WEBVTT 00:00:00.000 --> 00:00:01.440 align:middle line:90% [SQUEAKING] 00:00:01.440 --> 00:00:04.320 align:middle line:90% [RUSTLING] 00:00:04.320 --> 00:00:06.240 align:middle line:90% [CLICKING] 00:00:06.240 --> 00:00:10.487 align:middle line:90% 00:00:10.487 --> 00:00:12.570 align:middle line:84% PAIGE BRIGHT: Let's just go ahead and get started. 00:00:12.570 --> 00:00:17.060 align:middle line:84% So as a recap of what we've been up to, the first day, 00:00:17.060 --> 00:00:19.048 align:middle line:84% we talked about what a metric space is and went 00:00:19.048 --> 00:00:21.590 align:middle line:84% through a ton of examples, which was mostly what that day was 00:00:21.590 --> 00:00:22.645 align:middle line:90% supposed to be for. 00:00:22.645 --> 00:00:25.020 align:middle line:84% The next day, we went through some of the general theory, 00:00:25.020 --> 00:00:27.103 align:middle line:90% which was helpful, I believe. 00:00:27.103 --> 00:00:29.270 align:middle line:84% But it's really short, all things considered, right? 00:00:29.270 --> 00:00:32.990 align:middle line:84% It's just a single lecture of most of what real analysis took 00:00:32.990 --> 00:00:34.970 align:middle line:84% up for us, right-- what are sequences, what are 00:00:34.970 --> 00:00:36.620 align:middle line:90% Cauchy sequences, et cetera. 00:00:36.620 --> 00:00:39.200 align:middle line:84% That's what the beauty of metric spaces 00:00:39.200 --> 00:00:42.260 align:middle line:84% is to cover now, because because we 00:00:42.260 --> 00:00:44.180 align:middle line:84% have done all of that work in real analysis, 00:00:44.180 --> 00:00:47.090 align:middle line:84% now we can simply state and prove the theorems 00:00:47.090 --> 00:00:49.190 align:middle line:84% that we know and love for metric spaces 00:00:49.190 --> 00:00:51.410 align:middle line:84% because they're very, very much analogous. 00:00:51.410 --> 00:00:53.060 align:middle line:84% The one thing that wasn't analogous, 00:00:53.060 --> 00:00:55.700 align:middle line:84% though, was compact metric spaces. 00:00:55.700 --> 00:00:57.920 align:middle line:84% If you've never seen compact metric spaces, 00:00:57.920 --> 00:01:00.950 align:middle line:84% that's totally fair because if you only studied it-- 00:01:00.950 --> 00:01:03.590 align:middle line:84% if you only studied real analysis on Euclidean space, 00:01:03.590 --> 00:01:05.239 align:middle line:84% they're just closed and bounded sets. 00:01:05.239 --> 00:01:09.180 align:middle line:84% And those you can define without compactness. 00:01:09.180 --> 00:01:11.690 align:middle line:84% But there's clearly a ton of important statements 00:01:11.690 --> 00:01:13.460 align:middle line:90% you can make about compactness. 00:01:13.460 --> 00:01:16.220 align:middle line:84% We had the four-part huge theorem last time, 00:01:16.220 --> 00:01:18.830 align:middle line:84% where we showed that topologically compact is 00:01:18.830 --> 00:01:20.750 align:middle line:84% the same as sequentially compact, which 00:01:20.750 --> 00:01:23.270 align:middle line:84% was the same as totally bounded and Cauchy complete, 00:01:23.270 --> 00:01:26.340 align:middle line:84% and another property with the finite intersection property. 00:01:26.340 --> 00:01:28.130 align:middle line:84% But the main thing I want to talk 00:01:28.130 --> 00:01:30.620 align:middle line:84% about today is complete metric spaces 00:01:30.620 --> 00:01:33.020 align:middle line:84% because there's a ton can say about metric spaces 00:01:33.020 --> 00:01:34.640 align:middle line:84% once you know that they're complete. 00:01:34.640 --> 00:01:38.480 align:middle line:84% In particular, I want to talk about two specific examples-- 00:01:38.480 --> 00:01:40.400 align:middle line:84% one, the Banach fixed point theorem, 00:01:40.400 --> 00:01:43.730 align:middle line:84% which is a very useful theorem for differential equations. 00:01:43.730 --> 00:01:47.090 align:middle line:84% But secondly, I'm going to then talk about completions 00:01:47.090 --> 00:01:50.220 align:middle line:84% of metric spaces, which show up all of the time. 00:01:50.220 --> 00:01:55.360 align:middle line:84% So, yeah, let's go ahead and jump right in. 00:01:55.360 --> 00:01:59.345 align:middle line:84% So to motivate the conversation of the Banach fixed point 00:01:59.345 --> 00:02:01.470 align:middle line:84% theorem, I'm going to start by introducing what are 00:02:01.470 --> 00:02:03.120 align:middle line:90% known as Lipschitz functions-- 00:02:03.120 --> 00:02:08.380 align:middle line:90% 00:02:08.380 --> 00:02:11.200 align:middle line:90% Lipschitz. 00:02:11.200 --> 00:02:13.850 align:middle line:84% So a Lipschitz function is one that is continuous. 00:02:13.850 --> 00:02:14.560 align:middle line:90% It's pretty nice. 00:02:14.560 --> 00:02:17.280 align:middle line:90% It's a function f such that-- 00:02:17.280 --> 00:02:19.095 align:middle line:84% it's a function f from a metric space 00:02:19.095 --> 00:02:26.280 align:middle line:84% x to y such that there exists a K in the real numbers 00:02:26.280 --> 00:02:33.640 align:middle line:84% such that the distance between the image points, 00:02:33.640 --> 00:02:43.000 align:middle line:84% f of x, f of y, is bounded by K times the distance between x 00:02:43.000 --> 00:02:45.370 align:middle line:90% and y. 00:02:45.370 --> 00:02:47.680 align:middle line:84% Now, let's think to ourselves what values of k 00:02:47.680 --> 00:02:50.110 align:middle line:84% are actually interesting here for a moment. 00:02:50.110 --> 00:02:53.440 align:middle line:84% One, if k is negative, then positive definiteness 00:02:53.440 --> 00:02:55.810 align:middle line:84% tells us that everything is just 0, which 00:02:55.810 --> 00:02:59.630 align:middle line:90% is particularly nice, right? 00:02:59.630 --> 00:03:01.220 align:middle line:84% But if it's positive, then we have 00:03:01.220 --> 00:03:02.712 align:middle line:90% to do a little bit more work. 00:03:02.712 --> 00:03:04.670 align:middle line:84% But, yeah, this is the definition of Lipschitz, 00:03:04.670 --> 00:03:06.530 align:middle line:84% or K-Lipschitz, which is that there exists 00:03:06.530 --> 00:03:09.170 align:middle line:84% a K such that the distance between the image points 00:03:09.170 --> 00:03:11.270 align:middle line:84% is less than or equal to K times the distance 00:03:11.270 --> 00:03:13.700 align:middle line:90% of the original points. 00:03:13.700 --> 00:03:14.330 align:middle line:90% All right. 00:03:14.330 --> 00:03:16.580 align:middle line:84% So I claim that this is continuous, 00:03:16.580 --> 00:03:18.380 align:middle line:84% or functions like this are continuous. 00:03:18.380 --> 00:03:21.170 align:middle line:84% And that fact is not too hard to show. 00:03:21.170 --> 00:03:31.560 align:middle line:84% So proposition-- "Lipschitz implies continuous. 00:03:31.560 --> 00:03:34.380 align:middle line:84% The proof of this is not too bad because we're 00:03:34.380 --> 00:03:40.930 align:middle line:84% going to let epsilon be bigger than 0. 00:03:40.930 --> 00:03:42.510 align:middle line:84% And we want to choose a delta such 00:03:42.510 --> 00:03:44.260 align:middle line:84% that the distance between the image points 00:03:44.260 --> 00:03:46.780 align:middle line:84% is less than epsilon, where the distance between these two 00:03:46.780 --> 00:03:48.250 align:middle line:90% points is less than delta. 00:03:48.250 --> 00:03:57.850 align:middle line:84% But we can choose delta to be equal to epsilon over K, 00:03:57.850 --> 00:04:01.000 align:middle line:84% where K is the specific constant that works for our Lipschitz 00:04:01.000 --> 00:04:02.097 align:middle line:90% function. 00:04:02.097 --> 00:04:04.680 align:middle line:84% So then what we'll have is that the distance between the image 00:04:04.680 --> 00:04:13.110 align:middle line:84% points, f of x, f of y, will be less than or equal to K times 00:04:13.110 --> 00:04:17.459 align:middle line:90% the distance between x and y. 00:04:17.459 --> 00:04:19.890 align:middle line:84% But this, under assumption, is going 00:04:19.890 --> 00:04:25.202 align:middle line:84% to be less than K times delta, which is equal to epsilon. 00:04:25.202 --> 00:04:27.660 align:middle line:84% So this shows that the distance between points in the image 00:04:27.660 --> 00:04:30.390 align:middle line:84% gets arbitrarily small once you choose a small enough region 00:04:30.390 --> 00:04:31.440 align:middle line:90% around x. 00:04:31.440 --> 00:04:33.900 align:middle line:90% So this implies continuous. 00:04:33.900 --> 00:04:35.700 align:middle line:84% But, in fact, it implies a little bit more 00:04:35.700 --> 00:04:36.690 align:middle line:90% than continuous. 00:04:36.690 --> 00:04:39.870 align:middle line:84% Does anyone know what it implies, the stronger thing? 00:04:39.870 --> 00:04:41.370 align:middle line:84% Totally fair if not-- it's something 00:04:41.370 --> 00:04:44.060 align:middle line:90% you might not have seen before. 00:04:44.060 --> 00:04:45.620 align:middle line:90% Yeah, it's uniform continuity. 00:04:45.620 --> 00:04:55.550 align:middle line:84% So, in fact, Lipschitz implies uniform continuity 00:04:55.550 --> 00:04:57.490 align:middle line:84% because it doesn't depend on x at all. 00:04:57.490 --> 00:04:58.865 align:middle line:84% It only depends on f and epsilon. 00:04:58.865 --> 00:05:01.740 align:middle line:90% 00:05:01.740 --> 00:05:05.390 align:middle line:84% So that's our next section of discussion. 00:05:05.390 --> 00:05:07.760 align:middle line:84% I'm going to redefine what uniform continuity means 00:05:07.760 --> 00:05:09.680 align:middle line:90% for metric spaces. 00:05:09.680 --> 00:05:11.930 align:middle line:84% But most of the theory will still hold up. 00:05:11.930 --> 00:05:17.460 align:middle line:84% So definition, uniform continuity-- 00:05:17.460 --> 00:05:22.540 align:middle line:90% 00:05:22.540 --> 00:05:29.800 align:middle line:84% a function f from metric space x to metric space y 00:05:29.800 --> 00:05:40.020 align:middle line:84% is uniformly continuous if, for all epsilon bigger than 0, 00:05:40.020 --> 00:05:42.360 align:middle line:84% there exists a delta bigger than 0 such 00:05:42.360 --> 00:05:47.220 align:middle line:84% that if the distance between two points x and y in x 00:05:47.220 --> 00:05:50.400 align:middle line:84% is less than delta, then the distance between the image 00:05:50.400 --> 00:05:57.100 align:middle line:84% points in y, f of x, f of y, is less than epsilon. 00:05:57.100 --> 00:05:59.170 align:middle line:84% And the key difference between this definition 00:05:59.170 --> 00:06:02.460 align:middle line:84% and the definition of continuity is that this delta does not 00:06:02.460 --> 00:06:04.930 align:middle line:90% depend on the point x. 00:06:04.930 --> 00:06:08.590 align:middle line:84% The definition of continuity is that for every x, 00:06:08.590 --> 00:06:11.160 align:middle line:90% there exists this statement. 00:06:11.160 --> 00:06:14.490 align:middle line:90% 00:06:14.490 --> 00:06:14.990 align:middle line:90% OK. 00:06:14.990 --> 00:06:18.590 align:middle line:90% 00:06:18.590 --> 00:06:20.330 align:middle line:84% So let's prove some nice statements 00:06:20.330 --> 00:06:22.580 align:middle line:84% about uniform continuity, including 00:06:22.580 --> 00:06:28.100 align:middle line:84% but not limited to how it is affected by compactness. 00:06:28.100 --> 00:06:40.020 align:middle line:84% So proposition-- suppose f is a function from x to y 00:06:40.020 --> 00:06:44.415 align:middle line:90% continuous, and x is compact. 00:06:44.415 --> 00:06:48.220 align:middle line:90% 00:06:48.220 --> 00:06:50.140 align:middle line:84% Then what we can show is that, in fact, 00:06:50.140 --> 00:06:52.410 align:middle line:84% f is uniformly continuous, which is something 00:06:52.410 --> 00:06:54.150 align:middle line:84% that we should expect, that we had 00:06:54.150 --> 00:06:59.280 align:middle line:84% continuity on bounded intervals implying uniform continuity-- 00:06:59.280 --> 00:07:01.290 align:middle line:84% so let's keep going-- uniform continuity 00:07:01.290 --> 00:07:03.160 align:middle line:90% on the open interval. 00:07:03.160 --> 00:07:08.770 align:middle line:84% So then, f is uniformly continuous. 00:07:08.770 --> 00:07:12.148 align:middle line:90% 00:07:12.148 --> 00:07:14.440 align:middle line:84% The proof of this, I think, is particularly interesting 00:07:14.440 --> 00:07:16.773 align:middle line:84% because it's going to introduce something that we talked 00:07:16.773 --> 00:07:18.520 align:middle line:90% about a few lectures ago-- 00:07:18.520 --> 00:07:19.920 align:middle line:90% or reintroduce something. 00:07:19.920 --> 00:07:22.630 align:middle line:84% We're going to use the little big number lemma. 00:07:22.630 --> 00:07:25.960 align:middle line:84% So proof-- first we're going to start 00:07:25.960 --> 00:07:29.470 align:middle line:84% by covering our metric space x by a bunch of balls, 00:07:29.470 --> 00:07:31.120 align:middle line:84% just like we would do if we're doing 00:07:31.120 --> 00:07:33.280 align:middle line:90% a proof using compactness. 00:07:33.280 --> 00:07:39.220 align:middle line:84% So for all c in x, there exists a delta perhaps 00:07:39.220 --> 00:07:44.440 align:middle line:84% depending on c, which I'll note was delta c, bigger than 0, 00:07:44.440 --> 00:07:49.120 align:middle line:84% such that the distance between x and c is less than delta c. 00:07:49.120 --> 00:07:53.870 align:middle line:90% 00:07:53.870 --> 00:07:58.137 align:middle line:84% Let epsilon be bigger than 0, I should say. 00:07:58.137 --> 00:08:00.470 align:middle line:84% So then this implies that the distance between the image 00:08:00.470 --> 00:08:03.200 align:middle line:90% points is less than epsilon. 00:08:03.200 --> 00:08:05.692 align:middle line:90% 00:08:05.692 --> 00:08:08.150 align:middle line:84% So here epsilon is going to be fixed throughout this proof. 00:08:08.150 --> 00:08:10.220 align:middle line:84% In fact, while I'm at it, I'm going 00:08:10.220 --> 00:08:13.317 align:middle line:84% to say that this is less than epsilon over 2 00:08:13.317 --> 00:08:15.650 align:middle line:84% because we're going to use the triangle inequality later 00:08:15.650 --> 00:08:17.650 align:middle line:90% on in this proof. 00:08:17.650 --> 00:08:21.580 align:middle line:84% So what we know, then, is that these balls of radius delta c 00:08:21.580 --> 00:08:26.350 align:middle line:84% are going to cover our set x because we're just taking 00:08:26.350 --> 00:08:28.104 align:middle line:90% a ball centered at every point. 00:08:28.104 --> 00:08:36.830 align:middle line:84% So the balls of radius delta c around c cover x. 00:08:36.830 --> 00:08:42.350 align:middle line:84% So what we know is by the Lesbegue number lemma, which 00:08:42.350 --> 00:08:44.443 align:middle line:84% was true for sequentially compact spaces, 00:08:44.443 --> 00:08:46.610 align:middle line:84% but we know that sequentially compact is the same as 00:08:46.610 --> 00:08:48.830 align:middle line:90% topologically compact. 00:08:48.830 --> 00:08:56.150 align:middle line:84% So by the Lesbegue lemma, there exists a delta bigger 00:08:56.150 --> 00:09:03.710 align:middle line:84% than 0 such that for all x in x, there 00:09:03.710 --> 00:09:10.850 align:middle line:84% exists a c in x such that the ball of radius delta around x 00:09:10.850 --> 00:09:17.077 align:middle line:84% is fully contained in the ball of radius delta c around c. 00:09:17.077 --> 00:09:18.660 align:middle line:84% Now, does this make sense to everyone? 00:09:18.660 --> 00:09:20.202 align:middle line:84% I know that this was a huge statement 00:09:20.202 --> 00:09:23.040 align:middle line:90% with a bunch of quantifiers. 00:09:23.040 --> 00:09:25.380 align:middle line:84% But this is what the Lesbegue numbers lemma says. 00:09:25.380 --> 00:09:29.460 align:middle line:84% We are covering our set x, this compact set x, 00:09:29.460 --> 00:09:30.570 align:middle line:90% with these balls. 00:09:30.570 --> 00:09:34.050 align:middle line:84% And we can choose a delta such that for any point in x, 00:09:34.050 --> 00:09:36.060 align:middle line:84% I can squeeze this ball of radius delta 00:09:36.060 --> 00:09:38.970 align:middle line:84% into one of these sets in our open cover. 00:09:38.970 --> 00:09:41.070 align:middle line:84% That's what the Lesbegue number lemma says. 00:09:41.070 --> 00:09:43.500 align:middle line:90% Now, why is this so helpful? 00:09:43.500 --> 00:09:46.740 align:middle line:90% Because we know that to do-- 00:09:46.740 --> 00:09:51.720 align:middle line:84% then we know that if the distance between x and y 00:09:51.720 --> 00:09:56.040 align:middle line:84% is less than delta, then, in fact, 00:09:56.040 --> 00:09:59.550 align:middle line:84% y is already in the ball of radius delta 00:09:59.550 --> 00:10:08.260 align:middle line:84% c, which lets us know that the distance between the image 00:10:08.260 --> 00:10:12.760 align:middle line:84% point of f of y and f of c is less than epsilon over 2. 00:10:12.760 --> 00:10:16.450 align:middle line:84% And so now we can directly imply the triangle inequality, 00:10:16.450 --> 00:10:18.670 align:middle line:84% where here we'll have that the distance in y 00:10:18.670 --> 00:10:23.190 align:middle line:90% between f of x and f of y-- 00:10:23.190 --> 00:10:27.970 align:middle line:84% when the distance between x and y is less than delta, 00:10:27.970 --> 00:10:32.760 align:middle line:84% then this distance is less than the distance between-- 00:10:32.760 --> 00:10:37.455 align:middle line:84% or sorry - equal to the distance between f of x and f 00:10:37.455 --> 00:10:45.870 align:middle line:84% of c plus the distance between f of c and f of y. 00:10:45.870 --> 00:10:49.260 align:middle line:84% But both of these are less than epsilon over 2. 00:10:49.260 --> 00:10:52.770 align:middle line:90% 00:10:52.770 --> 00:10:56.370 align:middle line:84% So this delta, remember, from the Lesbegue number lemma, 00:10:56.370 --> 00:10:57.900 align:middle line:90% does not depend on x. 00:10:57.900 --> 00:11:01.500 align:middle line:84% So we've proven uniform continuity. 00:11:01.500 --> 00:11:03.420 align:middle line:90% Everyone follow? 00:11:03.420 --> 00:11:04.380 align:middle line:90% Cool. 00:11:04.380 --> 00:11:07.320 align:middle line:84% I think that this is a very interesting statement. 00:11:07.320 --> 00:11:11.160 align:middle line:84% And let's look at an example of how we can actually use this. 00:11:11.160 --> 00:11:14.940 align:middle line:84% It's going to be a very powerful application. 00:11:14.940 --> 00:11:16.410 align:middle line:84% Specifically, it's an application 00:11:16.410 --> 00:11:19.650 align:middle line:84% to integral operators, which the name of it 00:11:19.650 --> 00:11:22.420 align:middle line:84% will become apparent by the proposition itself. 00:11:22.420 --> 00:11:28.380 align:middle line:84% So proposition-- let f be a function from the interval a, 00:11:28.380 --> 00:11:36.820 align:middle line:90% b cross c, d to R continuous. 00:11:36.820 --> 00:11:38.460 align:middle line:84% So, in fact, what we know right now 00:11:38.460 --> 00:11:40.835 align:middle line:84% is that this implies that it's uniformly continuous, 00:11:40.835 --> 00:11:41.460 align:middle line:90% which is great. 00:11:41.460 --> 00:11:43.300 align:middle line:90% That would be helpful. 00:11:43.300 --> 00:11:45.630 align:middle line:84% But two, what we're going to look at 00:11:45.630 --> 00:11:51.180 align:middle line:84% is the function g of y being equal to the integral from a 00:11:51.180 --> 00:11:55.440 align:middle line:90% to b of f of x, y dx. 00:11:55.440 --> 00:11:59.250 align:middle line:84% So, in other words, we're considering the function g of y 00:11:59.250 --> 00:12:02.680 align:middle line:84% where I plug in y into this integral. 00:12:02.680 --> 00:12:06.130 align:middle line:84% And now what I claim is that this function g-- 00:12:06.130 --> 00:12:10.130 align:middle line:84% so consider g such that this is true. 00:12:10.130 --> 00:12:15.020 align:middle line:90% I claim that g is continuous. 00:12:15.020 --> 00:12:17.360 align:middle line:84% And the proof of this is going to be relatively short 00:12:17.360 --> 00:12:19.790 align:middle line:84% because we're going to use uniform continuity 00:12:19.790 --> 00:12:21.463 align:middle line:90% for the integral. 00:12:21.463 --> 00:12:23.880 align:middle line:84% Now, if you haven't seen this before, it is very powerful. 00:12:23.880 --> 00:12:25.700 align:middle line:90% It's a powerful statement. 00:12:25.700 --> 00:12:27.620 align:middle line:84% And I'll say what the real analysis 00:12:27.620 --> 00:12:30.150 align:middle line:84% statement is that we're using in just a moment. 00:12:30.150 --> 00:12:33.500 align:middle line:84% But let's just go through the proof. 00:12:33.500 --> 00:12:48.950 align:middle line:84% Let yn be a sequence in cd such that yn converges to y. 00:12:48.950 --> 00:12:53.240 align:middle line:84% To show continuity by our second lecture on the general theory, 00:12:53.240 --> 00:12:59.882 align:middle line:90% we want to show that g of xy-- 00:12:59.882 --> 00:13:05.030 align:middle line:84% sorry, yn-- goes to g of y as n goes 00:13:05.030 --> 00:13:08.810 align:middle line:84% to infinity because this will imply that g is continuous, 00:13:08.810 --> 00:13:09.380 align:middle line:90% right? 00:13:09.380 --> 00:13:11.172 align:middle line:84% This goes with our statement of continuity, 00:13:11.172 --> 00:13:14.210 align:middle line:84% otherwise known as sequential continuity. 00:13:14.210 --> 00:13:15.710 align:middle line:84% The proof of this is not going to be 00:13:15.710 --> 00:13:18.335 align:middle line:84% so bad because we're just going to write out what the limit is. 00:13:18.335 --> 00:13:24.950 align:middle line:84% We're going to have the limit as n goes to infinity of g of yn. 00:13:24.950 --> 00:13:27.080 align:middle line:84% Well, this is the limit as n goes 00:13:27.080 --> 00:13:35.960 align:middle line:84% to infinity of the integral a to b, f of x yn dx by definition. 00:13:35.960 --> 00:13:38.960 align:middle line:84% But by uniform continuity, for real analysis, 00:13:38.960 --> 00:13:42.387 align:middle line:84% we know that we can swap the limit and the integral, right? 00:13:42.387 --> 00:13:43.970 align:middle line:84% Has everyone seen this theorem before? 00:13:43.970 --> 00:13:45.560 align:middle line:90% It's very, very useful. 00:13:45.560 --> 00:13:49.760 align:middle line:84% If you need to relearn it, I would highly suggest doing so. 00:13:49.760 --> 00:13:54.710 align:middle line:84% But it's one that is mostly used with Lesbegue integration. 00:13:54.710 --> 00:13:59.550 align:middle line:84% Or, specifically, the lemma uses Riemannian integration, 00:13:59.550 --> 00:14:01.310 align:middle line:84% which makes it kind of annoying to prove. 00:14:01.310 --> 00:14:02.910 align:middle line:90% So I will not do so here. 00:14:02.910 --> 00:14:05.570 align:middle line:84% But the point is that we can switch these two limits as n 00:14:05.570 --> 00:14:08.108 align:middle line:90% goes to infinity of f of x, yn. 00:14:08.108 --> 00:14:11.600 align:middle line:90% 00:14:11.600 --> 00:14:13.580 align:middle line:84% But by the fact that f is continuous, 00:14:13.580 --> 00:14:17.780 align:middle line:84% we know that this is going to converge to f of x, y. 00:14:17.780 --> 00:14:20.330 align:middle line:84% So this is integral from a to b of f 00:14:20.330 --> 00:14:24.830 align:middle line:90% of x, y dx, which is g of y. 00:14:24.830 --> 00:14:26.050 align:middle line:90% So we're done with the proof. 00:14:26.050 --> 00:14:29.910 align:middle line:90% 00:14:29.910 --> 00:14:31.080 align:middle line:90% It was a very simple proof. 00:14:31.080 --> 00:14:32.910 align:middle line:84% But we highlighted the importance 00:14:32.910 --> 00:14:34.290 align:middle line:90% of uniform continuity. 00:14:34.290 --> 00:14:37.500 align:middle line:84% We can simply state that we can swap the limits using 00:14:37.500 --> 00:14:38.610 align:middle line:90% uniform continuity. 00:14:38.610 --> 00:14:41.490 align:middle line:90% And then our job is nearly done. 00:14:41.490 --> 00:14:43.170 align:middle line:84% But this integral operator is, in fact, 00:14:43.170 --> 00:14:46.030 align:middle line:84% going to come up a little bit later today. 00:14:46.030 --> 00:14:49.080 align:middle line:84% There's one on the face of it that is pretty straightforward 00:14:49.080 --> 00:14:53.470 align:middle line:84% but has very useful applications to differential equations. 00:14:53.470 --> 00:14:54.960 align:middle line:90% All right. 00:14:54.960 --> 00:14:58.040 align:middle line:84% So now we're turning back to the definition 00:14:58.040 --> 00:14:59.630 align:middle line:90% of Lipschitz functions. 00:14:59.630 --> 00:15:01.995 align:middle line:84% I'm going to define what a contraction is, which 00:15:01.995 --> 00:15:03.870 align:middle line:84% is going to lead us to the Banach fixed point 00:15:03.870 --> 00:15:08.040 align:middle line:84% theorem, which is our main goal of this first section. 00:15:08.040 --> 00:15:09.660 align:middle line:90% OK. 00:15:09.660 --> 00:15:19.860 align:middle line:84% Definition, contraction-- a function f from a metric space 00:15:19.860 --> 00:15:30.630 align:middle line:84% x to itself is a contraction if it 00:15:30.630 --> 00:15:39.210 align:middle line:84% is k-Lipschitz for k between 0 and 1, 00:15:39.210 --> 00:15:42.740 align:middle line:84% for k bigger than or equal to 0 and less than 1. 00:15:42.740 --> 00:15:49.760 align:middle line:84% In other words, the distance between two points f of x and f 00:15:49.760 --> 00:15:54.590 align:middle line:84% of y is less than or equal to k times the distance 00:15:54.590 --> 00:15:59.930 align:middle line:84% in x of x to y, where here k, small k, is between 0 and 1. 00:15:59.930 --> 00:16:01.910 align:middle line:84% Now, why is this called a contraction? 00:16:01.910 --> 00:16:06.930 align:middle line:84% Well, we can just sort of draw a picture of what's happening. 00:16:06.930 --> 00:16:10.920 align:middle line:84% If this is my metric space x, and this is my initial points x 00:16:10.920 --> 00:16:16.410 align:middle line:84% and y, I know that the distance between the images of them 00:16:16.410 --> 00:16:20.340 align:middle line:84% must be less than k times the distance from x to y. 00:16:20.340 --> 00:16:25.500 align:middle line:84% So if this is the distance from x to y, then for k 00:16:25.500 --> 00:16:29.970 align:middle line:84% really small, this will be the distance between f of x and f 00:16:29.970 --> 00:16:31.920 align:middle line:90% of y. 00:16:31.920 --> 00:16:33.570 align:middle line:84% And we can continue applying this 00:16:33.570 --> 00:16:35.050 align:middle line:90% over and over and over again. 00:16:35.050 --> 00:16:36.240 align:middle line:90% So what is this-- 00:16:36.240 --> 00:16:39.333 align:middle line:84% what do you think will actually happen in the end? 00:16:39.333 --> 00:16:41.750 align:middle line:84% Well, these points are going to get really close together. 00:16:41.750 --> 00:16:44.590 align:middle line:84% So our question is, does there exist a point such 00:16:44.590 --> 00:16:47.800 align:middle line:84% that its image is the same as its initial value? 00:16:47.800 --> 00:16:50.470 align:middle line:84% And I'll state that more explicitly in a moment. 00:16:50.470 --> 00:16:53.665 align:middle line:84% Specifically, we want to know if there exists a fixed point. 00:16:53.665 --> 00:16:56.197 align:middle line:90% 00:16:56.197 --> 00:16:58.655 align:middle line:84% I should have probably drawn that bigger, but that's fine-- 00:16:58.655 --> 00:16:59.780 align:middle line:90% a fixed point. 00:16:59.780 --> 00:17:04.400 align:middle line:90% 00:17:04.400 --> 00:17:08.450 align:middle line:84% If f is, again, a function from x to itself, 00:17:08.450 --> 00:17:19.066 align:middle line:84% x is a fixed point if f of x equals x. 00:17:19.066 --> 00:17:22.640 align:middle line:90% 00:17:22.640 --> 00:17:23.970 align:middle line:90% All right. 00:17:23.970 --> 00:17:26.339 align:middle line:84% So our claim for the Banach fixed point theorem, 00:17:26.339 --> 00:17:29.490 align:middle line:84% as you can probably guess from this image that I've drawn, 00:17:29.490 --> 00:17:32.910 align:middle line:84% is going to be that when I have a contraction on a Cauchy 00:17:32.910 --> 00:17:35.640 align:middle line:84% complete metric space, there is, in fact, going 00:17:35.640 --> 00:17:37.638 align:middle line:90% to be a fixed point. 00:17:37.638 --> 00:17:39.180 align:middle line:84% And let me write that out explicitly. 00:17:39.180 --> 00:17:41.388 align:middle line:84% Actually, this is more of a theorem-- sorry, theorem. 00:17:41.388 --> 00:17:43.690 align:middle line:90% 00:17:43.690 --> 00:17:45.440 align:middle line:84% It's called the Banach fixed point theorem 00:17:45.440 --> 00:17:50.840 align:middle line:84% but also sometimes called the contraction mapping theorem-- 00:17:50.840 --> 00:17:54.860 align:middle line:84% so whichever name you prefer to use. 00:17:54.860 --> 00:17:57.320 align:middle line:84% I usually use contraction fixed point-- 00:17:57.320 --> 00:17:59.570 align:middle line:84% sorry, contraction mapping theorem in the notes 00:17:59.570 --> 00:18:03.170 align:middle line:84% just because it's slightly shorter. 00:18:03.170 --> 00:18:26.070 align:middle line:84% So let x be a Cauchy complete metric space and f-- 00:18:26.070 --> 00:18:26.670 align:middle line:90% oh, sorry. 00:18:26.670 --> 00:18:28.800 align:middle line:84% It also needs to be nonempty-- because otherwise, 00:18:28.800 --> 00:18:33.630 align:middle line:84% if it's empty, then what does having a fixed point even mean? 00:18:33.630 --> 00:18:37.380 align:middle line:84% How can a point be fixed if there is no points in x?-- 00:18:37.380 --> 00:18:42.570 align:middle line:84% and then that f be a map from x to x, a contraction. 00:18:42.570 --> 00:18:46.830 align:middle line:90% 00:18:46.830 --> 00:18:50.520 align:middle line:84% Then the claim is that there exists a fixed point. 00:18:50.520 --> 00:18:54.600 align:middle line:90% 00:18:54.600 --> 00:18:57.620 align:middle line:84% In fact, it's going to be unique, 00:18:57.620 --> 00:18:59.180 align:middle line:84% which will be an interesting part 00:18:59.180 --> 00:19:03.000 align:middle line:84% of this proof, an interesting yet short part of this proof. 00:19:03.000 --> 00:19:05.100 align:middle line:84% Now let's restate what we already just 00:19:05.100 --> 00:19:08.130 align:middle line:84% so that we can make sure that we're all on the same page. 00:19:08.130 --> 00:19:10.740 align:middle line:84% We know that by definition, a contraction 00:19:10.740 --> 00:19:12.480 align:middle line:90% is a k-Lipschitz map. 00:19:12.480 --> 00:19:15.150 align:middle line:84% And as we've shown already up above, 00:19:15.150 --> 00:19:17.580 align:middle line:84% Lipschitz functions are uniformly continuous. 00:19:17.580 --> 00:19:19.320 align:middle line:84% Are we going to use this in our proof? 00:19:19.320 --> 00:19:20.640 align:middle line:90% Not explicitly. 00:19:20.640 --> 00:19:22.620 align:middle line:84% But it is helpful to just keep in mind, 00:19:22.620 --> 00:19:24.420 align:middle line:84% like, what are we actually doing here? 00:19:24.420 --> 00:19:26.820 align:middle line:84% We have a continuous map that just so happens 00:19:26.820 --> 00:19:28.740 align:middle line:84% to be a contraction, and we're going 00:19:28.740 --> 00:19:31.480 align:middle line:84% to show that there exists a fixed point. 00:19:31.480 --> 00:19:32.200 align:middle line:90% All right. 00:19:32.200 --> 00:19:34.930 align:middle line:84% The proof of this is relatively short. 00:19:34.930 --> 00:19:38.980 align:middle line:84% We're going to utilize Cauchy completeness. 00:19:38.980 --> 00:19:43.030 align:middle line:84% That's the main thing that we're going to utilize in this proof. 00:19:43.030 --> 00:19:45.580 align:middle line:84% We want to construct a function, or we 00:19:45.580 --> 00:19:46.960 align:middle line:84% want to construct a sequence that 00:19:46.960 --> 00:19:49.540 align:middle line:84% is Cauchy complete and therefore will have a limit. 00:19:49.540 --> 00:19:52.030 align:middle line:84% And that limit will be the fixed point. 00:19:52.030 --> 00:19:55.870 align:middle line:84% And to do so, we're just going to pick-- 00:19:55.870 --> 00:20:05.070 align:middle line:84% oh, proof-- pick arbitrary x0 big X. 00:20:05.070 --> 00:20:07.230 align:middle line:84% And this is where we use that it's nonempty. 00:20:07.230 --> 00:20:10.770 align:middle line:84% The fact that we can pick any x0 X has to do with the fact 00:20:10.770 --> 00:20:11.730 align:middle line:90% that it's nonempty. 00:20:11.730 --> 00:20:15.700 align:middle line:84% The fact that it's nonempty is relatively a small moot point. 00:20:15.700 --> 00:20:20.220 align:middle line:84% But I do want to iterate that this assumption is necessary. 00:20:20.220 --> 00:20:26.610 align:middle line:84% And then what we're going to do is define xn plus 1 00:20:26.610 --> 00:20:28.660 align:middle line:90% to be f of xn. 00:20:28.660 --> 00:20:38.870 align:middle line:84% So our sequence will be x0, f of x0, f of f of x0, and so on. 00:20:38.870 --> 00:20:40.760 align:middle line:84% This is going to be our sequence. 00:20:40.760 --> 00:20:42.980 align:middle line:84% And our hope is that this is going to converge, 00:20:42.980 --> 00:20:45.590 align:middle line:84% i.e., hopefully, it will be Cauchy. 00:20:45.590 --> 00:20:47.990 align:middle line:84% And the proof of this is pretty straightforward. 00:20:47.990 --> 00:20:53.060 align:middle line:84% Well, we'll note that the distance between xi 00:20:53.060 --> 00:20:55.700 align:middle line:90% plus 1 and xi-- 00:20:55.700 --> 00:20:57.660 align:middle line:90% what can we state about this? 00:20:57.660 --> 00:21:00.290 align:middle line:84% Well, firstly, we can plug in what the definitions of them 00:21:00.290 --> 00:21:01.490 align:middle line:90% are. 00:21:01.490 --> 00:21:05.210 align:middle line:84% This is the distance between f of xi 00:21:05.210 --> 00:21:08.510 align:middle line:84% and the distance between f of xi minus 1. 00:21:08.510 --> 00:21:12.080 align:middle line:90% 00:21:12.080 --> 00:21:13.940 align:middle line:84% And what we know is by the fact that it's 00:21:13.940 --> 00:21:18.500 align:middle line:84% a contraction is that this is less than or equal to k 00:21:18.500 --> 00:21:24.800 align:middle line:84% times the distance of xi to xi minus 1. 00:21:24.800 --> 00:21:26.810 align:middle line:84% And I can continue reiterating this 00:21:26.810 --> 00:21:30.380 align:middle line:84% over and over and over again, where here I'll get less than 00:21:30.380 --> 00:21:35.300 align:middle line:84% or equal to k to the i of the distance from-- 00:21:35.300 --> 00:21:36.500 align:middle line:90% i or xi? 00:21:36.500 --> 00:21:43.220 align:middle line:84% Yeah, i-- distance from x1 to x0. 00:21:43.220 --> 00:21:45.190 align:middle line:90% Everyone follow? 00:21:45.190 --> 00:21:47.430 align:middle line:84% And we're going to use this to consider-- 00:21:47.430 --> 00:21:49.470 align:middle line:84% to show that our sequence is in fact Cauchy, 00:21:49.470 --> 00:21:53.700 align:middle line:84% we're going to look at the distance between xn and xm. 00:21:53.700 --> 00:21:58.390 align:middle line:90% So I'll do that here. 00:21:58.390 --> 00:22:03.450 align:middle line:84% We're considering the distance from xn to xm, 00:22:03.450 --> 00:22:05.640 align:middle line:84% and we want to imply this simple fact 00:22:05.640 --> 00:22:08.730 align:middle line:84% between the distances between two nearby points, two neighbor 00:22:08.730 --> 00:22:09.660 align:middle line:90% points. 00:22:09.660 --> 00:22:11.160 align:middle line:84% Well, to do that, we can just look 00:22:11.160 --> 00:22:13.750 align:middle line:90% at the triangle inequality. 00:22:13.750 --> 00:22:16.920 align:middle line:84% We can write this as less than or equal to the distance 00:22:16.920 --> 00:22:26.090 align:middle line:84% from xi plus 1 to xi, where i goes from n to m minus 1. 00:22:26.090 --> 00:22:28.940 align:middle line:84% Does everyone see how I did this step? 00:22:28.940 --> 00:22:31.130 align:middle line:84% You can expand it out to see that it is precisely 00:22:31.130 --> 00:22:32.370 align:middle line:90% the triangle inequality. 00:22:32.370 --> 00:22:34.490 align:middle line:84% This is less than or equal to the distance from x 00:22:34.490 --> 00:22:38.090 align:middle line:84% to xn plus 1, xn plus 1 to xn plus 2, so on and so 00:22:38.090 --> 00:22:39.560 align:middle line:90% forth until we get to m. 00:22:39.560 --> 00:22:41.510 align:middle line:84% And here I'm assuming that m is bigger than n 00:22:41.510 --> 00:22:44.490 align:middle line:84% because it doesn't particularly matter. 00:22:44.490 --> 00:22:45.090 align:middle line:90% OK. 00:22:45.090 --> 00:22:48.690 align:middle line:84% Well, then, this is less than or equal to the sum of k 00:22:48.690 --> 00:22:54.780 align:middle line:84% to the i times the distance from x1 to x0, which is great. 00:22:54.780 --> 00:22:58.590 align:middle line:84% Notice already that this distance is just fixed. 00:22:58.590 --> 00:23:01.100 align:middle line:90% It doesn't depend on i at all. 00:23:01.100 --> 00:23:05.390 align:middle line:84% But now we want to figure out how we deal with the sum from n 00:23:05.390 --> 00:23:06.440 align:middle line:90% to mx1. 00:23:06.440 --> 00:23:09.120 align:middle line:84% And to do so, we can apply a pretty nice trick, 00:23:09.120 --> 00:23:11.780 align:middle line:90% which is just multiply by kn. 00:23:11.780 --> 00:23:14.540 align:middle line:84% This will be less than or equal to kn-- 00:23:14.540 --> 00:23:18.080 align:middle line:84% and I'll pull out the distance x1, x0-- 00:23:18.080 --> 00:23:24.215 align:middle line:84% of the sum from i equals 0 to m minus 1 minus n-- 00:23:24.215 --> 00:23:25.370 align:middle line:90% right, minus n? 00:23:25.370 --> 00:23:28.010 align:middle line:90% Yeah-- of k to the i. 00:23:28.010 --> 00:23:30.290 align:middle line:84% So here I'm just multiplying by k to the n such 00:23:30.290 --> 00:23:34.980 align:middle line:84% that when I plug it back in, I get the same exact sequence. 00:23:34.980 --> 00:23:37.840 align:middle line:90% And so what can we do? 00:23:37.840 --> 00:23:40.220 align:middle line:84% This is then just a geometric series. 00:23:40.220 --> 00:23:42.880 align:middle line:84% This is less than or equal to k to the n times 00:23:42.880 --> 00:23:46.930 align:middle line:84% the distance from x1 to x0 of the sum from i 00:23:46.930 --> 00:23:51.790 align:middle line:84% equals 0 to infinity of ki, which is simply 00:23:51.790 --> 00:23:57.190 align:middle line:84% k to the n over 1 minus k of the distance from x1 to x0, 00:23:57.190 --> 00:24:00.640 align:middle line:84% where here I'm applying the geometric series formula 00:24:00.640 --> 00:24:03.000 align:middle line:90% to this geometric series. 00:24:03.000 --> 00:24:06.970 align:middle line:84% And now we're going to use the fact that-- 00:24:06.970 --> 00:24:09.880 align:middle line:84% now we're going to use the fact that k is less than 1. 00:24:09.880 --> 00:24:13.240 align:middle line:84% Or, specifically, it's between 0 and 1, right? 00:24:13.240 --> 00:24:14.830 align:middle line:90% I can choose n large enough-- 00:24:14.830 --> 00:24:18.930 align:middle line:90% 00:24:18.930 --> 00:24:29.400 align:middle line:84% large enough so that this term is less than epsilon. 00:24:29.400 --> 00:24:32.220 align:middle line:84% I'm not going to state explicitly what this n is. 00:24:32.220 --> 00:24:34.200 align:middle line:84% You can figure this out independently. 00:24:34.200 --> 00:24:37.590 align:middle line:84% But we know that one minus exist, because k to the n 00:24:37.590 --> 00:24:39.480 align:middle line:90% is going to be going to 0. 00:24:39.480 --> 00:24:42.060 align:middle line:90% So does this complete our proof? 00:24:42.060 --> 00:24:42.745 align:middle line:90% Not fully. 00:24:42.745 --> 00:24:44.370 align:middle line:84% We have that this is a Cauchy sequence. 00:24:44.370 --> 00:24:45.600 align:middle line:90% So we know it converges. 00:24:45.600 --> 00:24:48.510 align:middle line:84% We need to, one, show that the point it converges to 00:24:48.510 --> 00:24:52.310 align:middle line:84% is our fixed point, and two, show it's unique. 00:24:52.310 --> 00:24:53.750 align:middle line:90% So let's go ahead and do that. 00:24:53.750 --> 00:25:23.040 align:middle line:90% 00:25:23.040 --> 00:25:23.540 align:middle line:90% OK. 00:25:23.540 --> 00:25:26.360 align:middle line:84% So we want to show that this is, in fact, our limit point. 00:25:26.360 --> 00:25:27.890 align:middle line:90% And to do so-- 00:25:27.890 --> 00:25:32.290 align:middle line:84% or specifically, first I'll state 00:25:32.290 --> 00:25:42.650 align:middle line:84% we know there exists an x and X such that xn converges to x. 00:25:42.650 --> 00:25:43.590 align:middle line:90% How do we know this? 00:25:43.590 --> 00:25:45.350 align:middle line:90% Because it's Cauchy complete. 00:25:45.350 --> 00:25:46.565 align:middle line:90% I'll reiterate this point. 00:25:46.565 --> 00:25:48.440 align:middle line:84% When you're going through proofs of theorems, 00:25:48.440 --> 00:25:50.190 align:middle line:84% it's important to double-check that you're 00:25:50.190 --> 00:25:52.070 align:middle line:84% using the statements of your theorem 00:25:52.070 --> 00:25:55.990 align:middle line:84% in every possible-- at least once in your proof, 00:25:55.990 --> 00:25:58.700 align:middle line:84% or at least the implications of your statements. 00:25:58.700 --> 00:26:00.550 align:middle line:84% So we know that there exists an x such 00:26:00.550 --> 00:26:03.620 align:middle line:84% that xn converges to X by Cauchy completeness. 00:26:03.620 --> 00:26:07.180 align:middle line:84% And we want to show that x is its own fixed point. 00:26:07.180 --> 00:26:08.240 align:middle line:90% But how do we do that? 00:26:08.240 --> 00:26:15.460 align:middle line:84% Well, x is the limit as n goes to infinity of f of xn. 00:26:15.460 --> 00:26:18.160 align:middle line:84% This is how we literally defined what the fixed point was. 00:26:18.160 --> 00:26:20.780 align:middle line:84% Or this is how we defined our sequence. 00:26:20.780 --> 00:26:23.660 align:middle line:90% So what can we do? 00:26:23.660 --> 00:26:26.020 align:middle line:84% We can imply the fact that f is continuous 00:26:26.020 --> 00:26:27.430 align:middle line:90% because it's Lipschitz. 00:26:27.430 --> 00:26:31.150 align:middle line:84% So this is the same as f of the limit as n 00:26:31.150 --> 00:26:35.710 align:middle line:84% goes to infinity of xn, which is f of x. 00:26:35.710 --> 00:26:38.053 align:middle line:84% So we've shown that it's a fixed point. 00:26:38.053 --> 00:26:39.220 align:middle line:90% It's not too bad of a proof. 00:26:39.220 --> 00:26:41.540 align:middle line:84% But it does utilize the fact that it's continuous. 00:26:41.540 --> 00:26:44.560 align:middle line:84% So that's why I talked about that earlier. 00:26:44.560 --> 00:26:48.490 align:middle line:84% Secondly, we want to show that this fixed point is, in fact, 00:26:48.490 --> 00:26:49.120 align:middle line:90% unique. 00:26:49.120 --> 00:26:51.400 align:middle line:84% And the way to do so is very similar to what 00:26:51.400 --> 00:26:53.060 align:middle line:90% we've done before. 00:26:53.060 --> 00:27:06.060 align:middle line:84% So, two, let y be an x such that y is equal to f of y. 00:27:06.060 --> 00:27:07.270 align:middle line:90% Then what will we do? 00:27:07.270 --> 00:27:10.740 align:middle line:84% Well, we will have that the distance from x to y 00:27:10.740 --> 00:27:17.040 align:middle line:84% will be equal to the distance from x to f of x, f of y. 00:27:17.040 --> 00:27:20.460 align:middle line:84% And then we can apply the fact that it's a contraction, right? 00:27:20.460 --> 00:27:25.110 align:middle line:84% This will be less than or equal to k times 00:27:25.110 --> 00:27:30.240 align:middle line:84% the distance from f-- sorry, of f x to y. 00:27:30.240 --> 00:27:31.710 align:middle line:90% And we can move this term over. 00:27:31.710 --> 00:27:37.310 align:middle line:84% This will be 1 minus k times the distance from x to y. 00:27:37.310 --> 00:27:40.260 align:middle line:90% It's less than or equal to 0. 00:27:40.260 --> 00:27:42.160 align:middle line:90% Now, what does this tell us? 00:27:42.160 --> 00:27:45.150 align:middle line:84% Well, 1 minus k is going to be nonzero, right? 00:27:45.150 --> 00:27:47.370 align:middle line:90% k is between 0 and 1. 00:27:47.370 --> 00:27:48.840 align:middle line:90% And it doesn't include 1. 00:27:48.840 --> 00:27:51.510 align:middle line:84% So what this tells us is then that the distance between x 00:27:51.510 --> 00:27:56.560 align:middle line:84% and y is 0, which, by our metric space theory, 00:27:56.560 --> 00:28:00.570 align:middle line:90% implies that x equals 5. 00:28:00.570 --> 00:28:03.300 align:middle line:84% Now, I'm going to quote the book Lebl for a moment 00:28:03.300 --> 00:28:05.940 align:middle line:84% here because I think that he states it very well. 00:28:05.940 --> 00:28:08.250 align:middle line:84% "Not only does this proof tell us 00:28:08.250 --> 00:28:10.620 align:middle line:84% that there exists a fixed point, but this proof 00:28:10.620 --> 00:28:11.370 align:middle line:90% is constructive." 00:28:11.370 --> 00:28:13.980 align:middle line:84% It tells you how to find the fixed point itself. 00:28:13.980 --> 00:28:16.290 align:middle line:84% You just reiterate the process of applying the function 00:28:16.290 --> 00:28:19.608 align:middle line:84% to itself over and over and over again, which, 00:28:19.608 --> 00:28:21.150 align:middle line:84% if you're in the math lecture series, 00:28:21.150 --> 00:28:23.025 align:middle line:84% is something that Professor Staffilani talked 00:28:23.025 --> 00:28:26.880 align:middle line:84% about in the conversation of the nonlinear Schwartz equation. 00:28:26.880 --> 00:28:30.510 align:middle line:84% I think it's very interesting and, in fact, 00:28:30.510 --> 00:28:35.850 align:middle line:84% has something to do with differential operators. 00:28:35.850 --> 00:28:38.910 align:middle line:84% So, to that point, I'm going to go through a proposition that 00:28:38.910 --> 00:28:43.860 align:middle line:84% is very directly tied to differential operators, 00:28:43.860 --> 00:28:46.180 align:middle line:90% or differential equations. 00:28:46.180 --> 00:28:56.720 align:middle line:84% So an example-- let lambda be in the real numbers and f 00:28:56.720 --> 00:29:00.110 align:middle line:90% and g be continuous on a to b. 00:29:00.110 --> 00:29:04.410 align:middle line:90% 00:29:04.410 --> 00:29:06.360 align:middle line:84% I'm going to define one more function. 00:29:06.360 --> 00:29:12.920 align:middle line:84% I'm going to let k be continuous on the interval a, b cross a, 00:29:12.920 --> 00:29:13.420 align:middle line:90% b. 00:29:13.420 --> 00:29:17.770 align:middle line:84% So k here is going to depend on two parameters 00:29:17.770 --> 00:29:18.430 align:middle line:90% OK. 00:29:18.430 --> 00:29:21.520 align:middle line:84% So then my question is, for which 00:29:21.520 --> 00:29:33.940 align:middle line:84% lambda is Tf of x equal to g of x plus the integral 00:29:33.940 --> 00:29:40.780 align:middle line:84% from a to b of k x, y f of x dx-- 00:29:40.780 --> 00:29:45.230 align:middle line:84% for which lambda-- sorry, lambda-- 00:29:45.230 --> 00:29:46.550 align:middle line:90% is this a contraction? 00:29:46.550 --> 00:29:51.770 align:middle line:90% 00:29:51.770 --> 00:29:53.160 align:middle line:90% I know it's a long statement. 00:29:53.160 --> 00:29:55.070 align:middle line:84% And I know I haven't largely motivated it. 00:29:55.070 --> 00:29:57.140 align:middle line:84% And I'll get to why the motivation is 00:29:57.140 --> 00:29:58.460 align:middle line:90% at the end of the proof. 00:29:58.460 --> 00:30:00.840 align:middle line:84% But let's think about this for a moment. 00:30:00.840 --> 00:30:03.320 align:middle line:84% I have this integral operator where what it's doing 00:30:03.320 --> 00:30:05.810 align:middle line:84% is it's taking in a function f and spitting out 00:30:05.810 --> 00:30:08.390 align:middle line:84% g of x plus lambda times f applied 00:30:08.390 --> 00:30:10.610 align:middle line:90% to this integral operator. 00:30:10.610 --> 00:30:15.710 align:middle line:84% This is pretty directly applied to differential equations. 00:30:15.710 --> 00:30:17.390 align:middle line:84% The second thing I want to know is 00:30:17.390 --> 00:30:20.640 align:middle line:84% that in order for it to be a contraction, by definition, 00:30:20.640 --> 00:30:24.570 align:middle line:84% it has to be a map from the set to itself, right? 00:30:24.570 --> 00:30:26.622 align:middle line:84% A contraction is a map from x to itself. 00:30:26.622 --> 00:30:28.080 align:middle line:84% So the first question we should ask 00:30:28.080 --> 00:30:31.950 align:middle line:84% ourselves is that if it takes in a function that's continuous, 00:30:31.950 --> 00:30:35.260 align:middle line:84% does it spit out something that's continuous? 00:30:35.260 --> 00:30:36.790 align:middle line:90% The answer will be yes. 00:30:36.790 --> 00:30:38.510 align:middle line:90% How do we see that? 00:30:38.510 --> 00:30:40.010 align:middle line:90% Well, g of x is continuous. 00:30:40.010 --> 00:30:41.380 align:middle line:90% So we know that that's all good. 00:30:41.380 --> 00:30:44.560 align:middle line:84% But secondly, we've already shown that the integral from a 00:30:44.560 --> 00:30:47.680 align:middle line:84% to b of a continuous function is going to be continuous 00:30:47.680 --> 00:30:50.370 align:middle line:90% when I plug in the parameter y. 00:30:50.370 --> 00:30:51.550 align:middle line:90% That was our-- 00:30:51.550 --> 00:30:54.940 align:middle line:90% I think I might have erased it. 00:30:54.940 --> 00:30:55.900 align:middle line:90% Yeah, I did. 00:30:55.900 --> 00:30:58.593 align:middle line:90% 00:30:58.593 --> 00:31:00.010 align:middle line:84% But that was one of our statements 00:31:00.010 --> 00:31:02.668 align:middle line:84% was that this is, in fact, going to be continuous. 00:31:02.668 --> 00:31:04.960 align:middle line:84% So we know that this is a map from continuous functions 00:31:04.960 --> 00:31:06.387 align:middle line:90% to itself. 00:31:06.387 --> 00:31:08.470 align:middle line:84% And that's what makes this question so interesting 00:31:08.470 --> 00:31:11.170 align:middle line:84% because now we're looking at specific functions. 00:31:11.170 --> 00:31:14.140 align:middle line:84% We're not looking at the distance between x and y. 00:31:14.140 --> 00:31:18.250 align:middle line:84% We're looking at the distance between functions f and g. 00:31:18.250 --> 00:31:20.350 align:middle line:90% I want to reiterate this point. 00:31:20.350 --> 00:31:22.870 align:middle line:84% Metric spaces give us the ability 00:31:22.870 --> 00:31:25.295 align:middle line:84% to view functions as single points. 00:31:25.295 --> 00:31:27.295 align:middle line:84% There are things that we can manipulate and plug 00:31:27.295 --> 00:31:29.740 align:middle line:84% in and figure out the distances between. 00:31:29.740 --> 00:31:31.330 align:middle line:84% Now we're no longer viewing functions 00:31:31.330 --> 00:31:33.970 align:middle line:84% as things that take in inputs and spit out outputs, 00:31:33.970 --> 00:31:35.740 align:middle line:84% but rather things that we can manipulate. 00:31:35.740 --> 00:31:39.040 align:middle line:84% And this viewpoint-- deeply influential for all parts 00:31:39.040 --> 00:31:41.660 align:middle line:90% of mathematics moving forward. 00:31:41.660 --> 00:31:43.210 align:middle line:90% But the question still stands. 00:31:43.210 --> 00:31:45.940 align:middle line:84% For which lambda is this a contraction? 00:31:45.940 --> 00:31:48.220 align:middle line:84% Well, to go about proving for which 00:31:48.220 --> 00:31:50.290 align:middle line:84% lambda it's a contraction, let's look 00:31:50.290 --> 00:31:56.470 align:middle line:84% at the distance between Tf of f1 of x 00:31:56.470 --> 00:32:01.600 align:middle line:84% and Tf2 of x, where, again, what I'm changing here is f. 00:32:01.600 --> 00:32:03.370 align:middle line:90% It's not x that we're changing. 00:32:03.370 --> 00:32:05.120 align:middle line:90% It's f. 00:32:05.120 --> 00:32:06.280 align:middle line:90% All right? 00:32:06.280 --> 00:32:08.630 align:middle line:84% Well, then this is equal to, by definition, 00:32:08.630 --> 00:32:10.400 align:middle line:84% the difference of these two functions. 00:32:10.400 --> 00:32:17.470 align:middle line:84% So this will be the integral from a to b k x, y f1 of x. 00:32:17.470 --> 00:32:19.570 align:middle line:84% And while I'm at it, I'm going to combine this 00:32:19.570 --> 00:32:21.400 align:middle line:84% with f2 of x just because I'm running out 00:32:21.400 --> 00:32:24.190 align:middle line:90% of room on this board. 00:32:24.190 --> 00:32:26.410 align:middle line:84% So it's going to be the integral of k 00:32:26.410 --> 00:32:29.950 align:middle line:84% applied to f1 minus the integral of k applied to f2. 00:32:29.950 --> 00:32:32.920 align:middle line:90% And we want to make this small. 00:32:32.920 --> 00:32:34.370 align:middle line:90% How do we do that? 00:32:34.370 --> 00:32:38.000 align:middle line:84% Well, firstly, let's note one very useful fact, 00:32:38.000 --> 00:32:41.770 align:middle line:84% which is that k is a continuous function on a bounded interval, 00:32:41.770 --> 00:32:45.140 align:middle line:84% or on a bounded interval across itself. 00:32:45.140 --> 00:32:49.832 align:middle line:84% So this tells us is that, in fact, k is bounded. 00:32:49.832 --> 00:32:51.290 align:middle line:84% Why don't we want to apply the fact 00:32:51.290 --> 00:32:53.852 align:middle line:84% that the difference between f1 and f2 is bounded? 00:32:53.852 --> 00:32:55.310 align:middle line:84% Well, because we want to write this 00:32:55.310 --> 00:32:57.650 align:middle line:84% in terms of the distance between f1 and f2. 00:32:57.650 --> 00:33:00.480 align:middle line:90% So we don't want to do that. 00:33:00.480 --> 00:33:01.607 align:middle line:90% So how do we do that? 00:33:01.607 --> 00:33:03.440 align:middle line:84% Well, first we apply the triangle inequality 00:33:03.440 --> 00:33:04.650 align:middle line:90% to this integral. 00:33:04.650 --> 00:33:06.860 align:middle line:84% So this is less than or equal to b 00:33:06.860 --> 00:33:16.280 align:middle line:84% of k of x, y times f1 of x minus f2 of x. 00:33:16.280 --> 00:33:19.550 align:middle line:84% And now we want to get in somehow the distance, 00:33:19.550 --> 00:33:22.220 align:middle line:84% or the metric, on the continuous functions. 00:33:22.220 --> 00:33:27.390 align:middle line:84% To do so, I can just move in the supremum here. 00:33:27.390 --> 00:33:29.910 align:middle line:84% I know that this integral will be less than 00:33:29.910 --> 00:33:34.320 align:middle line:84% or equal to the term k of x, y times the supremum 00:33:34.320 --> 00:33:38.120 align:middle line:84% of the distance between f of 1 minus f of 2. 00:33:38.120 --> 00:33:42.790 align:middle line:84% So then this will be left equal to an integral from a to b k x, 00:33:42.790 --> 00:33:56.300 align:middle line:84% y times the supremum over x of f1 of x minus f2 of x dx. 00:33:56.300 --> 00:33:57.410 align:middle line:90% All right? 00:33:57.410 --> 00:34:01.440 align:middle line:84% So how are we going to actually use this? 00:34:01.440 --> 00:34:05.190 align:middle line:84% Well, notice right now this is just a constant. 00:34:05.190 --> 00:34:07.270 align:middle line:90% So we can pull this term out. 00:34:07.270 --> 00:34:07.770 align:middle line:90% Oh, sorry. 00:34:07.770 --> 00:34:11.219 align:middle line:84% I should have had a lambda here this entire time-- 00:34:11.219 --> 00:34:16.359 align:middle line:90% lambda, lambda, lambda. 00:34:16.359 --> 00:34:16.859 align:middle line:90% OK. 00:34:16.859 --> 00:34:19.699 align:middle line:90% 00:34:19.699 --> 00:34:22.699 align:middle line:84% So this term is a con-- this term is just 00:34:22.699 --> 00:34:24.770 align:middle line:90% the distance between f1 and f2. 00:34:24.770 --> 00:34:26.330 align:middle line:90% How do we bound this term? 00:34:26.330 --> 00:34:28.460 align:middle line:84% Well, we're going to use the fact that k of x, y 00:34:28.460 --> 00:34:31.989 align:middle line:84% is bounded because it's continuous on a compact set. 00:34:31.989 --> 00:34:34.389 align:middle line:84% So suppose that it's bounded by c 00:34:34.389 --> 00:34:37.460 align:middle line:90% where c is just a real number. 00:34:37.460 --> 00:34:39.969 align:middle line:90% Then what we'll have-- 00:34:39.969 --> 00:34:43.150 align:middle line:84% doo-doo-doo-- is that the distance 00:34:43.150 --> 00:34:49.060 align:middle line:84% between T of f1 minus T of f2 is less than or equal to lambda 00:34:49.060 --> 00:34:57.070 align:middle line:84% times the distance in c0 of f1 to f2 times integral 00:34:57.070 --> 00:35:04.870 align:middle line:90% from a to b of k x, y dx. 00:35:04.870 --> 00:35:07.360 align:middle line:84% But, again, this is less than some constant c. 00:35:07.360 --> 00:35:12.720 align:middle line:84% So I can write this as c times the interval a to b-- 00:35:12.720 --> 00:35:16.030 align:middle line:90% so b minus a. 00:35:16.030 --> 00:35:18.840 align:middle line:90% Everyone see how I did that? 00:35:18.840 --> 00:35:23.120 align:middle line:84% So for which lambda is this going to be a contraction? 00:35:23.120 --> 00:35:28.940 align:middle line:84% Well, there will be the lambda such that lambda is less than 1 00:35:28.940 --> 00:35:32.960 align:middle line:90% over c times b minus a, right? 00:35:32.960 --> 00:35:36.860 align:middle line:84% Because once this is less than 1 over c times b minus a, 00:35:36.860 --> 00:35:40.430 align:middle line:84% then this suddenly states that the distance 00:35:40.430 --> 00:35:42.050 align:middle line:90% between f1 and f2-- 00:35:42.050 --> 00:35:44.150 align:middle line:84% i.e., the supremum of that distance, 00:35:44.150 --> 00:35:46.640 align:middle line:84% will be less than or equal to a constant that's 00:35:46.640 --> 00:35:50.960 align:middle line:84% less than 1 times the distance between f1 and f2. 00:35:50.960 --> 00:35:53.600 align:middle line:90% So this is now a contraction. 00:35:53.600 --> 00:35:54.980 align:middle line:90% Why is this so helpful? 00:35:54.980 --> 00:35:57.482 align:middle line:90% Why did we do this? 00:35:57.482 --> 00:35:59.870 align:middle line:84% This is the one part of today's lecture 00:35:59.870 --> 00:36:02.540 align:middle line:84% that I was having a hard time motivating specifically, 00:36:02.540 --> 00:36:04.010 align:middle line:90% or figuring out how to state. 00:36:04.010 --> 00:36:05.400 align:middle line:90% Why did we do this? 00:36:05.400 --> 00:36:09.100 align:middle line:84% Well, it's deeply important for differential equations 00:36:09.100 --> 00:36:10.900 align:middle line:84% on the first hand, which I'll just 00:36:10.900 --> 00:36:12.910 align:middle line:84% state without argument because that 00:36:12.910 --> 00:36:18.010 align:middle line:84% would require a study of partial differential equations. 00:36:18.010 --> 00:36:21.100 align:middle line:84% But the second thing I want to know 00:36:21.100 --> 00:36:28.320 align:middle line:84% is that then notice that by the Banach fixed point theorem, 00:36:28.320 --> 00:36:32.460 align:middle line:84% there exists a function f such that it's a fixed point. 00:36:32.460 --> 00:36:35.340 align:middle line:84% Not only that, but we know that it's unique. 00:36:35.340 --> 00:36:38.050 align:middle line:90% What does this actually tell us? 00:36:38.050 --> 00:36:43.170 align:middle line:84% This tells us that there exists an f for lambda less than 1 00:36:43.170 --> 00:36:47.100 align:middle line:84% over c times b minus a such that T 00:36:47.100 --> 00:36:51.750 align:middle line:84% of f, which is equal to g of x plus lambda 00:36:51.750 --> 00:36:59.370 align:middle line:84% integral a to b k x, y f of x dx. 00:36:59.370 --> 00:37:02.970 align:middle line:84% This tells us that Tf is, in fact, equal to f. 00:37:02.970 --> 00:37:06.750 align:middle line:90% 00:37:06.750 --> 00:37:09.450 align:middle line:84% We know this by the Banach fixed point thorem, where here we're 00:37:09.450 --> 00:37:12.430 align:middle line:84% using the fact that continuous functions are Cauchy complete. 00:37:12.430 --> 00:37:14.038 align:middle line:84% This was on your second problem set, 00:37:14.038 --> 00:37:15.330 align:middle line:90% which is why I had y'all do it. 00:37:15.330 --> 00:37:18.380 align:middle line:90% 00:37:18.380 --> 00:37:24.340 align:middle line:84% So we know that f of x is equal to this right-hand side, which, 00:37:24.340 --> 00:37:27.400 align:middle line:90% if you differentiate it or-- 00:37:27.400 --> 00:37:28.550 align:middle line:90% I'll make a smaller mark-- 00:37:28.550 --> 00:37:29.050 align:middle line:90% remark. 00:37:29.050 --> 00:37:33.360 align:middle line:90% 00:37:33.360 --> 00:37:37.710 align:middle line:84% If g is, in fact, differentiable-- so c1 a to b-- 00:37:37.710 --> 00:37:39.832 align:middle line:90% what does this tell us? 00:37:39.832 --> 00:37:41.790 align:middle line:84% Well, now, all of a sudden, I can differentiate 00:37:41.790 --> 00:37:43.050 align:middle line:90% the right-hand side. 00:37:43.050 --> 00:37:47.250 align:middle line:84% I'll get that f of x, or the derivative of f of x, 00:37:47.250 --> 00:37:48.930 align:middle line:90% is equal to-- 00:37:48.930 --> 00:37:50.430 align:middle line:90% how do I want to say this? 00:37:50.430 --> 00:37:51.780 align:middle line:90% Let me be more careful. 00:37:51.780 --> 00:37:54.780 align:middle line:84% One, we know that there exists an f that 00:37:54.780 --> 00:37:58.140 align:middle line:84% is continuous such that this is true. 00:37:58.140 --> 00:37:59.880 align:middle line:84% If g is, in fact, differentiable, 00:37:59.880 --> 00:38:01.470 align:middle line:84% you can think of it as a constant. 00:38:01.470 --> 00:38:04.920 align:middle line:84% If g is differentiable, then f must be differentiable, 00:38:04.920 --> 00:38:07.170 align:middle line:84% because then it's a differentiable function 00:38:07.170 --> 00:38:08.670 align:middle line:84% plus something that's differentiable 00:38:08.670 --> 00:38:10.360 align:middle line:90% because it's an integral. 00:38:10.360 --> 00:38:14.260 align:middle line:84% So this tells us that, in fact, f 00:38:14.260 --> 00:38:19.530 align:middle line:84% is in c1, which is important in differential equations, right? 00:38:19.530 --> 00:38:22.260 align:middle line:84% This tells us a way to prove that there exists 00:38:22.260 --> 00:38:24.780 align:middle line:84% a unique solution to the differential equation that 00:38:24.780 --> 00:38:27.010 align:middle line:90% defines this integral operator. 00:38:27.010 --> 00:38:29.280 align:middle line:84% Now, of course, not every differential equation 00:38:29.280 --> 00:38:31.590 align:middle line:84% will result in an integral operator like this. 00:38:31.590 --> 00:38:35.730 align:middle line:84% But a huge class of them can't be, which is very important. 00:38:35.730 --> 00:38:37.680 align:middle line:84% If you're interested in this question, 00:38:37.680 --> 00:38:43.140 align:middle line:84% I would highly suggest looking at section 7.6.2 00:38:43.140 --> 00:38:46.230 align:middle line:84% on Picard iteration in Lebl's book. 00:38:46.230 --> 00:38:48.300 align:middle line:90% It's a very useful book to have. 00:38:48.300 --> 00:38:51.180 align:middle line:84% Or it's a very useful section to read into. 00:38:51.180 --> 00:38:53.580 align:middle line:84% It just uses an application that I 00:38:53.580 --> 00:38:55.900 align:middle line:84% think will take up the rest of our time. 00:38:55.900 --> 00:38:59.280 align:middle line:84% And I want to talk about completions of metric spaces. 00:38:59.280 --> 00:39:01.920 align:middle line:84% So any questions before I move on? 00:39:01.920 --> 00:39:07.280 align:middle line:90% 00:39:07.280 --> 00:39:09.670 align:middle line:90% Totally fair. 00:39:09.670 --> 00:39:12.500 align:middle line:90% OK, so next section-- 00:39:12.500 --> 00:39:14.280 align:middle line:90% ah how do i put it? 00:39:14.280 --> 00:39:17.100 align:middle line:84% - 2 completions of metric spaces. 00:39:17.100 --> 00:39:26.600 align:middle line:90% 00:39:26.600 --> 00:39:29.070 align:middle line:84% Has anyone heard of this before, completion 00:39:29.070 --> 00:39:32.060 align:middle line:90% of the metric spaces? 00:39:32.060 --> 00:39:35.300 align:middle line:84% Well, one way to motivate completions of metric spaces 00:39:35.300 --> 00:39:39.290 align:middle line:84% is through an example that you all have already seen. 00:39:39.290 --> 00:39:52.927 align:middle line:84% Example-- the real numbers are a completion of the rationals. 00:39:52.927 --> 00:39:55.260 align:middle line:84% So remember how we started our journey in real analysis, 00:39:55.260 --> 00:39:55.870 align:middle line:90% right? 00:39:55.870 --> 00:39:58.380 align:middle line:84% We started by looking at rational numbers. 00:39:58.380 --> 00:40:00.540 align:middle line:84% We stated things like they're a field, 00:40:00.540 --> 00:40:04.680 align:middle line:84% and we stated things like there aren't every real number-- 00:40:04.680 --> 00:40:07.560 align:middle line:84% or, sorry, every real number is not a rational one, right? 00:40:07.560 --> 00:40:10.600 align:middle line:84% The square root of 2 is an example. 00:40:10.600 --> 00:40:12.000 align:middle line:90% And then we said, well, hold on. 00:40:12.000 --> 00:40:15.090 align:middle line:84% Let's just throw in the square root of 2, right? 00:40:15.090 --> 00:40:18.300 align:middle line:84% We completed the rationals by filling in all of the holes. 00:40:18.300 --> 00:40:22.560 align:middle line:84% And doing so, this method, is known as a completion. 00:40:22.560 --> 00:40:24.450 align:middle line:84% Now, there's a few ways you might have seen 00:40:24.450 --> 00:40:25.800 align:middle line:90% this for the rational numbers. 00:40:25.800 --> 00:40:27.660 align:middle line:90% And I'll state them right now. 00:40:27.660 --> 00:40:32.280 align:middle line:84% One, you might have seen a proof of it using Dedekind cuts. 00:40:32.280 --> 00:40:35.850 align:middle line:84% This is how Rudin uses it in his book, Baby Rudin, 00:40:35.850 --> 00:40:36.795 align:middle line:90% Dedekind cuts. 00:40:36.795 --> 00:40:39.643 align:middle line:90% 00:40:39.643 --> 00:40:41.560 align:middle line:84% If you're interested in how this method works, 00:40:41.560 --> 00:40:44.380 align:middle line:84% it's in his Appendix 1 of his first chapter. 00:40:44.380 --> 00:40:46.060 align:middle line:90% This one is not super used. 00:40:46.060 --> 00:40:47.740 align:middle line:84% It's sort of difficult to get around. 00:40:47.740 --> 00:40:50.710 align:middle line:84% But you could have viewed it this way. 00:40:50.710 --> 00:40:52.853 align:middle line:84% Two, you also could have viewed it 00:40:52.853 --> 00:40:54.520 align:middle line:84% by using the least upper bound property. 00:40:54.520 --> 00:41:03.140 align:middle line:90% 00:41:03.140 --> 00:41:06.985 align:middle line:84% Has anyone seen this method before? 00:41:06.985 --> 00:41:07.860 align:middle line:90% Totally fair if not-- 00:41:07.860 --> 00:41:11.810 align:middle line:84% I'll simply state what it is if not. 00:41:11.810 --> 00:41:14.990 align:middle line:84% For this method, what we do is we define the real numbers 00:41:14.990 --> 00:41:16.790 align:middle line:84% as the smallest set that contains 00:41:16.790 --> 00:41:20.180 align:middle line:84% the rational numbers such that every bounded set, 00:41:20.180 --> 00:41:23.560 align:middle line:90% its upper bound is in the set. 00:41:23.560 --> 00:41:24.730 align:middle line:90% Great. 00:41:24.730 --> 00:41:26.440 align:middle line:90% Does this make sense verbally? 00:41:26.440 --> 00:41:28.420 align:middle line:84% I can also write it up more explicitly. 00:41:28.420 --> 00:41:32.320 align:middle line:84% But this is another way you might have seen it. 00:41:32.320 --> 00:41:43.320 align:middle line:84% Three, you might have learned it using equivalence classes 00:41:43.320 --> 00:41:44.880 align:middle line:90% of Cauchy sequences. 00:41:44.880 --> 00:41:49.720 align:middle line:90% 00:41:49.720 --> 00:41:52.010 align:middle line:90% Did anyone see it this way? 00:41:52.010 --> 00:41:54.740 align:middle line:84% Or, in fact, let me just ask, what ways have y'all seen it, 00:41:54.740 --> 00:41:57.330 align:middle line:90% the completion of the rationals? 00:41:57.330 --> 00:41:59.470 align:middle line:84% The second one, the least upper bound property? 00:41:59.470 --> 00:42:01.101 align:middle line:90% What about you? 00:42:01.101 --> 00:42:02.360 align:middle line:90% It's really fair. 00:42:02.360 --> 00:42:03.440 align:middle line:90% It was a while ago. 00:42:03.440 --> 00:42:05.015 align:middle line:90% [INAUDIBLE student question] 00:42:05.015 --> 00:42:06.640 align:middle line:84% Oh, interesting-- the equivalence class 00:42:06.640 --> 00:42:07.960 align:middle line:90% has the Cauchy sequences. 00:42:07.960 --> 00:42:12.047 align:middle line:84% Well, it was at least a full semester ago, if not longer. 00:42:12.047 --> 00:42:13.630 align:middle line:84% But the third one is the one I'm going 00:42:13.630 --> 00:42:16.038 align:middle line:90% to iterate on today, all right? 00:42:16.038 --> 00:42:18.580 align:middle line:84% Where, today, we're looking at Cauchy complete metric spaces, 00:42:18.580 --> 00:42:20.890 align:middle line:84% it makes sense to look at Cauchy sequences 00:42:20.890 --> 00:42:22.430 align:middle line:90% as our starting point. 00:42:22.430 --> 00:42:27.180 align:middle line:84% But the equivalence classes is not too bad to motivate. 00:42:27.180 --> 00:42:37.510 align:middle line:84% The idea is we say a Cauchy sequence am is equivalent-- 00:42:37.510 --> 00:42:43.310 align:middle line:84% or I'll use this sym, note symbol. 00:42:43.310 --> 00:42:45.820 align:middle line:84% So here an and bn are Cauchy sequences. 00:42:45.820 --> 00:42:50.200 align:middle line:84% I'm going to say that they're equivalent if the distance 00:42:50.200 --> 00:42:56.982 align:middle line:84% between an and bn goes to 0 as n goes to infinity. 00:42:56.982 --> 00:42:58.690 align:middle line:84% This is how we define equivalence classes 00:42:58.690 --> 00:43:00.250 align:middle line:90% of Cauchy sequences. 00:43:00.250 --> 00:43:03.370 align:middle line:84% What this let us do is define all of our limit points 00:43:03.370 --> 00:43:04.420 align:middle line:90% to be the same. 00:43:04.420 --> 00:43:08.660 align:middle line:84% The issue-- let me say this differently. 00:43:08.660 --> 00:43:11.810 align:middle line:84% The idea is we want to take sequences of rational numbers 00:43:11.810 --> 00:43:13.878 align:middle line:84% that will converge to the real numbers we want. 00:43:13.878 --> 00:43:15.920 align:middle line:84% We want them to converge to the square root of 2. 00:43:15.920 --> 00:43:18.110 align:middle line:84% We want them to converge to the square root of 3, 00:43:18.110 --> 00:43:19.550 align:middle line:90% things like this. 00:43:19.550 --> 00:43:22.740 align:middle line:84% The issue is we don't want to overdefine our space. 00:43:22.740 --> 00:43:24.980 align:middle line:84% We don't want multiple sequences to go-- 00:43:24.980 --> 00:43:28.700 align:middle line:84% or we don't want multiple square roots of 2. 00:43:28.700 --> 00:43:30.770 align:middle line:90% We only want one specific one. 00:43:30.770 --> 00:43:33.080 align:middle line:84% If we use equivalence classes, let's just 00:43:33.080 --> 00:43:35.880 align:middle line:84% define the limit points as being the same. 00:43:35.880 --> 00:43:39.060 align:middle line:90% So this is what lets us do it. 00:43:39.060 --> 00:43:41.280 align:middle line:84% We have equivalence classes of real number-- 00:43:41.280 --> 00:43:44.100 align:middle line:90% of rational numbers. 00:43:44.100 --> 00:43:49.280 align:middle line:84% And then what we say is that R is 00:43:49.280 --> 00:43:52.880 align:middle line:90% the set of equivalence classes. 00:43:52.880 --> 00:44:00.540 align:middle line:90% 00:44:00.540 --> 00:44:02.520 align:middle line:84% The notion of an equivalence relationship, 00:44:02.520 --> 00:44:04.120 align:middle line:90% I've put on the lecture notes. 00:44:04.120 --> 00:44:05.910 align:middle line:84% So I would highly suggest looking at that on Wikipedia 00:44:05.910 --> 00:44:07.160 align:middle line:90% if you haven't seen it before. 00:44:07.160 --> 00:44:08.850 align:middle line:84% But the fact that this is true, you 00:44:08.850 --> 00:44:11.910 align:middle line:84% can simply write out and prove the fact that this 00:44:11.910 --> 00:44:15.270 align:middle line:90% is an equivalence relation. 00:44:15.270 --> 00:44:18.330 align:middle line:84% In fact, I'll just state what the three properties are now, 00:44:18.330 --> 00:44:20.683 align:middle line:84% the three properties you would have to show, 00:44:20.683 --> 00:44:23.100 align:middle line:84% because there are ones that, at least on the face of them, 00:44:23.100 --> 00:44:25.140 align:middle line:84% we should be able to think through a little bit. 00:44:25.140 --> 00:44:31.520 align:middle line:90% 00:44:31.520 --> 00:44:32.390 align:middle line:90% Doo-doo-doo. 00:44:32.390 --> 00:44:38.312 align:middle line:90% 00:44:38.312 --> 00:44:38.812 align:middle line:90% OK. 00:44:38.812 --> 00:44:43.260 align:middle line:90% 00:44:43.260 --> 00:44:47.033 align:middle line:84% So we have this set of equivalence classes. 00:44:47.033 --> 00:44:48.950 align:middle line:84% What we want to show is that this is, in fact, 00:44:48.950 --> 00:44:50.240 align:middle line:90% an equivalence relationship. 00:44:50.240 --> 00:44:54.280 align:middle line:84% So we want to show first that the sequence an, if this 00:44:54.280 --> 00:44:57.550 align:middle line:84% is Cauchy, this is equivalent to an. 00:44:57.550 --> 00:44:59.540 align:middle line:84% But how do we know that this is true? 00:44:59.540 --> 00:45:07.200 align:middle line:84% Well, it's simply by the fact that an minus an 00:45:07.200 --> 00:45:08.910 align:middle line:90% equals 0 for every single n. 00:45:08.910 --> 00:45:12.090 align:middle line:84% So it's definitely going to tend to 0. 00:45:12.090 --> 00:45:21.360 align:middle line:84% Secondly, you want to show that if an is equivalent to bn, 00:45:21.360 --> 00:45:27.030 align:middle line:84% this is true if and only if bn is equivalent to an. 00:45:27.030 --> 00:45:28.660 align:middle line:90% Why is this going to be true? 00:45:28.660 --> 00:45:31.120 align:middle line:84% Well, we know that absolute values are symmetric. 00:45:31.120 --> 00:45:33.560 align:middle line:84% So we can just swap those around. 00:45:33.560 --> 00:45:35.650 align:middle line:90% All good on that front. 00:45:35.650 --> 00:45:40.750 align:middle line:84% And three, we want to show that if an 00:45:40.750 --> 00:45:53.840 align:middle line:84% is equivalent to bn and bn is equivalent to cn, then 00:45:53.840 --> 00:45:55.035 align:middle line:90% an is equivalent to cn. 00:45:55.035 --> 00:45:59.650 align:middle line:90% 00:45:59.650 --> 00:46:01.480 align:middle line:90% This last one, I'll iterate on. 00:46:01.480 --> 00:46:04.660 align:middle line:90% I'll expand upon this. 00:46:04.660 --> 00:46:05.570 align:middle line:90% How do we do that? 00:46:05.570 --> 00:46:09.093 align:middle line:84% Well, we look at the distance from an to cn, 00:46:09.093 --> 00:46:10.510 align:middle line:84% and we know that this is less than 00:46:10.510 --> 00:46:15.655 align:middle line:84% or equal to the distance from an to bn plus the distance from bn 00:46:15.655 --> 00:46:16.960 align:middle line:90% to cn. 00:46:16.960 --> 00:46:19.540 align:middle line:84% And we can make both of these terms arbitrarily small 00:46:19.540 --> 00:46:22.690 align:middle line:84% because the distances between the points goes to 0. 00:46:22.690 --> 00:46:26.210 align:middle line:84% So then this is less than epsilon, 00:46:26.210 --> 00:46:29.860 align:middle line:84% which tells us that this is going to converge to 0. 00:46:29.860 --> 00:46:33.850 align:middle line:84% This is less than epsilon for all epsilon bigger than 0. 00:46:33.850 --> 00:46:39.120 align:middle line:84% There exists an n such that this is true. 00:46:39.120 --> 00:46:40.490 align:middle line:90% OK. 00:46:40.490 --> 00:46:44.660 align:middle line:84% So this tells us that, in fact, the distance between an and cn 00:46:44.660 --> 00:46:46.550 align:middle line:90% are going to tend to 0. 00:46:46.550 --> 00:46:48.980 align:middle line:84% And notice that all of the things I've stated so far, 00:46:48.980 --> 00:46:50.990 align:middle line:84% everything in this equivalence relationship, 00:46:50.990 --> 00:46:53.360 align:middle line:90% only uses the metric, right? 00:46:53.360 --> 00:46:56.970 align:middle line:84% Here we're just using the metric defined by absolute values. 00:46:56.970 --> 00:46:58.400 align:middle line:84% But you can picture replacing this 00:46:58.400 --> 00:47:03.590 align:middle line:84% with any metric you care to use, which 00:47:03.590 --> 00:47:05.750 align:middle line:84% is why we can view completions of metric spaces 00:47:05.750 --> 00:47:08.120 align:middle line:90% as a whole topic for today. 00:47:08.120 --> 00:47:22.830 align:middle line:84% So notice we have the same equivalence classes 00:47:22.830 --> 00:47:24.600 align:middle line:90% notion for metric spaces. 00:47:24.600 --> 00:47:35.110 align:middle line:90% 00:47:35.110 --> 00:47:36.530 align:middle line:90% Why is this helpful? 00:47:36.530 --> 00:47:39.670 align:middle line:84% Well I'll give one short example, at least verbally, 00:47:39.670 --> 00:47:43.420 align:middle line:84% which is this metric is not the only one that can 00:47:43.420 --> 00:47:45.250 align:middle line:90% exist on the rational numbers. 00:47:45.250 --> 00:47:48.410 align:middle line:84% We can impose it with a bunch of other metrics if you wanted to 00:47:48.410 --> 00:47:52.540 align:middle line:84% and then look at the equivalence classes of that. 00:47:52.540 --> 00:47:55.540 align:middle line:84% And that leads to what is known as p-adic numbers. 00:47:55.540 --> 00:47:57.940 align:middle line:84% I wish I had more time to talk about that today, 00:47:57.940 --> 00:48:01.430 align:middle line:84% but it's one of the examples that shows up in number theory. 00:48:01.430 --> 00:48:04.450 align:middle line:84% So I'll leave that for you to look at. 00:48:04.450 --> 00:48:07.570 align:middle line:84% But how do we add in these limit points? 00:48:07.570 --> 00:48:09.160 align:middle line:84% That's the whole question, right? 00:48:09.160 --> 00:48:11.930 align:middle line:90% 00:48:11.930 --> 00:48:13.820 align:middle line:84% Well, there's a theorem that tells us 00:48:13.820 --> 00:48:16.152 align:middle line:90% that we can simply do so. 00:48:16.152 --> 00:48:17.735 align:middle line:84% And let me state what that theorem is. 00:48:17.735 --> 00:48:21.080 align:middle line:90% 00:48:21.080 --> 00:48:24.860 align:middle line:84% This theorem, I'll just write as completions 00:48:24.860 --> 00:48:28.710 align:middle line:84% because it's what tells us that there exists a completion. 00:48:28.710 --> 00:48:30.880 align:middle line:90% OK. 00:48:30.880 --> 00:48:36.120 align:middle line:90% Let M, d be a metric space. 00:48:36.120 --> 00:48:40.208 align:middle line:90% 00:48:40.208 --> 00:48:42.250 align:middle line:84% And it doesn't need to be Cauchy complete, right? 00:48:42.250 --> 00:48:43.875 align:middle line:84% The idea is that we want to complete it 00:48:43.875 --> 00:48:46.750 align:middle line:90% so that it is Cauchy complete. 00:48:46.750 --> 00:48:56.530 align:middle line:84% Then there exists an M bar such that, one, 00:48:56.530 --> 00:49:02.450 align:middle line:90% M is a subset of M bar; two-- 00:49:02.450 --> 00:49:06.820 align:middle line:84% doo-doo-doo-- the distance on M bar-- 00:49:06.820 --> 00:49:09.325 align:middle line:84% I should note that this is a metric space. 00:49:09.325 --> 00:49:12.580 align:middle line:90% 00:49:12.580 --> 00:49:16.900 align:middle line:84% And I'll call the metric on it d bar for simplicity. 00:49:16.900 --> 00:49:24.140 align:middle line:84% The distance on M bar restricts to the regular distance on M. 00:49:24.140 --> 00:49:27.020 align:middle line:84% So what this tells you is that we have the subspace metric 00:49:27.020 --> 00:49:28.340 align:middle line:90% on our space, right? 00:49:28.340 --> 00:49:30.110 align:middle line:84% The distance between two points and M 00:49:30.110 --> 00:49:34.820 align:middle line:84% is going to stay the same in this new metric d bar. 00:49:34.820 --> 00:49:40.755 align:middle line:84% And three, M bar is Cauchy complete. 00:49:40.755 --> 00:49:47.720 align:middle line:90% 00:49:47.720 --> 00:50:01.790 align:middle line:84% And, finally, four, the closure of M 00:50:01.790 --> 00:50:05.360 align:middle line:84% is M bar, which is where this notation comes from. 00:50:05.360 --> 00:50:07.370 align:middle line:84% If you've ever seen the definition of closure 00:50:07.370 --> 00:50:12.890 align:middle line:84% before in a topology book or in Lebl's book, it's M bar. 00:50:12.890 --> 00:50:17.150 align:middle line:84% Does everyone remember what the definition of closure is? 00:50:17.150 --> 00:50:21.260 align:middle line:84% It's the smallest closed set that contains your smaller set. 00:50:21.260 --> 00:50:24.500 align:middle line:84% So the smallest closed set that contains M is, in fact, M bar. 00:50:24.500 --> 00:50:29.650 align:middle line:90% 00:50:29.650 --> 00:50:32.290 align:middle line:84% You can think of it as just including all the boundary 00:50:32.290 --> 00:50:33.490 align:middle line:90% points as the main thing. 00:50:33.490 --> 00:50:37.820 align:middle line:90% Doo-doo-doo 00:50:37.820 --> 00:50:40.160 align:middle line:84% The reason why that's the right notion to consider 00:50:40.160 --> 00:50:42.830 align:middle line:84% is because we know that closed sets contain 00:50:42.830 --> 00:50:45.630 align:middle line:90% all of your limit points. 00:50:45.630 --> 00:50:49.020 align:middle line:84% Specifically, the issue are the ones at the boundary. 00:50:49.020 --> 00:50:53.550 align:middle line:84% But if you add those points in, then we're all good. 00:50:53.550 --> 00:50:55.530 align:middle line:84% Before we go into this proof, however, I 00:50:55.530 --> 00:50:57.960 align:middle line:84% need to simply state one small lemma. 00:50:57.960 --> 00:51:00.900 align:middle line:90% 00:51:00.900 --> 00:51:02.650 align:middle line:84% I won't go through the proof, because it's 00:51:02.650 --> 00:51:04.317 align:middle line:84% very similar to what's on your homework, 00:51:04.317 --> 00:51:06.190 align:middle line:84% but I'll definitely talk through it, which 00:51:06.190 --> 00:51:17.950 align:middle line:84% is the set C at lower infinity of M, which 00:51:17.950 --> 00:51:26.680 align:middle line:84% is the set of functions f, which are continuous, 00:51:26.680 --> 00:51:32.180 align:middle line:84% with the supremum of m and M of f of m bounded-- 00:51:32.180 --> 00:51:35.500 align:middle line:90% so this is less than infinity-- 00:51:35.500 --> 00:51:36.385 align:middle line:90% as a metric space. 00:51:36.385 --> 00:51:41.940 align:middle line:90% 00:51:41.940 --> 00:51:46.910 align:middle line:84% So what this is is the set of continuous bounded functions. 00:51:46.910 --> 00:51:49.790 align:middle line:84% And denoted it's c infinity of M. Notice 00:51:49.790 --> 00:51:52.280 align:middle line:84% it's in the small part because it's not smooth. 00:51:52.280 --> 00:51:54.690 align:middle line:90% Smooth goes in the upper part. 00:51:54.690 --> 00:51:59.450 align:middle line:84% So this is a metric space, specifically with metric given 00:51:59.450 --> 00:52:04.070 align:middle line:84% by the supremum or distance d infinity-- 00:52:04.070 --> 00:52:10.295 align:middle line:84% or always d upper infinity of f to g equal to-- 00:52:10.295 --> 00:52:13.170 align:middle line:90% 00:52:13.170 --> 00:52:16.410 align:middle line:90% no, I'll use lower infinity-- 00:52:16.410 --> 00:52:21.480 align:middle line:84% the supremum of m and M of the distance 00:52:21.480 --> 00:52:26.060 align:middle line:90% between f of x minus g of x. 00:52:26.060 --> 00:52:26.840 align:middle line:90% Oh, sorry. 00:52:26.840 --> 00:52:27.770 align:middle line:90% These should be m's. 00:52:27.770 --> 00:52:30.400 align:middle line:90% 00:52:30.400 --> 00:52:33.550 align:middle line:84% Firstly, I should ask how do we know that this supremum exists 00:52:33.550 --> 00:52:36.000 align:middle line:90% and is finite. 00:52:36.000 --> 00:52:36.500 align:middle line:90% Anyone? 00:52:36.500 --> 00:52:41.437 align:middle line:90% 00:52:41.437 --> 00:52:42.770 align:middle line:90% I'll let y'all think through it. 00:52:42.770 --> 00:52:49.300 align:middle line:90% 00:52:49.300 --> 00:52:53.250 align:middle line:84% Here we want to use the fact that f and g are bounded. 00:52:53.250 --> 00:52:56.885 align:middle line:84% So does that immediately imply that the supremum is bounded? 00:52:56.885 --> 00:53:00.720 align:middle line:90% 00:53:00.720 --> 00:53:01.320 align:middle line:90% Yeah. 00:53:01.320 --> 00:53:02.220 align:middle line:90% The answer is yes. 00:53:02.220 --> 00:53:04.920 align:middle line:84% You can just apply the triangle inequality, right? 00:53:04.920 --> 00:53:06.955 align:middle line:84% The absolute value of f of m minus g of m 00:53:06.955 --> 00:53:08.580 align:middle line:84% is less than or equal to absolute value 00:53:08.580 --> 00:53:10.800 align:middle line:90% f plus absolute value g. 00:53:10.800 --> 00:53:12.517 align:middle line:84% And we can take the supremum of that, 00:53:12.517 --> 00:53:14.350 align:middle line:84% and that will have to be less than infinity. 00:53:14.350 --> 00:53:16.140 align:middle line:84% So this supremum definitely exists, 00:53:16.140 --> 00:53:18.330 align:middle line:84% and it's finite, which is helpful. 00:53:18.330 --> 00:53:22.560 align:middle line:84% We want to be able to say metrics must be finite, 00:53:22.560 --> 00:53:25.860 align:middle line:84% unless you're looking at some weird version of a metric. 00:53:25.860 --> 00:53:28.080 align:middle line:84% So we know that this is, in fact, a metric. 00:53:28.080 --> 00:53:30.930 align:middle line:84% The proof of this is just using the same proof of supremums 00:53:30.930 --> 00:53:32.670 align:middle line:84% that we did for continuous functions 00:53:32.670 --> 00:53:34.140 align:middle line:90% on a bounded interval. 00:53:34.140 --> 00:53:37.740 align:middle line:84% The only difference is instead of the extreme value theorem, 00:53:37.740 --> 00:53:39.487 align:middle line:90% we used boundedness, right? 00:53:39.487 --> 00:53:41.070 align:middle line:84% That's the main difference, because we 00:53:41.070 --> 00:53:42.840 align:middle line:90% don't know that it's compact. 00:53:42.840 --> 00:53:45.990 align:middle line:84% So we don't know that there exists an extreme value. 00:53:45.990 --> 00:53:47.410 align:middle line:84% But we do know that it's bounded. 00:53:47.410 --> 00:53:50.480 align:middle line:84% So that's all we needed from the extreme value theorem. 00:53:50.480 --> 00:53:54.360 align:middle line:84% So that shows that this is, in fact, a metric. 00:53:54.360 --> 00:53:57.430 align:middle line:84% What's the next thing that we might want to say about this? 00:53:57.430 --> 00:54:08.215 align:middle line:84% Well, in fact, C infinity of M is Cauchy complete. 00:54:08.215 --> 00:54:14.830 align:middle line:90% 00:54:14.830 --> 00:54:16.720 align:middle line:84% The proof of this is just the same 00:54:16.720 --> 00:54:18.100 align:middle line:90% as it was on your homework. 00:54:18.100 --> 00:54:21.650 align:middle line:84% You take a Cauchy sequence of functions. 00:54:21.650 --> 00:54:24.100 align:middle line:84% So the function such that the distance between two points 00:54:24.100 --> 00:54:26.310 align:middle line:90% gets small. 00:54:26.310 --> 00:54:29.640 align:middle line:84% And then what we know is we can take the limit points of them 00:54:29.640 --> 00:54:30.750 align:middle line:90% by continuity. 00:54:30.750 --> 00:54:33.390 align:middle line:84% And then you have to show that, in fact, the limit 00:54:33.390 --> 00:54:34.320 align:middle line:90% point is bounded. 00:54:34.320 --> 00:54:36.300 align:middle line:84% But the proof of this is nearly exactly 00:54:36.300 --> 00:54:38.560 align:middle line:84% the same as your second problem set. 00:54:38.560 --> 00:54:40.860 align:middle line:84% So unless anyone wants me to work through the details 00:54:40.860 --> 00:54:44.243 align:middle line:84% right here, I would highly suggest doing it yourself. 00:54:44.243 --> 00:54:45.660 align:middle line:84% But I think it's the sort of thing 00:54:45.660 --> 00:54:47.670 align:middle line:84% that, because you've all done the problem set, 00:54:47.670 --> 00:54:50.660 align:middle line:84% you can believe me that this is true. 00:54:50.660 --> 00:54:52.410 align:middle line:90% All right. 00:54:52.410 --> 00:54:55.730 align:middle line:84% So how are you going to use this fact? 00:54:55.730 --> 00:54:58.190 align:middle line:84% Well, the fact that this is Cauchy complete 00:54:58.190 --> 00:54:59.550 align:middle line:90% is super helpful for us. 00:54:59.550 --> 00:55:01.760 align:middle line:84% We want to somehow view our metric space 00:55:01.760 --> 00:55:08.680 align:middle line:84% M as a subset of C infinity of M. Now, how do we do that? 00:55:08.680 --> 00:55:12.700 align:middle line:84% So I'm going to prove this theorem now. 00:55:12.700 --> 00:55:17.150 align:middle line:84% The way that we do that is we simply map points to-- 00:55:17.150 --> 00:55:23.490 align:middle line:84% we simply map points in our metric space to a function. 00:55:23.490 --> 00:55:40.650 align:middle line:84% So fix m prime M arbitrary and define the map from M and m. 00:55:40.650 --> 00:55:44.040 align:middle line:84% We're going to map this to the function g of m which 00:55:44.040 --> 00:55:47.430 align:middle line:84% is defined pointewise at p as the distance of p 00:55:47.430 --> 00:55:54.940 align:middle line:84% to m minus the distance from p to m prime. 00:55:54.940 --> 00:55:57.440 align:middle line:90% So-- oh, sorry-- 00:55:57.440 --> 00:55:59.270 align:middle line:90% yeah. 00:55:59.270 --> 00:56:02.840 align:middle line:84% So fix sum m prime and M, which is arbitrary. 00:56:02.840 --> 00:56:05.810 align:middle line:84% And we're defining the map from our point in a metric space 00:56:05.810 --> 00:56:07.790 align:middle line:90% to a continuous function. 00:56:07.790 --> 00:56:09.740 align:middle line:84% How do we know this is continuous? 00:56:09.740 --> 00:56:13.200 align:middle line:84% Well, firstly, you can show that a metric is continuous, 00:56:13.200 --> 00:56:15.350 align:middle line:84% which is what we did on our second day, right? 00:56:15.350 --> 00:56:21.950 align:middle line:84% We had that the distance between xn and y 00:56:21.950 --> 00:56:25.610 align:middle line:90% goes to the distance of x and y. 00:56:25.610 --> 00:56:33.710 align:middle line:84% And, in fact, distance of x and ym converges to d x, y. 00:56:33.710 --> 00:56:36.620 align:middle line:84% So we have already briefly, in loose terms, 00:56:36.620 --> 00:56:40.290 align:middle line:90% shown continuity of a metric. 00:56:40.290 --> 00:56:42.172 align:middle line:84% But how do we show that this is bounded? 00:56:42.172 --> 00:56:44.630 align:middle line:84% Well, the fact that this is bounded has to do with the fact 00:56:44.630 --> 00:56:48.050 align:middle line:84% that both of these metrics are less than infinity, right? 00:56:48.050 --> 00:56:51.000 align:middle line:84% That's by definition of the metric. 00:56:51.000 --> 00:56:53.520 align:middle line:84% So we're never going to get something that's nonfinite. 00:56:53.520 --> 00:56:57.300 align:middle line:84% So this tells us that we can take the supremum of it. 00:56:57.300 --> 00:57:02.800 align:middle line:84% So we now have a way to view our metric M 00:57:02.800 --> 00:57:09.930 align:middle line:84% as a subset of C infinity of M because you've 00:57:09.930 --> 00:57:12.710 align:middle line:84% mapped it to a continuous and bounded function. 00:57:12.710 --> 00:57:16.310 align:middle line:84% Now, our goal is simply to take the closure of this subset. 00:57:16.310 --> 00:57:20.390 align:middle line:84% But to do so, we need to clarify what we mean by this, quotation 00:57:20.390 --> 00:57:23.100 align:middle line:90% mark, "subset," right? 00:57:23.100 --> 00:57:28.240 align:middle line:84% What I mean by this is notice that gm of p-- 00:57:28.240 --> 00:57:31.480 align:middle line:90% 00:57:31.480 --> 00:57:33.480 align:middle line:90% how do I want to say this? 00:57:33.480 --> 00:57:39.600 align:middle line:84% Notice, then, that the distance from m1 to m2, 00:57:39.600 --> 00:57:44.940 align:middle line:84% this is, in fact, equal to the supremum of m 00:57:44.940 --> 00:57:53.580 align:middle line:84% and M of the distance between gm1 of p minus gm2 of p, 00:57:53.580 --> 00:57:55.710 align:middle line:90% because you can simply plug in-- 00:57:55.710 --> 00:58:01.110 align:middle line:84% oh, sorry this should be supremum over p-- 00:58:01.110 --> 00:58:03.420 align:middle line:84% because you can simply plug in m1 and m2 00:58:03.420 --> 00:58:04.830 align:middle line:90% and take their difference. 00:58:04.830 --> 00:58:08.440 align:middle line:84% And then you just want the maximum distance between them. 00:58:08.440 --> 00:58:11.310 align:middle line:84% And notice that this is precisely the same metric 00:58:11.310 --> 00:58:14.680 align:middle line:90% as is on our C infinity space. 00:58:14.680 --> 00:58:18.480 align:middle line:84% So this is equal to the distance to infinity 00:58:18.480 --> 00:58:23.640 align:middle line:90% between gm1 and gm2. 00:58:23.640 --> 00:58:26.220 align:middle line:84% What this tells us is that this map-- 00:58:26.220 --> 00:58:30.790 align:middle line:84% this map that goes from m to gm-- 00:58:30.790 --> 00:58:38.418 align:middle line:84% is an isometry, which means that it just preserves distances. 00:58:38.418 --> 00:58:40.210 align:middle line:84% The distance between two points is the same 00:58:40.210 --> 00:58:42.100 align:middle line:84% as the distance between two functions 00:58:42.100 --> 00:58:43.690 align:middle line:90% in this different space. 00:58:43.690 --> 00:58:46.180 align:middle line:84% And, in fact, not only is it an isometry. 00:58:46.180 --> 00:58:48.040 align:middle line:84% It's going to be bijective, right? 00:58:48.040 --> 00:58:50.440 align:middle line:84% For each m, there exists a unique gm 00:58:50.440 --> 00:58:53.240 align:middle line:90% defined as this distance. 00:58:53.240 --> 00:58:59.380 align:middle line:84% So this tells us that we really can view m as a subset of C 00:58:59.380 --> 00:59:01.570 align:middle line:84% infinity because we're just viewing it as a subset 00:59:01.570 --> 00:59:04.730 align:middle line:84% such that distances are preserved. 00:59:04.730 --> 00:59:08.710 align:middle line:84% Does everyone get why this fact is true? 00:59:08.710 --> 00:59:09.610 align:middle line:90% OK. 00:59:09.610 --> 00:59:12.840 align:middle line:90% So move over to the next board. 00:59:12.840 --> 00:59:17.320 align:middle line:84% The proof from here is pretty straightforward. 00:59:17.320 --> 00:59:20.700 align:middle line:84% We have a subset of our metric space inside of a Cauchy 00:59:20.700 --> 00:59:21.630 align:middle line:90% complete metric space. 00:59:21.630 --> 00:59:24.390 align:middle line:90% 00:59:24.390 --> 00:59:26.680 align:middle line:90% Doo-doo-doo. 00:59:26.680 --> 00:59:32.557 align:middle line:84% So what we have is that M is a subset of C infinity of M. 00:59:32.557 --> 00:59:34.640 align:middle line:84% And notice that this is complete, infinite, Cauchy 00:59:34.640 --> 00:59:35.140 align:middle line:90% complete. 00:59:35.140 --> 00:59:40.710 align:middle line:90% 00:59:40.710 --> 00:59:42.360 align:middle line:84% So it contains all of its limit points. 00:59:42.360 --> 00:59:47.350 align:middle line:84% And secondly, we can just take the closure of M. 00:59:47.350 --> 00:59:52.000 align:middle line:84% So let M bar be equal to C infinity-- oh, sorry, 00:59:52.000 --> 00:59:54.340 align:middle line:90% not C infinity-- 00:59:54.340 --> 01:00:05.650 align:middle line:84% be the closure of M in C infinity of M. 01:00:05.650 --> 01:00:10.425 align:middle line:84% So is this the completion that we want? 01:00:10.425 --> 01:00:12.550 align:middle line:84% Well, to show that it's the completion that we want 01:00:12.550 --> 01:00:13.900 align:middle line:90% is not too bad. 01:00:13.900 --> 01:00:16.462 align:middle line:84% You just have to check each of these properties. 01:00:16.462 --> 01:00:18.670 align:middle line:84% I'll say that again in case it was a little bit loud. 01:00:18.670 --> 01:00:20.950 align:middle line:84% We just have to check these four properties. 01:00:20.950 --> 01:00:23.170 align:middle line:84% Well, one, we know that M is definitely 01:00:23.170 --> 01:00:24.700 align:middle line:90% a subset of its closure. 01:00:24.700 --> 01:00:26.980 align:middle line:90% That's by definition. 01:00:26.980 --> 01:00:30.640 align:middle line:84% And, in fact, it also satisfies four. 01:00:30.640 --> 01:00:32.760 align:middle line:84% Secondly, we know the distances are preserved 01:00:32.760 --> 01:00:35.460 align:middle line:84% because the distances are preserved on the infinity of M 01:00:35.460 --> 01:00:37.080 align:middle line:90% when we take the closure. 01:00:37.080 --> 01:00:38.688 align:middle line:90% That's by the subspace metric. 01:00:38.688 --> 01:00:40.980 align:middle line:84% The last thing you want to check is that, in fact, this 01:00:40.980 --> 01:00:42.660 align:middle line:90% is Cauchy complete. 01:00:42.660 --> 01:00:44.580 align:middle line:84% But that fact follows from the fact 01:00:44.580 --> 01:00:48.300 align:middle line:84% that it's a closed subset of a Cauchy complete space. 01:00:48.300 --> 01:00:54.410 align:middle line:90% Recall the closed subs-- 01:00:54.410 --> 01:01:13.270 align:middle line:84% or a closed subset of a Cauchy complete metric space 01:01:13.270 --> 01:01:14.095 align:middle line:90% is Cauchy complete. 01:01:14.095 --> 01:01:22.360 align:middle line:90% 01:01:22.360 --> 01:01:23.870 align:middle line:90% How do I check this? 01:01:23.870 --> 01:01:26.585 align:middle line:84% Well, take any Cauchy sequence in your space. 01:01:26.585 --> 01:01:28.960 align:middle line:84% We know that it has to converge to something because it's 01:01:28.960 --> 01:01:30.410 align:middle line:90% in a Cauchy complete space. 01:01:30.410 --> 01:01:36.010 align:middle line:84% But that thing it converges to has to be in your closed subset 01:01:36.010 --> 01:01:37.300 align:middle line:90% because it's closed. 01:01:37.300 --> 01:01:39.420 align:middle line:84% So it contains all of its limit points. 01:01:39.420 --> 01:01:41.170 align:middle line:84% So what this tells us is that every Cauchy 01:01:41.170 --> 01:01:44.380 align:middle line:84% sequence in our closed subset converges in our closed subset. 01:01:44.380 --> 01:01:48.160 align:middle line:84% So this implies that it's Cauchy complete. 01:01:48.160 --> 01:01:50.590 align:middle line:84% So then we're basically done, right? 01:01:50.590 --> 01:01:55.190 align:middle line:84% M bar is a subset of C infinity of M. But by this isometry, 01:01:55.190 --> 01:01:57.760 align:middle line:84% we can map it back to just the metric space. 01:01:57.760 --> 01:01:59.260 align:middle line:90% So the proof is then done. 01:01:59.260 --> 01:02:04.910 align:middle line:90% 01:02:04.910 --> 01:02:07.610 align:middle line:90% So why is this so helpful? 01:02:07.610 --> 01:02:09.610 align:middle line:84% I've already given the example of the rationals. 01:02:09.610 --> 01:02:11.735 align:middle line:84% The rationals are a completion of the real numbers. 01:02:11.735 --> 01:02:13.810 align:middle line:84% But that's one that we've already known, right? 01:02:13.810 --> 01:02:16.030 align:middle line:84% That that's one that has been known 01:02:16.030 --> 01:02:21.160 align:middle line:84% for quite some time in our real analysis studies. 01:02:21.160 --> 01:02:24.730 align:middle line:84% But there's quite a few more examples that we can consider. 01:02:24.730 --> 01:02:28.510 align:middle line:84% These examples show up, for instance, in 18.102, 01:02:28.510 --> 01:02:30.950 align:middle line:90% in very, very important ways. 01:02:30.950 --> 01:02:36.155 align:middle line:84% So example-- consider normed spaces. 01:02:36.155 --> 01:02:39.590 align:middle line:90% 01:02:39.590 --> 01:02:42.350 align:middle line:84% These are vector spaces with a norm put on them. 01:02:42.350 --> 01:02:45.670 align:middle line:84% Recall this from our third lecture. 01:02:45.670 --> 01:02:47.140 align:middle line:84% What if this isn't Cauchy complete? 01:02:47.140 --> 01:02:49.515 align:middle line:84% We want Cauchy completeness for a number of our theorems, 01:02:49.515 --> 01:02:51.910 align:middle line:84% and it's very important that we do. 01:02:51.910 --> 01:02:54.790 align:middle line:84% Well, we can just take the closure of them 01:02:54.790 --> 01:02:56.650 align:middle line:90% and end up with-- 01:02:56.650 --> 01:02:58.180 align:middle line:90% or the completion of them-- 01:02:58.180 --> 01:03:02.200 align:middle line:84% and end up with what are known as Banach spaces. 01:03:02.200 --> 01:03:05.380 align:middle line:84% And Banach space theory is literally 01:03:05.380 --> 01:03:08.360 align:middle line:90% the majority of 18.102. 01:03:08.360 --> 01:03:10.070 align:middle line:84% In fact, we could do slightly more 01:03:10.070 --> 01:03:13.340 align:middle line:84% if you've heard of this from perhaps 18.06. 01:03:13.340 --> 01:03:17.412 align:middle line:84% You can complete inner product spaces, 01:03:17.412 --> 01:03:18.495 align:middle line:90% which I won't define here. 01:03:18.495 --> 01:03:21.820 align:middle line:90% 01:03:21.820 --> 01:03:23.950 align:middle line:84% But I'll simply note that the completion of them 01:03:23.950 --> 01:03:25.135 align:middle line:90% are known as Hilbert spaces. 01:03:25.135 --> 01:03:35.240 align:middle line:90% 01:03:35.240 --> 01:03:37.100 align:middle line:90% And these are just names, right? 01:03:37.100 --> 01:03:39.615 align:middle line:84% These are just names that aren't necessarily too 01:03:39.615 --> 01:03:40.740 align:middle line:90% important to you right now. 01:03:40.740 --> 01:03:43.310 align:middle line:84% But I think that they're super cool because they show up 01:03:43.310 --> 01:03:44.510 align:middle line:90% in 18.102. 01:03:44.510 --> 01:03:47.030 align:middle line:84% But perhaps an example that would be more interesting right 01:03:47.030 --> 01:03:52.070 align:middle line:84% now is how we end up with integrable functions, right? 01:03:52.070 --> 01:04:00.520 align:middle line:84% Recall Riemann integrable functions have a ton of holes-- 01:04:00.520 --> 01:04:04.360 align:middle line:90% 01:04:04.360 --> 01:04:09.100 align:middle line:90% "has a ton of holes." 01:04:09.100 --> 01:04:12.170 align:middle line:90% 01:04:12.170 --> 01:04:13.810 align:middle line:84% So, for instance, an example of this 01:04:13.810 --> 01:04:18.880 align:middle line:84% would be to take the indicator function on rationals, where 01:04:18.880 --> 01:04:22.600 align:middle line:84% the indicator function at the point x is 1 01:04:22.600 --> 01:04:26.830 align:middle line:84% where x is in the rationals and 0 otherwise, 01:04:26.830 --> 01:04:30.463 align:middle line:90% i.e., if it's irrational. 01:04:30.463 --> 01:04:31.880 align:middle line:84% How do you integrate this function 01:04:31.880 --> 01:04:33.590 align:middle line:90% using Riemann integration? 01:04:33.590 --> 01:04:35.330 align:middle line:90% The answer is that you can't. 01:04:35.330 --> 01:04:36.380 align:middle line:90% It doesn't quite work. 01:04:36.380 --> 01:04:39.860 align:middle line:84% Somehow, this should both have an integral 0 01:04:39.860 --> 01:04:42.050 align:middle line:84% because most terms are irrational, 01:04:42.050 --> 01:04:43.747 align:middle line:84% but it should also have an integral 1 01:04:43.747 --> 01:04:45.080 align:middle line:90% because most terms are rational. 01:04:45.080 --> 01:04:47.270 align:middle line:90% It's a huge issue, right? 01:04:47.270 --> 01:04:49.880 align:middle line:84% Riemann integration has a ton of flaws, 01:04:49.880 --> 01:04:52.160 align:middle line:90% this just being one of them. 01:04:52.160 --> 01:04:55.400 align:middle line:84% But, in fact, I can complete them. 01:04:55.400 --> 01:04:57.720 align:middle line:84% I can complete Riemann integrable functions. 01:04:57.720 --> 01:05:02.300 align:middle line:84% So, specifically, we can consider 01:05:02.300 --> 01:05:04.130 align:middle line:90% compactly supported functions. 01:05:04.130 --> 01:05:08.120 align:middle line:90% 01:05:08.120 --> 01:05:20.010 align:middle line:84% So, specifically, consider C C0 functions on, let's say, 01:05:20.010 --> 01:05:21.883 align:middle line:90% a metric space-- 01:05:21.883 --> 01:05:22.800 align:middle line:90% what do I want to say? 01:05:22.800 --> 01:05:25.560 align:middle line:90% I'll say the metric space x. 01:05:25.560 --> 01:05:27.570 align:middle line:90% No, I'll just say on R. 01:05:27.570 --> 01:05:30.180 align:middle line:84% So this is a set of compactly supported functions 01:05:30.180 --> 01:05:33.872 align:middle line:84% on the real numbers that are continuous. 01:05:33.872 --> 01:05:36.330 align:middle line:84% We've already talked about this before on our third lecture 01:05:36.330 --> 01:05:38.310 align:middle line:84% when we were talking about integration. 01:05:38.310 --> 01:05:40.230 align:middle line:84% Here what's so nice about this set 01:05:40.230 --> 01:05:41.760 align:middle line:84% is that the integral of a function 01:05:41.760 --> 01:05:46.320 align:middle line:90% f in the space over R-- 01:05:46.320 --> 01:05:49.710 align:middle line:84% the integral of f of x dx will be less than infinity 01:05:49.710 --> 01:05:53.120 align:middle line:84% because I know that the terms have to eventually go to 0. 01:05:53.120 --> 01:05:56.670 align:middle line:84% But, in fact, this gives us a metric on our space. 01:05:56.670 --> 01:05:59.600 align:middle line:84% I can take absolute values of f of x. 01:05:59.600 --> 01:06:05.360 align:middle line:84% And then this is suddenly our L1 metric from day one, right? 01:06:05.360 --> 01:06:09.760 align:middle line:84% Our L1 metric was such that the distance-- 01:06:09.760 --> 01:06:12.640 align:middle line:90% I think I called it I1-- 01:06:12.640 --> 01:06:16.360 align:middle line:84% I1 of f, g is equal to the integral 01:06:16.360 --> 01:06:24.730 align:middle line:84% over R of f of x minus g of x dx, where it's called L1 01:06:24.730 --> 01:06:27.384 align:middle line:84% because it's raised to the first power, then raised to the 1 01:06:27.384 --> 01:06:30.230 align:middle line:90% over 1 power. 01:06:30.230 --> 01:06:34.150 align:middle line:84% So this is our metric on the set of compactly 01:06:34.150 --> 01:06:35.560 align:middle line:90% supported functions. 01:06:35.560 --> 01:06:39.130 align:middle line:84% I can simply complete the subset of the compactly supported 01:06:39.130 --> 01:06:43.180 align:middle line:84% functions under this metric because, again, our definition 01:06:43.180 --> 01:06:46.450 align:middle line:84% of completion depends on what metric we're choosing. 01:06:46.450 --> 01:06:49.390 align:middle line:84% Right here, our metric has to restrict to the metric 01:06:49.390 --> 01:06:52.090 align:middle line:90% on our original metric space. 01:06:52.090 --> 01:06:54.060 align:middle line:84% So if I change this metric, it's going 01:06:54.060 --> 01:06:56.160 align:middle line:84% to change what the completion looks like. 01:06:56.160 --> 01:07:01.350 align:middle line:84% In our case, what we would end up with-- 01:07:01.350 --> 01:07:04.100 align:middle line:84% so, in other words, the completion 01:07:04.100 --> 01:07:09.150 align:middle line:84% of the metric space C0 C of R with respect 01:07:09.150 --> 01:07:13.410 align:middle line:84% to the I1 metric, what this will give us 01:07:13.410 --> 01:07:15.480 align:middle line:90% is the set of LP functions-- 01:07:15.480 --> 01:07:17.370 align:middle line:90% or L1 functions, sorry-- 01:07:17.370 --> 01:07:21.750 align:middle line:84% on R, which are simply the set of functions 01:07:21.750 --> 01:07:22.800 align:middle line:90% that have integral 1. 01:07:22.800 --> 01:07:29.715 align:middle line:84% So this is functions f such that the integral of f is 1. 01:07:29.715 --> 01:07:32.640 align:middle line:90% 01:07:32.640 --> 01:07:35.310 align:middle line:84% Now, here, our functions can look quite a bit weirder. 01:07:35.310 --> 01:07:37.780 align:middle line:84% They don't have to be Riemann integrable anymore. 01:07:37.780 --> 01:07:39.780 align:middle line:84% They can be Lesbegue integrable, which 01:07:39.780 --> 01:07:42.030 align:middle line:90% is what this L stands for. 01:07:42.030 --> 01:07:48.940 align:middle line:84% This integral is then known as the Lesbegue integral. 01:07:48.940 --> 01:07:52.290 align:middle line:90% 01:07:52.290 --> 01:07:54.570 align:middle line:90% And this is used all the time. 01:07:54.570 --> 01:07:56.197 align:middle line:90% This is-- oh, sorry. 01:07:56.197 --> 01:07:57.030 align:middle line:90% This shouldn't be 1. 01:07:57.030 --> 01:08:00.160 align:middle line:84% This should be less than infinity. 01:08:00.160 --> 01:08:01.808 align:middle line:84% This is used all the time, right? 01:08:01.808 --> 01:08:03.850 align:middle line:84% Because we want to be able to integrate functions 01:08:03.850 --> 01:08:06.550 align:middle line:90% that actually integrate well. 01:08:06.550 --> 01:08:09.520 align:middle line:84% And Riemann integrable functions are not it. 01:08:09.520 --> 01:08:12.370 align:middle line:90% They have a ton of holes, right? 01:08:12.370 --> 01:08:14.660 align:middle line:84% In fact, we can generalize this quite a bit more. 01:08:14.660 --> 01:08:17.229 align:middle line:84% Remember on that first day that I 01:08:17.229 --> 01:08:20.560 align:middle line:84% defined Ip, which is the integral of the distance 01:08:20.560 --> 01:08:23.620 align:middle line:84% between them to the pth power, all of which raised to the 1 01:08:23.620 --> 01:08:24.250 align:middle line:90% over p. 01:08:24.250 --> 01:08:26.790 align:middle line:90% 01:08:26.790 --> 01:08:30.029 align:middle line:84% If I consider completing the metric of continuous functions 01:08:30.029 --> 01:08:33.390 align:middle line:84% under Ip, then what I'll end up with 01:08:33.390 --> 01:08:36.630 align:middle line:84% are Lp functions, where all of a sudden this 01:08:36.630 --> 01:08:39.160 align:middle line:90% is raised to the pth power. 01:08:39.160 --> 01:08:41.970 align:middle line:84% So this is the space of Lp functions-- 01:08:41.970 --> 01:08:44.950 align:middle line:90% deeply, deeply important. 01:08:44.950 --> 01:08:46.920 align:middle line:90% This also shows up in 18.102. 01:08:46.920 --> 01:08:50.539 align:middle line:84% But it's perhaps slightly easier to see why it's so important. 01:08:50.539 --> 01:08:53.210 align:middle line:90% 01:08:53.210 --> 01:08:56.100 align:middle line:84% So I'll note one more thing, I think, 01:08:56.100 --> 01:08:58.100 align:middle line:84% and then that will mostly be it, which 01:08:58.100 --> 01:09:02.770 align:middle line:84% is recall that this set was, in fact, a normed space, right? 01:09:02.770 --> 01:09:06.109 align:middle line:84% I can define this distance as a norm. 01:09:06.109 --> 01:09:10.160 align:middle line:84% What this tells us is that the Lp spaces are Cauchy complete, 01:09:10.160 --> 01:09:11.870 align:middle line:84% right, because they're Banach spaces. 01:09:11.870 --> 01:09:15.350 align:middle line:84% They are completion of a normed space. 01:09:15.350 --> 01:09:18.319 align:middle line:84% Now, this doesn't say that we've said all the things that there 01:09:18.319 --> 01:09:20.090 align:middle line:90% are to say about Lp functions. 01:09:20.090 --> 01:09:22.580 align:middle line:84% There's still questions like, what theorems about limits 01:09:22.580 --> 01:09:23.149 align:middle line:90% do we have? 01:09:23.149 --> 01:09:26.270 align:middle line:84% What theorems about approximations do we have? 01:09:26.270 --> 01:09:28.790 align:middle line:84% But this does give us a useful intuition 01:09:28.790 --> 01:09:30.950 align:middle line:90% behind what's happening. 01:09:30.950 --> 01:09:34.880 align:middle line:84% All that's happening is we're looking at equivalence classes 01:09:34.880 --> 01:09:37.365 align:middle line:90% of Cauchy sequences. 01:09:37.365 --> 01:09:38.990 align:middle line:84% It makes a lot of sense to do this when 01:09:38.990 --> 01:09:42.399 align:middle line:90% it comes to the real numbers. 01:09:42.399 --> 01:09:44.580 align:middle line:84% But it might be weirder to think about with respect 01:09:44.580 --> 01:09:48.479 align:middle line:84% to functions or with respect to topologies and weird things 01:09:48.479 --> 01:09:49.720 align:middle line:90% like that. 01:09:49.720 --> 01:09:53.340 align:middle line:84% So, yeah, I'll just leave y'all with that thought is 01:09:53.340 --> 01:09:57.510 align:middle line:84% equivalence classes of Cauchy sequences is a useful intuition 01:09:57.510 --> 01:09:59.730 align:middle line:90% behind all of these things. 01:09:59.730 --> 01:10:01.560 align:middle line:84% I should note that this is the Lp metric. 01:10:01.560 --> 01:10:02.850 align:middle line:90% I forgot to change that one. 01:10:02.850 --> 01:10:04.710 align:middle line:90% OK. 01:10:04.710 --> 01:10:08.315 align:middle line:84% Is there anything else I want to say? 01:10:08.315 --> 01:10:10.038 align:middle line:90% I think that's it. 01:10:10.038 --> 01:10:11.830 align:middle line:84% Yeah, so unless you all have any questions, 01:10:11.830 --> 01:10:15.720 align:middle line:84% then we can end today slightly early. 01:10:15.720 --> 01:10:17.850 align:middle line:90% But, yeah, this was very neat. 01:10:17.850 --> 01:10:20.160 align:middle line:84% Next time, I'm going to talk about more applications 01:10:20.160 --> 01:10:25.720 align:middle line:90% to 18.101, 102, 103, 152. 01:10:25.720 --> 01:10:27.220 align:middle line:84% There's a ton of applications of all 01:10:27.220 --> 01:10:30.090 align:middle line:84% of this material to this later class-- oh, 901, 01:10:30.090 --> 01:10:32.190 align:middle line:84% if you're planning to take 18.901. 01:10:32.190 --> 01:10:33.780 align:middle line:84% There's a ton of applications of what 01:10:33.780 --> 01:10:36.630 align:middle line:84% we've talked about in this class so far to these later classes. 01:10:36.630 --> 01:10:39.040 align:middle line:84% And I'm going to bring up how that actually works. 01:10:39.040 --> 01:10:42.110 align:middle line:90% So see y'all next time. 01:10:42.110 --> 01:10:48.000 align:middle line:90%