:58:27.550 --> 00:58:30.570 align:middle line:84% I will have to do some further restriction. 00:58:30.570 --> 00:58:34.240 align:middle line:84% But I'm going to look at this interval initially. 00:58:34.240 --> 00:58:37.540 align:middle line:84% And the 1 here, it just comes in that I 00:58:37.540 --> 00:58:42.340 align:middle line:84% want this here to be a subset of this unit interval. 00:58:42.340 --> 00:58:50.104 align:middle line:90% 00:58:50.104 --> 00:58:52.640 align:middle line:90% So now, let's try. 00:58:52.640 --> 00:58:54.435 align:middle line:84% So now we are in good shape here. 00:58:54.435 --> 00:59:04.240 align:middle line:90% 00:59:04.240 --> 00:59:06.400 align:middle line:90% So now, let's look at Ty. 00:59:06.400 --> 00:59:09.790 align:middle line:90% 00:59:09.790 --> 00:59:13.270 align:middle line:84% And we need to estimate this at each x. 00:59:13.270 --> 00:59:19.750 align:middle line:84% And we want to say that the norm is less than capital-- 00:59:19.750 --> 00:59:21.680 align:middle line:90% less or equal to capital R. 00:59:21.680 --> 00:59:24.160 align:middle line:84% Well, let's just look at what this is. 00:59:24.160 --> 00:59:28.090 align:middle line:84% This thing here is equal to this function. 00:59:28.090 --> 00:59:32.810 align:middle line:84% T of y is defined to be a plus the integral from zero 00:59:32.810 --> 00:59:49.640 align:middle line:84% to x of f of y of s, plus g of s, ds. 00:59:49.640 --> 00:59:53.450 align:middle line:84% And then I have to have an absolute value like that. 00:59:53.450 --> 00:59:56.570 align:middle line:84% And now I can just use the triangle inequality, of course, 00:59:56.570 --> 00:59:57.830 align:middle line:90% trivially. 00:59:57.830 --> 01:00:01.700 align:middle line:84% I get absolute value of this, and then plus, and then 01:00:01.700 --> 01:00:05.215 align:middle line:90% the integral here, like that. 01:00:05.215 --> 01:00:09.330 align:middle line:90% 01:00:09.330 --> 01:00:09.830 align:middle line:90% Oops. 01:00:09.830 --> 01:00:15.650 align:middle line:90% 01:00:15.650 --> 01:00:20.750 align:middle line:84% And then I can just take this absolute value in under. 01:00:20.750 --> 01:00:27.990 align:middle line:84% So I get less than or equal to a plus, and then the integral, 01:00:27.990 --> 01:00:33.780 align:middle line:84% and then the absolute value of f of y of s. 01:00:33.780 --> 01:00:37.740 align:middle line:84% And here I can also take the absolute value 01:00:37.740 --> 01:00:40.930 align:middle line:84% using the triangle inequality on each individually. 01:00:40.930 --> 01:00:45.240 align:middle line:90% 01:00:45.240 --> 01:00:48.480 align:middle line:90% So I can write it like this. 01:00:48.480 --> 01:00:52.750 align:middle line:90% Now, this is where I use-- 01:00:52.750 --> 01:00:59.550 align:middle line:84% so s here will have to lie between zero and x. 01:00:59.550 --> 01:01:01.783 align:middle line:90% And so x is between-- 01:01:01.783 --> 01:01:05.940 align:middle line:90% 01:01:05.940 --> 01:01:07.990 align:middle line:90% let me just draw it here. 01:01:07.990 --> 01:01:11.610 align:middle line:90% So here I have zero. 01:01:11.610 --> 01:01:13.890 align:middle line:90% Here I have delta zero. 01:01:13.890 --> 01:01:18.270 align:middle line:84% And then I have-- this is zero, and here I have x somewhere. 01:01:18.270 --> 01:01:21.910 align:middle line:90% 01:01:21.910 --> 01:01:25.500 align:middle line:90% So this means that-- 01:01:25.500 --> 01:01:33.280 align:middle line:84% this means that so here you have x, but x itself, x has to lie-- 01:01:33.280 --> 01:01:40.180 align:middle line:84% x has to lie between delta zero minus delta zero, 01:01:40.180 --> 01:01:41.510 align:middle line:90% and delta zero. 01:01:41.510 --> 01:01:45.290 align:middle line:84% And s has to lie between zero and x. 01:01:45.290 --> 01:01:49.730 align:middle line:84% So this means that s here lies between-- 01:01:49.730 --> 01:01:53.505 align:middle line:84% x will lie in this interval, also. 01:01:53.505 --> 01:01:58.889 align:middle line:90% 01:01:58.889 --> 01:02:01.466 align:middle line:90% But now you see that-- 01:02:01.466 --> 01:02:06.400 align:middle line:90% 01:02:06.400 --> 01:02:13.040 align:middle line:84% so now if we're looking at-- so we have T of y in x. 01:02:13.040 --> 01:02:18.700 align:middle line:84% Again, remember that x here is less than 01:02:18.700 --> 01:02:20.860 align:middle line:90% or equal to delta zero. 01:02:20.860 --> 01:02:24.040 align:middle line:84% So we have that this thing here is 01:02:24.040 --> 01:02:28.700 align:middle line:84% less than or equal to a plus, and then the integral from zero 01:02:28.700 --> 01:02:44.510 align:middle line:84% to x of f of y of s, plus g of s ds. 01:02:44.510 --> 01:02:48.500 align:middle line:84% And the first thing I observe is that s here 01:02:48.500 --> 01:02:53.270 align:middle line:84% is also, just because x was in that interval, 01:02:53.270 --> 01:02:56.690 align:middle line:90% you also have that s-- 01:02:56.690 --> 01:03:01.850 align:middle line:84% because x is in that interval, s is also in this interval. 01:03:01.850 --> 01:03:06.860 align:middle line:84% The second thing I observe is that I'm 01:03:06.860 --> 01:03:12.320 align:middle line:84% assuming that the function that I start with at each point 01:03:12.320 --> 01:03:20.450 align:middle line:84% have no less than R. Now, if I'm using these two things, 01:03:20.450 --> 01:03:22.800 align:middle line:90% then I have that-- 01:03:22.800 --> 01:03:30.520 align:middle line:84% so this means that g of s, this one here, 01:03:30.520 --> 01:03:35.820 align:middle line:84% this is how we defined L2 because this interval here 01:03:35.820 --> 01:03:40.270 align:middle line:84% is contained in the interval between minus 1 and 1. 01:03:40.270 --> 01:03:42.970 align:middle line:84% This here is contained in the interval between negative, 01:03:42.970 --> 01:03:45.720 align:middle line:90% minus 1 and 1. 01:03:45.720 --> 01:03:51.690 align:middle line:84% So we have that g of s is less than this L2, 01:03:51.690 --> 01:03:52.865 align:middle line:90% less than or equal to L2. 01:03:52.865 --> 01:03:56.502 align:middle line:90% 01:03:56.502 --> 01:04:03.240 align:middle line:84% The second thing we observe is that because we have this here, 01:04:03.240 --> 01:04:06.140 align:middle line:90% for each s, we have this. 01:04:06.140 --> 01:04:09.420 align:middle line:90% 01:04:09.420 --> 01:04:16.360 align:middle line:84% Then it means, you see, by the first-- 01:04:16.360 --> 01:04:25.460 align:middle line:84% by the definition of L1 that f of y of s, 01:04:25.460 --> 01:04:28.150 align:middle line:84% this thing here, is less than L1. 01:04:28.150 --> 01:04:30.920 align:middle line:90% 01:04:30.920 --> 01:04:35.572 align:middle line:90% L1 and L2 are now fixed numbers. 01:04:35.572 --> 01:04:39.350 align:middle line:84% And so you see that this second thing here, I 01:04:39.350 --> 01:04:42.660 align:middle line:90% can bound this thing here by-- 01:04:42.660 --> 01:04:45.657 align:middle line:84% so this is less than or equal to a. 01:04:45.657 --> 01:04:47.490 align:middle line:84% And that should have been an absolute value. 01:04:47.490 --> 01:04:52.610 align:middle line:84% Somehow I lost it when I copied it from this line. 01:04:52.610 --> 01:04:55.190 align:middle line:90% a, absolute value of a. 01:04:55.190 --> 01:04:59.390 align:middle line:84% And then on the first one here, the length of this 01:04:59.390 --> 01:05:05.100 align:middle line:84% interval, but that is, of course, bounded by delta zero. 01:05:05.100 --> 01:05:11.450 align:middle line:84% So this is plus delta zero, and then the maximum possible value 01:05:11.450 --> 01:05:14.360 align:middle line:90% of this, that would be L1. 01:05:14.360 --> 01:05:17.900 align:middle line:84% And then likewise, the length of this interval you're integrating 01:05:17.900 --> 01:05:20.160 align:middle line:90% over is at most delta zero. 01:05:20.160 --> 01:05:23.700 align:middle line:84% And this one here cannot be larger than L2. 01:05:23.700 --> 01:05:24.855 align:middle line:90% So you have this. 01:05:24.855 --> 01:05:27.660 align:middle line:90% 01:05:27.660 --> 01:05:31.900 align:middle line:84% And now, remember what delta zero was. 01:05:31.900 --> 01:05:33.510 align:middle line:90% And I will just-- 01:05:33.510 --> 01:05:37.570 align:middle line:84% I have just on the bottom, below here. 01:05:37.570 --> 01:05:39.630 align:middle line:90% But let me just write it here. 01:05:39.630 --> 01:05:44.220 align:middle line:84% This here was what delta zero was chosen to be, 01:05:44.220 --> 01:05:54.110 align:middle line:84% the minimum, 1 over L1 plus 1, 1 over L2 plus 1. 01:05:54.110 --> 01:06:03.520 align:middle line:90% 01:06:03.520 --> 01:06:11.910 align:middle line:84% So now, we have that this thing here is less than this here. 01:06:11.910 --> 01:06:16.260 align:middle line:90% We just keep this thing here. 01:06:16.260 --> 01:06:24.010 align:middle line:84% Well, delta is, in particular, less than this thing here. 01:06:24.010 --> 01:06:31.420 align:middle line:84% So you see that if you plug in this, so it's like plus 1, L1 01:06:31.420 --> 01:06:34.300 align:middle line:90% plus 1 times L1. 01:06:34.300 --> 01:06:37.910 align:middle line:84% Likewise, over here, you're using this. 01:06:37.910 --> 01:06:42.500 align:middle line:84% So you're getting 1 over L2 plus 1 L2. 01:06:42.500 --> 01:06:45.710 align:middle line:84% And now you see that this thing here is less than 1. 01:06:45.710 --> 01:06:47.480 align:middle line:90% This is less than 1. 01:06:47.480 --> 01:06:49.780 align:middle line:90% So this is less than a. 01:06:49.780 --> 01:06:52.450 align:middle line:84% It's actually strictly less, but it doesn't really matter. 01:06:52.450 --> 01:06:59.230 align:middle line:90% a plus 2 like this. 01:06:59.230 --> 01:07:05.320 align:middle line:90% 01:07:05.320 --> 01:07:07.840 align:middle line:90% So this means that-- 01:07:07.840 --> 01:07:12.670 align:middle line:84% so then, but this here, this was exactly 01:07:12.670 --> 01:07:17.750 align:middle line:84% the definition of capital R. And so you have that. 01:07:17.750 --> 01:07:20.060 align:middle line:84% And when I copied it over, I also 01:07:20.060 --> 01:07:23.780 align:middle line:84% forgot to put absolute value sign here. 01:07:23.780 --> 01:07:27.270 align:middle line:84% So you have that if-- so have that 01:07:27.270 --> 01:07:31.310 align:middle line:84% if-- so the conclusion here is that if y here 01:07:31.310 --> 01:07:41.060 align:middle line:84% is a continuous function on this interval here, 01:07:41.060 --> 01:07:47.420 align:middle line:84% and the norm of y at each point is less or equal to r, well, 01:07:47.420 --> 01:07:58.010 align:middle line:84% then you have that T of y at each point is less than R. 01:07:58.010 --> 01:08:02.780 align:middle line:84% It's actually strictly less, but we don't care about that. 01:08:02.780 --> 01:08:09.920 align:middle line:90% So what you have now is-- 01:08:09.920 --> 01:08:14.960 align:middle line:84% so what we now have proven is that we have this map T. 01:08:14.960 --> 01:08:17.250 align:middle line:84% And now I'm going to think about this map-- 01:08:17.250 --> 01:08:26.220 align:middle line:84% sorry-- on the space of y that is continuous on this interval 01:08:26.220 --> 01:08:26.720 align:middle line:90% here. 01:08:26.720 --> 01:08:32.220 align:middle line:90% 01:08:32.220 --> 01:08:35.340 align:middle line:84% And what you have is-- and also, sorry. 01:08:35.340 --> 01:08:40.149 align:middle line:84% And I'm thinking about with the norm at each point. 01:08:40.149 --> 01:08:41.819 align:middle line:90% This here is less than R. 01:08:41.819 --> 01:08:45.340 align:middle line:84% And again, this is really a ball in this metric space, 01:08:45.340 --> 01:08:48.569 align:middle line:84% but let me just write it out like this. 01:08:48.569 --> 01:08:50.970 align:middle line:84% And what you have is that-- sorry. 01:08:50.970 --> 01:08:56.340 align:middle line:84% This is the space, and you have that T of y. 01:08:56.340 --> 01:09:01.090 align:middle line:84% If y is in this space, then T of y is in the same space. 01:09:01.090 --> 01:09:06.919 align:middle line:84% So T of y is also continuous on that interval. 01:09:06.919 --> 01:09:09.490 align:middle line:90% 01:09:09.490 --> 01:09:16.200 align:middle line:84% And T of y at each point x is less or equal to R. 01:09:16.200 --> 01:09:19.275 align:middle line:84% So the map mapped this space into itself. 01:09:19.275 --> 01:09:22.495 align:middle line:90% 01:09:22.495 --> 01:09:28.120 align:middle line:84% Now, the next thing we want to prove 01:09:28.120 --> 01:09:30.496 align:middle line:90% is that the map is contracting. 01:09:30.496 --> 01:09:46.900 align:middle line:90% 01:09:46.900 --> 01:09:49.029 align:middle line:90% You see that you need-- 01:09:49.029 --> 01:09:52.640 align:middle line:84% you know if this space is not stable. 01:09:52.640 --> 01:09:57.550 align:middle line:84% So imagine that you take-- so we're starting 01:09:57.550 --> 01:09:59.210 align:middle line:90% with some giant metric space. 01:09:59.210 --> 01:10:02.550 align:middle line:84% This is the continuous function, say, on this interval. 01:10:02.550 --> 01:10:09.040 align:middle line:90% 01:10:09.040 --> 01:10:12.470 align:middle line:84% And then we're looking at some subspace of that. 01:10:12.470 --> 01:10:14.785 align:middle line:84% So these are continuous on a smaller interval. 01:10:14.785 --> 01:10:19.740 align:middle line:90% 01:10:19.740 --> 01:10:23.600 align:middle line:90% And then we have to find a map. 01:10:23.600 --> 01:10:29.090 align:middle line:84% But if somehow the map was taking us 01:10:29.090 --> 01:10:31.950 align:middle line:84% out of the space where it's contracting, 01:10:31.950 --> 01:10:35.970 align:middle line:84% then we couldn't keep applying the contracting map. 01:10:35.970 --> 01:10:38.420 align:middle line:90% It wouldn't anymore be-- 01:10:38.420 --> 01:10:41.370 align:middle line:84% And the thing is that you want to just iterate. 01:10:41.370 --> 01:10:42.290 align:middle line:90% You want to keep. 01:10:42.290 --> 01:10:44.360 align:middle line:84% You want to pick some point, and then you 01:10:44.360 --> 01:10:47.610 align:middle line:84% want to keep applying the contracting map. 01:10:47.610 --> 01:10:51.360 align:middle line:84% So you want it to stay in the space where it is contracting. 01:10:51.360 --> 01:10:53.750 align:middle line:84% And so this is what we did in one. 01:10:53.750 --> 01:10:56.660 align:middle line:84% But we haven't yet proved that the map is contracting. 01:10:56.660 --> 01:11:00.440 align:middle line:84% And so now, to prove that the map is contracting, 01:11:00.440 --> 01:11:05.175 align:middle line:84% we're going to further restrict the interval. 01:11:05.175 --> 01:11:08.600 align:middle line:84% And so this is-- so I'm going to set. 01:11:08.600 --> 01:11:11.400 align:middle line:84% I'm going to define now one more constant. 01:11:11.400 --> 01:11:13.130 align:middle line:90% Sorry about that. 01:11:13.130 --> 01:11:15.750 align:middle line:90% So this is M here. 01:11:15.750 --> 01:11:21.660 align:middle line:84% And M is going to be just the max of f prime-- 01:11:21.660 --> 01:11:24.450 align:middle line:90% 01:11:24.450 --> 01:11:29.460 align:middle line:84% of f prime of z, absolute value of this, 01:11:29.460 --> 01:11:35.563 align:middle line:84% where z here is less or equal to R. So this is my-- 01:11:35.563 --> 01:11:41.730 align:middle line:90% 01:11:41.730 --> 01:11:44.315 align:middle line:84% well, almost the second last constant function. 01:11:44.315 --> 01:11:48.090 align:middle line:90% 01:11:48.090 --> 01:11:50.640 align:middle line:84% And now I'm going to define this. 01:11:50.640 --> 01:11:54.540 align:middle line:84% So the last constant is going to be the interval 01:11:54.540 --> 01:11:55.870 align:middle line:90% that I'm going to work with. 01:11:55.870 --> 01:11:58.170 align:middle line:84% So remember, what I'm trying to do 01:11:58.170 --> 01:12:03.970 align:middle line:84% is, I have this differential equation. 01:12:03.970 --> 01:12:11.230 align:middle line:84% And I want to say that, near zero, there is a solution. 01:12:11.230 --> 01:12:14.830 align:middle line:84% But near zero just means that there exists some tiny interval 01:12:14.830 --> 01:12:17.090 align:middle line:84% around zero where I have a solution. 01:12:17.090 --> 01:12:20.350 align:middle line:84% And again, if I have time, I also will do it next time. 01:12:20.350 --> 01:12:24.610 align:middle line:84% I will give an example where you can see that, in general, you 01:12:24.610 --> 01:12:27.430 align:middle line:84% would need to look at a very tiny interval for this 01:12:27.430 --> 01:12:28.250 align:middle line:90% to have a solution. 01:12:28.250 --> 01:12:30.720 align:middle line:84% Otherwise, it doesn't have a solution on a very large set. 01:12:30.720 --> 01:12:34.370 align:middle line:90% 01:12:34.370 --> 01:12:36.500 align:middle line:90% So I define it like this. 01:12:36.500 --> 01:12:38.960 align:middle line:90% And then I define delta here. 01:12:38.960 --> 01:12:44.680 align:middle line:84% So delta is now going to be the minimum of delta zero, 01:12:44.680 --> 01:12:50.340 align:middle line:84% the one from before, and then 1 over 2M plus 1. 01:12:50.340 --> 01:12:53.320 align:middle line:90% 01:12:53.320 --> 01:12:58.360 align:middle line:84% And now I want to look at the space. 01:12:58.360 --> 01:13:02.320 align:middle line:84% So I want to look at T. And then T 01:13:02.320 --> 01:13:07.400 align:middle line:84% is going to be a map from the continuous functions. 01:13:07.400 --> 01:13:10.720 align:middle line:84% So y here is a continuous function 01:13:10.720 --> 01:13:14.250 align:middle line:90% on, now, this little interval. 01:13:14.250 --> 01:13:19.010 align:middle line:84% So this interval here, because you have the minimum of this, 01:13:19.010 --> 01:13:22.040 align:middle line:84% this interval is contained in the corresponding interval 01:13:22.040 --> 01:13:23.730 align:middle line:90% with delta zero. 01:13:23.730 --> 01:13:28.880 align:middle line:84% So it's y in this space, such that the norm here at each point 01:13:28.880 --> 01:13:31.490 align:middle line:84% is less than or equal to R. That's 01:13:31.490 --> 01:13:33.330 align:middle line:90% the space I'm going to look at. 01:13:33.330 --> 01:13:39.540 align:middle line:84% And we already know that T maps this space into itself, 01:13:39.540 --> 01:13:41.570 align:middle line:84% because there's really no new statement. 01:13:41.570 --> 01:13:43.860 align:middle line:90% If the function is-- 01:13:43.860 --> 01:13:48.390 align:middle line:84% so as long as you have this here, and you apply T on it, 01:13:48.390 --> 01:13:50.490 align:middle line:84% T, when you apply T on a function, 01:13:50.490 --> 01:13:52.200 align:middle line:84% it's defined on the same interval. 01:13:52.200 --> 01:13:55.280 align:middle line:84% So it will automatically define on this smaller interval. 01:13:55.280 --> 01:13:58.890 align:middle line:84% And we already have this property. 01:13:58.890 --> 01:14:03.170 align:middle line:84% So T maps this space here into itself. 01:14:03.170 --> 01:14:06.260 align:middle line:90% 01:14:06.260 --> 01:14:08.780 align:middle line:90% And now I want to prove. 01:14:08.780 --> 01:14:10.570 align:middle line:90% I want to estimate. 01:14:10.570 --> 01:14:14.850 align:middle line:84% So I take two elements in this space. 01:14:14.850 --> 01:14:17.230 align:middle line:90% So I have y1 and y2. 01:14:17.230 --> 01:14:18.850 align:middle line:90% So you're looking at-- 01:14:18.850 --> 01:14:25.800 align:middle line:90% so y1 and y2 are in this space. 01:14:25.800 --> 01:14:35.240 align:middle line:84% And you want to look at what is the distance between y1 and y2. 01:14:35.240 --> 01:14:39.450 align:middle line:90% 01:14:39.450 --> 01:14:42.960 align:middle line:84% So we want this here, and we want-- 01:14:42.960 --> 01:14:47.400 align:middle line:84% what I want to prove is that it's contracting. 01:14:47.400 --> 01:14:48.755 align:middle line:90% So we want to show. 01:14:48.755 --> 01:14:54.030 align:middle line:90% 01:14:54.030 --> 01:14:55.470 align:middle line:90% And I want to prove. 01:14:55.470 --> 01:14:58.020 align:middle line:90% I'm going to prove it. 01:14:58.020 --> 01:15:00.700 align:middle line:90% My constant c will be explicit. 01:15:00.700 --> 01:15:06.510 align:middle line:84% It will be just one half, so that certainly is less than 1. 01:15:06.510 --> 01:15:07.970 align:middle line:90% And this is what I want. 01:15:07.970 --> 01:15:12.601 align:middle line:90% 01:15:12.601 --> 01:15:20.595 align:middle line:90% So now, let's look at them. 01:15:20.595 --> 01:15:26.320 align:middle line:90% 01:15:26.320 --> 01:15:31.380 align:middle line:90% So T of y1 minus T of y2. 01:15:31.380 --> 01:15:34.090 align:middle line:90% 01:15:34.090 --> 01:15:38.110 align:middle line:84% This thing here, T of y1 is defined 01:15:38.110 --> 01:15:40.850 align:middle line:90% to be a plus the integral. 01:15:40.850 --> 01:15:41.440 align:middle line:90% Sorry. 01:15:41.440 --> 01:15:42.470 align:middle line:90% And let me write. 01:15:42.470 --> 01:15:44.290 align:middle line:84% So I'm going to-- let me just make 01:15:44.290 --> 01:15:45.770 align:middle line:90% it a little more space here. 01:15:45.770 --> 01:15:49.150 align:middle line:84% I worry that I'm running out of time, but let's see. 01:15:49.150 --> 01:15:53.320 align:middle line:84% So this thing here, minus T of y2. 01:15:53.320 --> 01:15:56.690 align:middle line:84% And I'm just looking at the difference in some x. 01:15:56.690 --> 01:15:59.890 align:middle line:90% And again, x here is now in-- 01:15:59.890 --> 01:16:03.880 align:middle line:90% x here will be in the integral. 01:16:03.880 --> 01:16:11.580 align:middle line:84% You were looking at x in the integral, in this integral. 01:16:11.580 --> 01:16:15.580 align:middle line:84% And so I'm just writing down first here what this is. 01:16:15.580 --> 01:16:29.030 align:middle line:84% This is the integral from a to x of f of y1, s plus g of s. 01:16:29.030 --> 01:16:33.660 align:middle line:90% 01:16:33.660 --> 01:16:36.420 align:middle line:90% And then you have minus. 01:16:36.420 --> 01:16:39.750 align:middle line:90% This is what you get for y2. 01:16:39.750 --> 01:16:43.040 align:middle line:90% a, so minus this thing here. 01:16:43.040 --> 01:16:56.760 align:middle line:90% 01:16:56.760 --> 01:17:00.690 align:middle line:90% So this is this difference. 01:17:00.690 --> 01:17:02.730 align:middle line:84% And what you see is the first thing 01:17:02.730 --> 01:17:06.370 align:middle line:84% you observe is that this a here cancels. 01:17:06.370 --> 01:17:11.090 align:middle line:84% But not only does this cancel, but this term here also cancel. 01:17:11.090 --> 01:17:13.480 align:middle line:84% And actually, there was a good reason 01:17:13.480 --> 01:17:15.460 align:middle line:90% that there are lower order. 01:17:15.460 --> 01:17:20.380 align:middle line:84% And the point here is that by-- the main point 01:17:20.380 --> 01:17:23.660 align:middle line:84% is in the unknown function anyway. 01:17:23.660 --> 01:17:29.290 align:middle line:84% And it's what's called a non-linear equation. 01:17:29.290 --> 01:17:33.598 align:middle line:90% So that's a difficulty. 01:17:33.598 --> 01:17:37.150 align:middle line:84% But what you then get when you write down, 01:17:37.150 --> 01:17:40.150 align:middle line:84% when you have-- after the cancellation, what 01:17:40.150 --> 01:17:55.520 align:middle line:84% you get is f of y1 of s minus f of y2 of s, ds. 01:17:55.520 --> 01:17:57.700 align:middle line:90% So you have that. 01:17:57.700 --> 01:18:05.970 align:middle line:84% Now, I will look at this thing here and that thing here. 01:18:05.970 --> 01:18:07.220 align:middle line:90% And I'm going to-- 01:18:07.220 --> 01:18:10.530 align:middle line:90% and you have to now abstract. 01:18:10.530 --> 01:18:13.790 align:middle line:84% So now I'm going to look at this difference here. 01:18:13.790 --> 01:18:18.170 align:middle line:84% And then now, the thing is that you have to abstract from it, 01:18:18.170 --> 01:18:23.540 align:middle line:84% and you have to think about s as being-- 01:18:23.540 --> 01:18:28.040 align:middle line:84% if you think about s as being fixed, 01:18:28.040 --> 01:18:31.790 align:middle line:90% then this is just two numbers. 01:18:31.790 --> 01:18:37.970 align:middle line:84% You could think about this as z1 and this here as z2. 01:18:37.970 --> 01:18:40.950 align:middle line:90% This is z1, and this is z2. 01:18:40.950 --> 01:18:46.105 align:middle line:84% So it's like f of z1 minus f of z2. 01:18:46.105 --> 01:18:48.671 align:middle line:90% 01:18:48.671 --> 01:18:52.910 align:middle line:84% Now, the mean value theorem tells us that this thing here 01:18:52.910 --> 01:18:59.810 align:middle line:84% is equal to the derivative at some intermediate point, 01:18:59.810 --> 01:19:12.570 align:middle line:84% let's say, z3, where z3 is between z1 and z2, 01:19:12.570 --> 01:19:17.640 align:middle line:90% and then times z1 minus z2. 01:19:17.640 --> 01:19:23.400 align:middle line:90% 01:19:23.400 --> 01:19:30.210 align:middle line:84% So this is just for any xs, you have this. 01:19:30.210 --> 01:19:33.150 align:middle line:90% But this means that-- 01:19:33.150 --> 01:19:36.240 align:middle line:84% this means that-- are we are almost out of time? 01:19:36.240 --> 01:19:42.960 align:middle line:84% So this means that when you're looking at the norm of this, 01:19:42.960 --> 01:19:54.850 align:middle line:84% f of y1 of s minus f of y2 of s, that the norm here is-- 01:19:54.850 --> 01:20:04.390 align:middle line:84% so the norm is equal to f prime at z3 times z1. 01:20:04.390 --> 01:20:05.840 align:middle line:90% And z1 I can write in. 01:20:05.840 --> 01:20:07.010 align:middle line:90% It's really this. 01:20:07.010 --> 01:20:13.030 align:middle line:90% And this is z2. 01:20:13.030 --> 01:20:16.210 align:middle line:84% This thing here, remember, z is some point 01:20:16.210 --> 01:20:18.380 align:middle line:90% in between those two points. 01:20:18.380 --> 01:20:22.430 align:middle line:84% But both of these points, both of these points here, 01:20:22.430 --> 01:20:28.810 align:middle line:84% both this one here and that one here have norm less than R. 01:20:28.810 --> 01:20:31.850 align:middle line:90% So this here is-- 01:20:31.850 --> 01:20:37.390 align:middle line:84% but now where the way that M was defined, 01:20:37.390 --> 01:20:41.380 align:middle line:84% and it should have been on the same board. 01:20:41.380 --> 01:20:49.570 align:middle line:84% So M here was defined to be the maximum of f prime of z, 01:20:49.570 --> 01:20:53.860 align:middle line:90% where z was less or equal to 1. 01:20:53.860 --> 01:20:57.520 align:middle line:84% And so you see that this thing here where 01:20:57.520 --> 01:21:01.850 align:middle line:84% you take the derivative that is bounded by M. 01:21:01.850 --> 01:21:12.635 align:middle line:84% So this thing here is less than M y1 s minus y2 of s, like that. 01:21:12.635 --> 01:21:14.720 align:middle line:84% So I'm not sure where we should stop. 01:21:14.720 --> 01:21:17.280 align:middle line:84% We over, and I probably need a few more minutes. 01:21:17.280 --> 01:21:18.530 align:middle line:90% Maybe we should stop. 01:21:18.530 --> 01:21:25.520 align:middle line:84% And it's not a great place to stop, but that's how it goes. 01:21:25.520 --> 01:21:29.990 align:middle line:84% So I will keep you in suspension, 01:21:29.990 --> 01:21:37.020 align:middle line:84% and so we'll pick up from here on Tuesday. 01:21:37.020 --> 01:21:41.480 align:middle line:84% And so I want anyway to explain to you why, actually, 01:21:41.480 --> 01:21:45.530 align:middle line:84% this method, which seems a little crazy in a way 01:21:45.530 --> 01:21:48.650 align:middle line:84% because it's bringing in all these things, why it turned out 01:21:48.650 --> 01:21:51.980 align:middle line:90% to be basically optimal. 01:21:51.980 --> 01:21:56.830 align:middle line:84% And then we'll do a review the rest of the time on Tuesday. 01:21:56.830 --> 01:21:59.000 align:middle line:90%