WEBVTT 00:00:00.000 --> 00:00:02.934 align:middle line:90% [SQUEAKING] 00:00:02.934 --> 00:00:04.401 align:middle line:90% [RUSTLING] 00:00:04.401 --> 00:00:05.868 align:middle line:90% [CLICKING] 00:00:05.868 --> 00:00:11.740 align:middle line:90% 00:00:11.740 --> 00:00:14.620 align:middle line:84% PAIGE BRIGHT: So welcome to 18.S190, 00:00:14.620 --> 00:00:18.010 align:middle line:90% Intro to Metric Spaces. 00:00:18.010 --> 00:00:20.230 align:middle line:84% My name is Paige BRIGHT, though sometimes you'll 00:00:20.230 --> 00:00:23.050 align:middle line:90% see it as Paige Bright online. 00:00:23.050 --> 00:00:26.050 align:middle line:84% That's just because I'm going to change my name, 00:00:26.050 --> 00:00:28.035 align:middle line:90% but you can just call me Paige. 00:00:28.035 --> 00:00:31.500 align:middle line:90% 00:00:31.500 --> 00:00:33.690 align:middle line:84% And, yeah, this is Intro to Metric Spaces, 00:00:33.690 --> 00:00:36.690 align:middle line:84% where today we're going to talk about the connection between 00:00:36.690 --> 00:00:40.320 align:middle line:84% what's covered in 18.100A and what's covered in 18.100B, 00:00:40.320 --> 00:00:41.880 align:middle line:90% or, respectively, p and q. 00:00:41.880 --> 00:00:43.440 align:middle line:84% And for those of you who don't know, 00:00:43.440 --> 00:00:45.023 align:middle line:84% there's this nice little rubric that I 00:00:45.023 --> 00:00:47.400 align:middle line:84% like to draw every time I teach this class, 00:00:47.400 --> 00:00:49.373 align:middle line:84% because it was drawn for me in my first year, 00:00:49.373 --> 00:00:50.790 align:middle line:84% and it helped me really understand 00:00:50.790 --> 00:00:52.457 align:middle line:84% what the differences between the courses 00:00:52.457 --> 00:00:58.320 align:middle line:84% were, which were not CI-M, the communications class, 00:00:58.320 --> 00:01:05.430 align:middle line:84% communications class, real analysis on Euclidean space, 00:01:05.430 --> 00:01:09.480 align:middle line:84% and-- or I guess I should say more than Euclidean space. 00:01:09.480 --> 00:01:12.240 align:middle line:90% 00:01:12.240 --> 00:01:19.260 align:middle line:84% And then the classes go as follows 18.100A, B, P, and Q. 00:01:19.260 --> 00:01:21.942 align:middle line:84% But realistically what I hope to highlight today is the fact 00:01:21.942 --> 00:01:24.150 align:middle line:84% that there's not too much different between these two 00:01:24.150 --> 00:01:24.660 align:middle line:90% courses. 00:01:24.660 --> 00:01:26.160 align:middle line:90% It's just a conceptual leap. 00:01:26.160 --> 00:01:28.140 align:middle line:84% But this conceptual leap is really important 00:01:28.140 --> 00:01:33.450 align:middle line:84% to have for next courses, so 18.101, 102, 103, 901. 00:01:33.450 --> 00:01:34.980 align:middle line:84% There's any number of classes that 00:01:34.980 --> 00:01:38.100 align:middle line:84% come after this that having intuition of metric spaces 00:01:38.100 --> 00:01:40.680 align:middle line:90% will be deeply helpful for. 00:01:40.680 --> 00:01:42.240 align:middle line:84% So, yeah, metric workspaces are going 00:01:42.240 --> 00:01:47.820 align:middle line:84% to be what lies on the interim between A, B, and P and Q. 00:01:47.820 --> 00:01:50.100 align:middle line:84% A little bit about me before I jump right in. 00:01:50.100 --> 00:01:51.720 align:middle line:90% I'm a third year at MIT. 00:01:51.720 --> 00:01:54.190 align:middle line:84% I've been teaching this class for two years. 00:01:54.190 --> 00:01:55.890 align:middle line:90% This is my second year. 00:01:55.890 --> 00:01:58.440 align:middle line:84% And I started teaching it because one of my friends 00:01:58.440 --> 00:02:02.670 align:middle line:84% was in 18.102 and having a really hard time with things 00:02:02.670 --> 00:02:06.630 align:middle line:84% like norms, which is a version of a metric space which we'll 00:02:06.630 --> 00:02:08.340 align:middle line:84% talk about a little bit in this class. 00:02:08.340 --> 00:02:11.160 align:middle line:84% But it's rather unfortunate if you get to the next class, 00:02:11.160 --> 00:02:13.740 align:middle line:84% and it seems like you're unprepared for one 00:02:13.740 --> 00:02:16.470 align:middle line:84% of the introductory tools, but one 00:02:16.470 --> 00:02:18.150 align:middle line:84% that comes with a lot of baggage, 00:02:18.150 --> 00:02:20.490 align:middle line:84% a lot of conceptual baggage, to keep in mind. 00:02:20.490 --> 00:02:22.900 align:middle line:84% And hopefully today, through today's lecture, 00:02:22.900 --> 00:02:25.880 align:middle line:84% you'll see what some of those concepts are. 00:02:25.880 --> 00:02:27.530 align:middle line:84% But let's start with a simpler example 00:02:27.530 --> 00:02:30.440 align:middle line:84% before we jump right into metric spaces. 00:02:30.440 --> 00:02:35.300 align:middle line:84% We're just going to talk about what makes real analysis work. 00:02:35.300 --> 00:02:39.800 align:middle line:84% What's the basic tool of real analysis that makes it work? 00:02:39.800 --> 00:02:42.215 align:middle line:84% And as I'm certain y'all have realized through 00:02:42.215 --> 00:02:44.090 align:middle line:84% the definitions that we use in real analysis, 00:02:44.090 --> 00:02:46.760 align:middle line:84% like convergence sequences, continuous functions, 00:02:46.760 --> 00:02:52.070 align:middle line:84% all of it relies on a notion of absolute values and Euclidean 00:02:52.070 --> 00:02:53.180 align:middle line:90% distance. 00:02:53.180 --> 00:02:56.390 align:middle line:84% And so let me write out what that Euclidean distance is. 00:02:56.390 --> 00:03:02.760 align:middle line:84% So given two points x and y in Euclidean space, 00:03:02.760 --> 00:03:10.540 align:middle line:84% we call the distance between them x minus y and Rn 00:03:10.540 --> 00:03:13.420 align:middle line:84% to be the sum of the distances between each 00:03:13.420 --> 00:03:17.290 align:middle line:84% of the components squared to the 1/2. 00:03:17.290 --> 00:03:19.150 align:middle line:84% This is just the Pythagorean theorem. 00:03:19.150 --> 00:03:23.920 align:middle line:84% And this definition is known as Euclidean distance, 00:03:23.920 --> 00:03:29.560 align:middle line:84% as I'm certain y'all know from 18.02. 00:03:29.560 --> 00:03:31.480 align:middle line:84% Mostly speaking, though, we focus on R, 00:03:31.480 --> 00:03:33.820 align:middle line:84% just because most of the theory breaks down 00:03:33.820 --> 00:03:37.058 align:middle line:84% to studying everything in R. So if you don't prefer thinking 00:03:37.058 --> 00:03:39.350 align:middle line:84% in n dimensions, you can always think in one dimension, 00:03:39.350 --> 00:03:41.350 align:middle line:84% and, for the most part, everything will be fine. 00:03:41.350 --> 00:03:43.990 align:middle line:84% And in fact, we'll see why it will be fine in a moment. 00:03:43.990 --> 00:03:48.620 align:middle line:84% But what are the most important properties of this distance? 00:03:48.620 --> 00:03:56.690 align:middle line:84% Well, firstly, we have that it's symmetric, 00:03:56.690 --> 00:04:00.520 align:middle line:84% meaning that the distance from x to y 00:04:00.520 --> 00:04:06.905 align:middle line:84% is the same as the distance from y to x over Rn. 00:04:06.905 --> 00:04:08.530 align:middle line:84% And this is something we should expect. 00:04:08.530 --> 00:04:10.090 align:middle line:84% We should expect the distance for me 00:04:10.090 --> 00:04:12.970 align:middle line:84% to you is the same as the distance from you to me. 00:04:12.970 --> 00:04:19.860 align:middle line:84% We also have that it's positive or positive definite, 00:04:19.860 --> 00:04:23.460 align:middle line:84% specifically that the distances between points 00:04:23.460 --> 00:04:26.550 align:middle line:84% is always bigger than or equal to 0. 00:04:26.550 --> 00:04:31.920 align:middle line:84% And if the distance is exactly equal to 0, 00:04:31.920 --> 00:04:35.010 align:middle line:84% this is true only if-- if, and only if-- 00:04:35.010 --> 00:04:37.240 align:middle line:90% the two points are the same. 00:04:37.240 --> 00:04:39.990 align:middle line:84% So the distance between you and someone else 00:04:39.990 --> 00:04:42.150 align:middle line:84% is only 0 if you are the same person. 00:04:42.150 --> 00:04:45.063 align:middle line:90% 00:04:45.063 --> 00:04:47.230 align:middle line:84% And if any of the notation I use doesn't make sense, 00:04:47.230 --> 00:04:48.188 align:middle line:90% I'm happy to elaborate. 00:04:48.188 --> 00:04:50.760 align:middle line:84% But this is the notation for if and only if. 00:04:50.760 --> 00:04:53.340 align:middle line:84% And then finally, the most important one, arguably, 00:04:53.340 --> 00:04:56.940 align:middle line:84% though the hardest one to show most of the time, 00:04:56.940 --> 00:05:05.040 align:middle line:84% is the triangle inequality, that the distance between x and z 00:05:05.040 --> 00:05:08.370 align:middle line:84% is less than or equal to the distance from x to y 00:05:08.370 --> 00:05:10.110 align:middle line:90% plus the distance from y to z. 00:05:10.110 --> 00:05:12.810 align:middle line:90% 00:05:12.810 --> 00:05:16.710 align:middle line:84% These are the three properties that make real analysis work. 00:05:16.710 --> 00:05:18.750 align:middle line:84% Now if you think back to all those definitions 00:05:18.750 --> 00:05:21.083 align:middle line:84% that I was talking about earlier, convergence sequences, 00:05:21.083 --> 00:05:23.250 align:middle line:84% Cauchy sequences, and so on, all of them 00:05:23.250 --> 00:05:26.980 align:middle line:84% have something to do with these absolute values. 00:05:26.980 --> 00:05:28.770 align:middle line:84% And that's because these absolute values 00:05:28.770 --> 00:05:30.060 align:middle line:90% are, in fact, a metric. 00:05:30.060 --> 00:05:32.700 align:middle line:84% When I define what a metric space is, which will just 00:05:32.700 --> 00:05:34.620 align:middle line:84% be in a moment, it's simply going 00:05:34.620 --> 00:05:37.525 align:middle line:84% to be a set with these three properties. 00:05:37.525 --> 00:05:39.150 align:middle line:84% So let me go ahead and write that down. 00:05:39.150 --> 00:05:45.600 align:middle line:90% 00:05:45.600 --> 00:05:56.150 align:middle line:84% A metric space is a set x with a function 00:05:56.150 --> 00:05:58.900 align:middle line:84% d, which will be called our metric, 00:05:58.900 --> 00:06:02.830 align:middle line:84% which takes in two points in x and spits out a real number. 00:06:02.830 --> 00:06:05.240 align:middle line:84% In fact, it doesn't just spit out a real number. 00:06:05.240 --> 00:06:08.980 align:middle line:84% It spits out one that is between 0 and infinity, 00:06:08.980 --> 00:06:11.830 align:middle line:90% and not including infinity. 00:06:11.830 --> 00:06:19.960 align:middle line:84% And it satisfies the three properties listed here. 00:06:19.960 --> 00:06:21.390 align:middle line:90% And d satisfies-- 00:06:21.390 --> 00:06:28.183 align:middle line:90% 00:06:28.183 --> 00:06:30.350 align:middle line:84% I can rewrite these definitions if y'all would like, 00:06:30.350 --> 00:06:35.120 align:middle line:84% but just replace the-- what's really happening is you replace 00:06:35.120 --> 00:06:39.350 align:middle line:84% the absolute values between x and y with the distance from x 00:06:39.350 --> 00:06:41.630 align:middle line:90% to y. 00:06:41.630 --> 00:06:43.820 align:middle line:84% Does this make sense to everyone? 00:06:43.820 --> 00:06:46.450 align:middle line:84% And one point of confusion last year was, 00:06:46.450 --> 00:06:48.610 align:middle line:84% what does the notation with this times thing mean? 00:06:48.610 --> 00:06:50.193 align:middle line:84% It just means that we're taking in one 00:06:50.193 --> 00:06:54.402 align:middle line:84% point in x and another point in x, so distance from x to y. 00:06:54.402 --> 00:06:56.110 align:middle line:84% This is the definition of a metric space. 00:06:56.110 --> 00:06:58.240 align:middle line:90% It's not terribly insane. 00:06:58.240 --> 00:07:01.270 align:middle line:84% But what it allows us to do is study all the tools 00:07:01.270 --> 00:07:04.900 align:middle line:84% that we did in real analysis all of a sudden in a completely 00:07:04.900 --> 00:07:06.440 align:middle line:90% new setting. 00:07:06.440 --> 00:07:09.310 align:middle line:84% So now the first few examples I'm going to talk about 00:07:09.310 --> 00:07:10.930 align:middle line:84% are ones on Euclidean space, just 00:07:10.930 --> 00:07:13.360 align:middle line:84% because we have a lot of intuition about that already. 00:07:13.360 --> 00:07:17.200 align:middle line:84% But soon, as we'll see, we can study this on plain old sets. 00:07:17.200 --> 00:07:20.020 align:middle line:84% You can study this on topological spaces. 00:07:20.020 --> 00:07:23.800 align:middle line:90% You can study this on functions. 00:07:23.800 --> 00:07:27.970 align:middle line:84% It's a very powerful tool that generalized a lot of math 00:07:27.970 --> 00:07:30.360 align:middle line:90% when it was first invented. 00:07:30.360 --> 00:07:32.760 align:middle line:84% All right, so let's start off with some more 00:07:32.760 --> 00:07:33.510 align:middle line:90% of these examples. 00:07:33.510 --> 00:07:36.580 align:middle line:90% 00:07:36.580 --> 00:07:43.930 align:middle line:84% So another distance on our end that we can consider-- 00:07:43.930 --> 00:07:49.740 align:middle line:84% hi-- is one that takes in two points, x and y. 00:07:49.740 --> 00:07:54.000 align:middle line:84% In fact, I'll call this one d infinity for the metric-- 00:07:54.000 --> 00:07:56.550 align:middle line:84% that takes in x and y and spits out 00:07:56.550 --> 00:08:07.590 align:middle line:84% the maximum over every component of the distances. 00:08:07.590 --> 00:08:13.730 align:middle line:84% So in other words, if I have a vector here 00:08:13.730 --> 00:08:18.080 align:middle line:84% in 3-dimensional space x, and one that goes here, 00:08:18.080 --> 00:08:23.630 align:middle line:84% the largest distance would probably be the vertical one, 00:08:23.630 --> 00:08:24.682 align:middle line:90% I will just say. 00:08:24.682 --> 00:08:26.390 align:middle line:84% And that would be the Euclidean distance. 00:08:26.390 --> 00:08:28.950 align:middle line:84% It's a maximum distance between the components. 00:08:28.950 --> 00:08:34.909 align:middle line:84% So if I write x as x1 through xn, 00:08:34.909 --> 00:08:37.190 align:middle line:84% symmetric just gives out the maximum value 00:08:37.190 --> 00:08:39.900 align:middle line:84% of the differences of these components. 00:08:39.900 --> 00:08:42.225 align:middle line:90% 00:08:42.225 --> 00:08:44.100 align:middle line:84% Since you just walked in, let me just briefly 00:08:44.100 --> 00:08:46.770 align:middle line:90% explain what's happening. 00:08:46.770 --> 00:08:52.650 align:middle line:84% So we define what a metric space is as simply as 00:08:52.650 --> 00:08:55.650 align:middle line:84% set with a function that acts like a distance. 00:08:55.650 --> 00:08:57.990 align:middle line:84% And the three properties we want it to have is 00:08:57.990 --> 00:08:59.320 align:middle line:90% we want it to be symmetric. 00:08:59.320 --> 00:09:01.872 align:middle line:84% So the distance from me to you is the same as you to me. 00:09:01.872 --> 00:09:03.330 align:middle line:84% We want it to be positive definite. 00:09:03.330 --> 00:09:06.120 align:middle line:84% We don't want our distances to be negative, 00:09:06.120 --> 00:09:07.560 align:middle line:84% and the triangle inequality, which 00:09:07.560 --> 00:09:09.850 align:middle line:84% we should be a little bit more familiar with. 00:09:09.850 --> 00:09:14.280 align:middle line:84% So, yeah, that's the definition of a metric space, 00:09:14.280 --> 00:09:16.150 align:middle line:84% and I'm just defining a new example. 00:09:16.150 --> 00:09:18.720 align:middle line:84% But if I ask you on a problem set, which I will do, 00:09:18.720 --> 00:09:20.220 align:middle line:84% to prove that something is a metric, 00:09:20.220 --> 00:09:22.095 align:middle line:84% you have to prove the three properties, which 00:09:22.095 --> 00:09:26.430 align:middle line:84% isn't, in this case, terrible to do, because-- 00:09:26.430 --> 00:09:29.070 align:middle line:90% I guess prove, technically-- 00:09:29.070 --> 00:09:33.960 align:middle line:84% one, it's definitely positive-- or it's definitely positive 00:09:33.960 --> 00:09:35.520 align:middle line:84% because we're using absolute values. 00:09:35.520 --> 00:09:37.395 align:middle line:84% It's not going to give us something negative. 00:09:37.395 --> 00:09:40.050 align:middle line:84% But the thing that's a little bit harder to check sometimes 00:09:40.050 --> 00:09:47.230 align:middle line:84% is you have to check it's positive definite, so definite. 00:09:47.230 --> 00:09:53.440 align:middle line:84% So if the distance d infinity from x to y is 0, 00:09:53.440 --> 00:09:55.700 align:middle line:90% what does that tell us? 00:09:55.700 --> 00:10:00.100 align:middle line:84% Well, that tells us that for all i, 00:10:00.100 --> 00:10:05.530 align:middle line:84% the distance between xy and yi must be 0. 00:10:05.530 --> 00:10:06.920 align:middle line:90% Why is this true? 00:10:06.920 --> 00:10:09.430 align:middle line:84% Well, if it wasn't 0, if it was something slightly bigger 00:10:09.430 --> 00:10:14.500 align:middle line:84% than 0, then the supremum metric on this d infinity 00:10:14.500 --> 00:10:17.920 align:middle line:84% would make this have to be bigger than zero, right? 00:10:17.920 --> 00:10:21.100 align:middle line:84% So this implies the x is, in fact equal, to y. 00:10:21.100 --> 00:10:25.150 align:middle line:84% That's how we define equality on Euclidean space. 00:10:25.150 --> 00:10:28.060 align:middle line:84% And the opposite direction, usually a little bit easier, 00:10:28.060 --> 00:10:32.780 align:middle line:84% if x is equal to y, then the same conclusion holds. 00:10:32.780 --> 00:10:34.780 align:middle line:84% So usually that direction is a little bit easier 00:10:34.780 --> 00:10:37.035 align:middle line:84% if you assume equality showing that it's 0. 00:10:37.035 --> 00:10:39.160 align:middle line:84% But the other direction can be a little bit harder. 00:10:39.160 --> 00:10:40.030 align:middle line:90% Assume it's 0. 00:10:40.030 --> 00:10:40.960 align:middle line:90% What can you say? 00:10:40.960 --> 00:10:42.100 align:middle line:90% Or assume it's not 0. 00:10:42.100 --> 00:10:47.390 align:middle line:84% What can you say to do a proof by contradiction? 00:10:47.390 --> 00:10:50.080 align:middle line:84% And then lastly, we have to-- or, sorry, no, two more. 00:10:50.080 --> 00:10:58.343 align:middle line:84% It's definitely Symmetric the fact 00:10:58.343 --> 00:11:00.260 align:middle line:84% that it's symmetric just follows from the fact 00:11:00.260 --> 00:11:04.640 align:middle line:84% that you can just swap any of the two in the maximum. 00:11:04.640 --> 00:11:07.110 align:middle line:90% It's not too bad. 00:11:07.110 --> 00:11:09.575 align:middle line:84% And three, you have to check the triangle inequality. 00:11:09.575 --> 00:11:11.450 align:middle line:84% And this is where it can be a little bit more 00:11:11.450 --> 00:11:13.910 align:middle line:84% tricky because you might be inclined to just say, 00:11:13.910 --> 00:11:16.850 align:middle line:84% oh, it's the maximum of things that satisfy the triangle 00:11:16.850 --> 00:11:17.610 align:middle line:90% inequality. 00:11:17.610 --> 00:11:19.860 align:middle line:84% But we have to be a little bit more careful than that, 00:11:19.860 --> 00:11:21.600 align:middle line:84% and I'll explain why in a moment. 00:11:21.600 --> 00:11:23.990 align:middle line:84% So let's consider two-- three points, 00:11:23.990 --> 00:11:29.440 align:middle line:84% x, y, and z in Euclidean space, Rn. 00:11:29.440 --> 00:11:31.870 align:middle line:84% And let's suppose we're considering the distance 00:11:31.870 --> 00:11:35.530 align:middle line:84% d infinity from x to z, which I'll write down again 00:11:35.530 --> 00:11:40.000 align:middle line:84% is the maximum of i between 1 and n 00:11:40.000 --> 00:11:44.110 align:middle line:84% of the different distances between the components. 00:11:44.110 --> 00:11:46.530 align:middle line:84% Now we want to go from these to information about y. 00:11:46.530 --> 00:11:49.140 align:middle line:84% And what we want to do is exploit the fact 00:11:49.140 --> 00:11:51.630 align:middle line:84% that we know that the absolute value satisfies the triangle 00:11:51.630 --> 00:11:52.540 align:middle line:90% inequality. 00:11:52.540 --> 00:11:56.130 align:middle line:84% But this maximum function makes it a little bit more difficult. 00:11:56.130 --> 00:11:59.340 align:middle line:84% Any ideas what we could do before we use the triangle 00:11:59.340 --> 00:12:00.467 align:middle line:90% inequality? 00:12:00.467 --> 00:12:06.080 align:middle line:90% 00:12:06.080 --> 00:12:08.572 align:middle line:84% Well, in this case, there's only finitely many terms 00:12:08.572 --> 00:12:09.530 align:middle line:90% that we're considering. 00:12:09.530 --> 00:12:11.960 align:middle line:84% So we know that a maximum has to exist. 00:12:11.960 --> 00:12:13.840 align:middle line:90% So I'll just call that term j. 00:12:13.840 --> 00:12:16.630 align:middle line:84% This is as opposed to looking at the maximum, 00:12:16.630 --> 00:12:20.320 align:middle line:84% I'll just consider the distances of xj to zj. 00:12:20.320 --> 00:12:22.640 align:middle line:84% And then I can apply the triangle inequality. 00:12:22.640 --> 00:12:24.640 align:middle line:84% And this is where it's important to double-check 00:12:24.640 --> 00:12:25.960 align:middle line:90% that everything is working out. 00:12:25.960 --> 00:12:29.110 align:middle line:84% We had to know that a maximum existed before we could 00:12:29.110 --> 00:12:31.610 align:middle line:90% apply the triangle inequality. 00:12:31.610 --> 00:12:35.020 align:middle line:84% So then this is less than or equal to the distance from xj 00:12:35.020 --> 00:12:42.470 align:middle line:84% to yj plus the distance from yj to zj. 00:12:42.470 --> 00:12:45.260 align:middle line:84% And now we can apply the maximum operator again 00:12:45.260 --> 00:12:48.620 align:middle line:84% to both of these because we know that this one is 00:12:48.620 --> 00:12:53.450 align:middle line:84% going to be less than or equal to the maximum over i of xi 00:12:53.450 --> 00:12:55.370 align:middle line:90% to yi. 00:12:55.370 --> 00:12:57.230 align:middle line:84% And we know that this term will be less than 00:12:57.230 --> 00:12:59.390 align:middle line:84% or equal to the maximum of that side, 00:12:59.390 --> 00:13:06.170 align:middle line:90% so plus the maximum yi to zi. 00:13:06.170 --> 00:13:09.430 align:middle line:90% And then we're done. 00:13:09.430 --> 00:13:10.590 align:middle line:90% Any questions? 00:13:10.590 --> 00:13:12.570 align:middle line:84% Happy to reiterate any of these points. 00:13:12.570 --> 00:13:16.957 align:middle line:90% 00:13:16.957 --> 00:13:18.790 align:middle line:84% So that shows that it is, in fact, a metric. 00:13:18.790 --> 00:13:21.910 align:middle line:84% And it's a pretty useful one at that. 00:13:21.910 --> 00:13:25.090 align:middle line:84% What this is telling you is if I want to study things on Rn, 00:13:25.090 --> 00:13:27.550 align:middle line:84% I can essentially just study them on R, 00:13:27.550 --> 00:13:29.800 align:middle line:84% and things will work out the same, because we can just 00:13:29.800 --> 00:13:31.540 align:middle line:84% study the maximum of the differences 00:13:31.540 --> 00:13:32.860 align:middle line:90% between their components. 00:13:32.860 --> 00:13:35.690 align:middle line:90% 00:13:35.690 --> 00:13:42.280 align:middle line:84% Let's look at one more example on Euclidean space. 00:13:42.280 --> 00:13:44.900 align:middle line:84% Now as opposed to looking at the maximum, 00:13:44.900 --> 00:13:47.990 align:middle line:84% I could instead sum over all of the terms, which will, in fact, 00:13:47.990 --> 00:13:49.260 align:middle line:90% be a little bit easier. 00:13:49.260 --> 00:13:51.800 align:middle line:84% So I'll call this the d1 distance 00:13:51.800 --> 00:13:55.580 align:middle line:84% between x and y in Euclidean space. 00:13:55.580 --> 00:13:58.730 align:middle line:84% This is the sum of the distances between the components, 00:13:58.730 --> 00:14:01.475 align:middle line:90% so yi, xi to yi. 00:14:01.475 --> 00:14:05.240 align:middle line:90% 00:14:05.240 --> 00:14:07.740 align:middle line:84% And I'll leave you to prove that this is, in fact, a metric. 00:14:07.740 --> 00:14:10.140 align:middle line:84% But let's run through the checklist one time. 00:14:10.140 --> 00:14:13.890 align:middle line:84% One, we know that it's definitely positive. 00:14:13.890 --> 00:14:16.617 align:middle line:84% Positive definite is a little bit harder, right? 00:14:16.617 --> 00:14:18.450 align:middle line:84% I mean, if they're equal, then we definitely 00:14:18.450 --> 00:14:22.784 align:middle line:84% have that the distances between these ones are all 0. 00:14:22.784 --> 00:14:26.370 align:middle line:84% If the distance is 0, then note that we're 00:14:26.370 --> 00:14:29.440 align:middle line:84% taking the sum over non-negative things. 00:14:29.440 --> 00:14:32.160 align:middle line:84% So if the thing on the left-hand side is 0, 00:14:32.160 --> 00:14:34.740 align:middle line:84% every single term in the sum must be 0. 00:14:34.740 --> 00:14:36.843 align:middle line:84% And that allows us to use the same conclusion. 00:14:36.843 --> 00:14:38.760 align:middle line:84% And then you can apply the triangle inequality 00:14:38.760 --> 00:14:41.200 align:middle line:84% right now because we are just summing over terms. 00:14:41.200 --> 00:14:43.960 align:middle line:84% So the triangle inequality automatically applies. 00:14:43.960 --> 00:14:46.620 align:middle line:84% So if you want to write through that, feel free to. 00:14:46.620 --> 00:14:50.220 align:middle line:84% The thing that's more important that I want to note here 00:14:50.220 --> 00:14:52.350 align:middle line:84% is that this is known as the l1 metric. 00:14:52.350 --> 00:14:54.860 align:middle line:90% 00:14:54.860 --> 00:14:57.910 align:middle line:84% In fact, if we wanted to, I could just 00:14:57.910 --> 00:15:04.390 align:middle line:84% replace this d1 with a dp, raise this to the pth power, 00:15:04.390 --> 00:15:06.160 align:middle line:90% and raise this to the 1 over p. 00:15:06.160 --> 00:15:08.260 align:middle line:84% And that would give me the lp metric, 00:15:08.260 --> 00:15:13.030 align:middle line:84% which is deeply important in functional analysis. 00:15:13.030 --> 00:15:14.890 align:middle line:90% It's a tool that-- 00:15:14.890 --> 00:15:18.920 align:middle line:84% it's a surprise tool that will help us later. 00:15:18.920 --> 00:15:20.950 align:middle line:84% So if you take a class on functional analysis, 00:15:20.950 --> 00:15:22.450 align:middle line:84% this is likely one of the first ones 00:15:22.450 --> 00:15:25.190 align:middle line:84% that you'll consider, though to check that the lp metric is, 00:15:25.190 --> 00:15:27.410 align:middle line:84% in fact, a metric is a little bit more difficult. 00:15:27.410 --> 00:15:28.690 align:middle line:84% There's a problem on the first problem 00:15:28.690 --> 00:15:31.340 align:middle line:84% set that's optional if you want to work through those details. 00:15:31.340 --> 00:15:33.257 align:middle line:84% And there's a lot of hints, so that's helpful. 00:15:33.257 --> 00:15:36.680 align:middle line:90% 00:15:36.680 --> 00:15:40.810 align:middle line:84% So now that we've gone through all of these examples 00:15:40.810 --> 00:15:44.470 align:middle line:84% on Euclidean space, before I go much 00:15:44.470 --> 00:15:47.590 align:middle line:84% further, I want to explain how this actually relates 00:15:47.590 --> 00:15:49.930 align:middle line:84% to the definitions that we already have, 00:15:49.930 --> 00:15:52.370 align:middle line:84% because you might be sitting here, thinking to yourself, 00:15:52.370 --> 00:15:53.300 align:middle line:90% why does this matter? 00:15:53.300 --> 00:15:55.120 align:middle line:84% We've been studying real analysis, 00:15:55.120 --> 00:15:57.490 align:middle line:84% and now we're going back to studying a function that 00:15:57.490 --> 00:15:58.700 align:middle line:90% acts like a distance. 00:15:58.700 --> 00:16:00.090 align:middle line:90% So what? 00:16:00.090 --> 00:16:01.530 align:middle line:84% Well, these four definitions we're 00:16:01.530 --> 00:16:03.840 align:middle line:84% going to be able to rewrite in terms 00:16:03.840 --> 00:16:06.400 align:middle line:90% of sequences of a metric space. 00:16:06.400 --> 00:16:10.950 align:middle line:84% So let's write out what a conversion sequence is. 00:16:10.950 --> 00:16:24.160 align:middle line:84% Let xi be a sequence the metric space xd. 00:16:24.160 --> 00:16:31.030 align:middle line:84% And let x be just a point in the metric space. 00:16:31.030 --> 00:16:33.530 align:middle line:84% Here our sequence is just as we defined it in real analysis, 00:16:33.530 --> 00:16:37.400 align:middle line:84% it's a bijection between this set of points 00:16:37.400 --> 00:16:40.320 align:middle line:90% and the natural numbers. 00:16:40.320 --> 00:16:43.580 align:middle line:84% And then we defined convergent sequence. 00:16:43.580 --> 00:16:46.070 align:middle line:84% For all epsilon bigger than 0, there 00:16:46.070 --> 00:16:49.490 align:middle line:84% exists an n in the natural numbers 00:16:49.490 --> 00:16:54.320 align:middle line:84% such that for all n bigger than or equal to n, 00:16:54.320 --> 00:17:00.890 align:middle line:84% the distance between xn and x is less than epsilon. 00:17:00.890 --> 00:17:03.140 align:middle line:84% So this is what we mean for converge-- a sequence that 00:17:03.140 --> 00:17:06.079 align:middle line:90% converges to the point x. 00:17:06.079 --> 00:17:08.599 align:middle line:84% And this is essentially just the same definition 00:17:08.599 --> 00:17:10.819 align:middle line:84% that we've been dealing with in real analysis. 00:17:10.819 --> 00:17:14.736 align:middle line:84% Just replace the distance with absolute values. 00:17:14.736 --> 00:17:17.069 align:middle line:84% And I'm just going to go through these other definitions 00:17:17.069 --> 00:17:18.970 align:middle line:90% as well. 00:17:18.970 --> 00:17:22.319 align:middle line:84% So, same setup let xi be a sequence in xd. 00:17:22.319 --> 00:17:25.890 align:middle line:84% Then a Cauchy sequence is such that for all epsilon 00:17:25.890 --> 00:17:32.540 align:middle line:84% bigger than 0, there exists an n in the natural numbers 00:17:32.540 --> 00:17:38.810 align:middle line:84% such that for all n and m bigger than or equal to n, 00:17:38.810 --> 00:17:42.620 align:middle line:84% the distance between individual points of a sequence 00:17:42.620 --> 00:17:45.980 align:middle line:90% is less than epsilon. 00:17:45.980 --> 00:17:49.390 align:middle line:84% So here's the definition of a Cauchy sequence. 00:17:49.390 --> 00:17:51.730 align:middle line:84% Who here hasn't heard of a definition of an open set 00:17:51.730 --> 00:17:52.780 align:middle line:90% before? 00:17:52.780 --> 00:17:53.530 align:middle line:90% No worries if not. 00:17:53.530 --> 00:17:55.120 align:middle line:90% Some classes don't cover it. 00:17:55.120 --> 00:17:58.120 align:middle line:84% Yeah, let me just briefly explain up here what an open-- 00:17:58.120 --> 00:18:00.100 align:middle line:90% actually, I'll do it down here-- 00:18:00.100 --> 00:18:01.027 align:middle line:90% what an open set is. 00:18:01.027 --> 00:18:03.610 align:middle line:84% Actually, no, I'll do it up here so that y'all can keep taking 00:18:03.610 --> 00:18:06.790 align:middle line:90% notes if you need to. 00:18:06.790 --> 00:18:09.070 align:middle line:84% So an open set is just a generalization 00:18:09.070 --> 00:18:11.110 align:middle line:90% of what an open interval is. 00:18:11.110 --> 00:18:16.840 align:middle line:84% So an example of an open set in R is, for instance, 0 to 1. 00:18:16.840 --> 00:18:20.490 align:middle line:90% 00:18:20.490 --> 00:18:22.700 align:middle line:84% It's such that if I'm considering the interval from 0 00:18:22.700 --> 00:18:27.420 align:middle line:84% to 1, the thing that makes it an open interval 00:18:27.420 --> 00:18:29.520 align:middle line:84% is the fact that for any point I consider, 00:18:29.520 --> 00:18:36.510 align:middle line:84% let's say, that one, that I can choose a ball of radius epsilon 00:18:36.510 --> 00:18:40.390 align:middle line:84% around it, so a distance epsilon on both sides, 00:18:40.390 --> 00:18:45.070 align:middle line:84% such that the ball of radius epsilon around the point x 00:18:45.070 --> 00:18:48.190 align:middle line:90% is contained between 0 and 1. 00:18:48.190 --> 00:18:51.590 align:middle line:84% And I can do this for every single point in the interval 0 00:18:51.590 --> 00:18:52.730 align:middle line:90% to 1. 00:18:52.730 --> 00:18:56.080 align:middle line:84% This is what makes it an open interval. 00:18:56.080 --> 00:18:57.820 align:middle line:84% And the definition of an open set 00:18:57.820 --> 00:19:00.250 align:middle line:84% is going to be essentially the same, where 00:19:00.250 --> 00:19:03.160 align:middle line:84% instead, the definition of the ball of radius 00:19:03.160 --> 00:19:10.930 align:middle line:84% epsilon, this is defined as the set of points 00:19:10.930 --> 00:19:16.900 align:middle line:84% in your metric space such that the distance from x to y 00:19:16.900 --> 00:19:18.007 align:middle line:90% is less than epsilon. 00:19:18.007 --> 00:19:19.840 align:middle line:84% So that's the only difference between what's 00:19:19.840 --> 00:19:22.960 align:middle line:84% happening on a metric space rather than Euclidean space. 00:19:22.960 --> 00:19:27.170 align:middle line:84% Before, again, this would be absolute values. 00:19:27.170 --> 00:19:37.550 align:middle line:84% So to define an open set, a set A 00:19:37.550 --> 00:19:41.090 align:middle line:84% contained in x where x is a metric space 00:19:41.090 --> 00:19:49.351 align:middle line:84% is open if for all points A and A, 00:19:49.351 --> 00:19:52.360 align:middle line:84% there exists an epsilon bigger than 0 such 00:19:52.360 --> 00:19:57.130 align:middle line:84% that the ball of radius epsilon around A 00:19:57.130 --> 00:20:00.767 align:middle line:84% is contained in A. Pictorially, if you prefer 00:20:00.767 --> 00:20:02.600 align:middle line:84% to think about it this way, this essentially 00:20:02.600 --> 00:20:04.890 align:middle line:84% means that there is no boundary on your set. 00:20:04.890 --> 00:20:08.090 align:middle line:84% So if I were to draw a conceptual diagram of what's 00:20:08.090 --> 00:20:11.780 align:middle line:90% happening here, here's my set x. 00:20:11.780 --> 00:20:15.900 align:middle line:84% Here's my subset A that lies in it. 00:20:15.900 --> 00:20:18.220 align:middle line:84% And I'm just cutting out the boundary. 00:20:18.220 --> 00:20:21.430 align:middle line:84% And this allows us to say, OK, I can get for any point, 00:20:21.430 --> 00:20:24.210 align:middle line:84% even as close as we want to the boundary of A, 00:20:24.210 --> 00:20:26.850 align:middle line:84% I can still squeeze in a little ball there. 00:20:26.850 --> 00:20:29.370 align:middle line:84% And in fact, thinking about it in this conceptual drawing 00:20:29.370 --> 00:20:30.510 align:middle line:90% is deeply important. 00:20:30.510 --> 00:20:33.740 align:middle line:84% And I'll give an example of this in a moment. 00:20:33.740 --> 00:20:35.490 align:middle line:84% But this is the definition of an open set. 00:20:35.490 --> 00:20:39.040 align:middle line:84% Does everyone feel a little bit comfortable with what this is? 00:20:39.040 --> 00:20:39.540 align:middle line:90% Cool. 00:20:39.540 --> 00:20:40.620 align:middle line:84% I know it's just a little bit weird, 00:20:40.620 --> 00:20:42.120 align:middle line:84% because some classes don't cover it, 00:20:42.120 --> 00:20:45.195 align:middle line:84% because on 100A, you don't particularly need it as much. 00:20:45.195 --> 00:20:47.840 align:middle line:90% 00:20:47.840 --> 00:20:49.550 align:middle line:84% Now continuous functions are going 00:20:49.550 --> 00:20:52.700 align:middle line:84% to be a little bit weirder, because before when we studied 00:20:52.700 --> 00:20:56.780 align:middle line:84% continuous functions, we studied ones from R to R. 00:20:56.780 --> 00:20:58.580 align:middle line:84% It takes in values in the real numbers, 00:20:58.580 --> 00:21:00.260 align:middle line:90% and it spits out a real number. 00:21:00.260 --> 00:21:02.630 align:middle line:84% But now that we have a notion of a metric space, 00:21:02.630 --> 00:21:05.750 align:middle line:84% we can now study continuous functions between them, 00:21:05.750 --> 00:21:07.430 align:middle line:84% because all we really need again, 00:21:07.430 --> 00:21:10.100 align:middle line:90% is absolute values, right? 00:21:10.100 --> 00:21:15.770 align:middle line:84% So let f be a function between x and y where both of these 00:21:15.770 --> 00:21:16.820 align:middle line:90% are metric spaces. 00:21:16.820 --> 00:21:18.320 align:middle line:84% In fact, I'll write that out now. 00:21:18.320 --> 00:21:23.540 align:middle line:84% I'm going to say that x has metric dx, and y has metric dy. 00:21:23.540 --> 00:21:25.040 align:middle line:84% And this is the appropriate notation 00:21:25.040 --> 00:21:29.220 align:middle line:84% for noting which one goes with which one. 00:21:29.220 --> 00:21:39.180 align:middle line:84% Then I say that f is continuous, continuous, 00:21:39.180 --> 00:21:42.750 align:middle line:84% if for all epsilon bigger than 0 there exists a delta bigger 00:21:42.750 --> 00:21:47.220 align:middle line:84% than 0, such that if the distance between two points x 00:21:47.220 --> 00:21:49.770 align:middle line:90% and y is less than delta-- 00:21:49.770 --> 00:21:53.220 align:middle line:84% or, sorry, dx-- then the distances 00:21:53.220 --> 00:22:01.523 align:middle line:84% in the space y of f of x and f of y is less than epsilon. 00:22:01.523 --> 00:22:03.690 align:middle line:84% Now let's double-check that this definition actually 00:22:03.690 --> 00:22:04.297 align:middle line:90% makes sense. 00:22:04.297 --> 00:22:05.880 align:middle line:84% This is a very helpful way to remember 00:22:05.880 --> 00:22:07.950 align:middle line:90% what metric goes where. 00:22:07.950 --> 00:22:12.070 align:middle line:84% x and y, again, are two points in x, right? 00:22:12.070 --> 00:22:13.800 align:middle line:84% So it only makes sense to consider 00:22:13.800 --> 00:22:16.920 align:middle line:84% the distance on the metric space x acting on them. 00:22:16.920 --> 00:22:20.310 align:middle line:84% And then f takes points in x to points in y 00:22:20.310 --> 00:22:24.530 align:middle line:84% So here on the right-hand side, we should have the metric on y. 00:22:24.530 --> 00:22:26.300 align:middle line:84% And this is the definition of continuous. 00:22:26.300 --> 00:22:29.600 align:middle line:84% I'm going to prove that a very specific special operator is 00:22:29.600 --> 00:22:31.160 align:middle line:90% continuous today. 00:22:31.160 --> 00:22:35.030 align:middle line:84% And that will be helpful conceptually. 00:22:35.030 --> 00:22:36.590 align:middle line:90% But this is far more general. 00:22:36.590 --> 00:22:39.510 align:middle line:84% If you prefer, you can just let y be the set of real numbers. 00:22:39.510 --> 00:22:42.980 align:middle line:84% And this is still a really powerful tool already 00:22:42.980 --> 00:22:46.130 align:middle line:84% because now we can study them between metric spaces x 00:22:46.130 --> 00:22:47.375 align:middle line:90% and the real numbers. 00:22:47.375 --> 00:22:50.835 align:middle line:90% 00:22:50.835 --> 00:22:53.210 align:middle line:84% So I just wanted to highlight what these four definitions 00:22:53.210 --> 00:22:56.810 align:middle line:90% convert over to in this class. 00:22:56.810 --> 00:22:59.930 align:middle line:90% Tomorrow-- not tomorrow, sorry. 00:22:59.930 --> 00:23:03.080 align:middle line:84% On Thursday, we'll be proving quite a few theorems 00:23:03.080 --> 00:23:06.770 align:middle line:90% about all of these spaces. 00:23:06.770 --> 00:23:09.050 align:middle line:84% Yeah, on Thursday we'll be proving 00:23:09.050 --> 00:23:13.307 align:middle line:84% quite a bit of properties about these definitions. 00:23:13.307 --> 00:23:15.140 align:middle line:84% In fact, I'm calling that the general theory 00:23:15.140 --> 00:23:16.160 align:middle line:90% of metric spaces. 00:23:16.160 --> 00:23:19.490 align:middle line:84% It will be a very intense class in terms of proof writing. 00:23:19.490 --> 00:23:23.270 align:middle line:84% But I wanted to bring them up today so that y'all all saw 00:23:23.270 --> 00:23:25.220 align:middle line:84% that it wasn't just random stuff. 00:23:25.220 --> 00:23:30.280 align:middle line:90% 00:23:30.280 --> 00:23:33.880 align:middle line:84% So though this is all good and dandy, 00:23:33.880 --> 00:23:36.970 align:middle line:84% we've only talked about Euclidean metrics. 00:23:36.970 --> 00:23:38.410 align:middle line:90% And this is fine. 00:23:38.410 --> 00:23:40.233 align:middle line:84% But things get a little bit weirder 00:23:40.233 --> 00:23:41.650 align:middle line:84% once you go from finite dimensions 00:23:41.650 --> 00:23:43.180 align:middle line:90% to infinite dimensions. 00:23:43.180 --> 00:23:45.860 align:middle line:84% And in fact, the definitions of, for instance, 00:23:45.860 --> 00:23:50.740 align:middle line:84% a continuous function will look a little bit different, right? 00:23:50.740 --> 00:23:52.390 align:middle line:84% So let's give an example of one that's 00:23:52.390 --> 00:23:57.040 align:middle line:84% just a little bit weirder than a set of points 00:23:57.040 --> 00:23:58.120 align:middle line:90% in Euclidean space. 00:23:58.120 --> 00:24:01.690 align:middle line:84% Actually, I'm going to rearrange this a little bit. 00:24:01.690 --> 00:24:04.900 align:middle line:84% First, I want to give a really weird example that's 00:24:04.900 --> 00:24:07.900 align:middle line:84% pretty simple, but interesting, nonetheless. 00:24:07.900 --> 00:24:11.155 align:middle line:90% Let x be any set. 00:24:11.155 --> 00:24:15.620 align:middle line:90% 00:24:15.620 --> 00:24:19.830 align:middle line:84% Then I define the metric on x, so 00:24:19.830 --> 00:24:24.420 align:middle line:84% the metric between two points x and y, could be as follows-- 00:24:24.420 --> 00:24:30.240 align:middle line:84% 1 if x is not equal to y, and 0 otherwise. 00:24:30.240 --> 00:24:36.000 align:middle line:90% 00:24:36.000 --> 00:24:38.630 align:middle line:84% So if x is equal to y, then it's going to be 0. 00:24:38.630 --> 00:24:39.780 align:middle line:90% Why is this important? 00:24:39.780 --> 00:24:42.110 align:middle line:84% Because this is telling you that every single set 00:24:42.110 --> 00:24:43.610 align:middle line:90% you can give a metric to. 00:24:43.610 --> 00:24:47.000 align:middle line:84% Now that-- this metric isn't all too interesting, 00:24:47.000 --> 00:24:50.780 align:middle line:84% but it is a useful example to keep in mind, right? 00:24:50.780 --> 00:24:53.930 align:middle line:84% And let's prove that this is a metric, because even 00:24:53.930 --> 00:24:55.520 align:middle line:84% though it's a simple function, it's 00:24:55.520 --> 00:24:57.350 align:middle line:90% going to be a little bit weird. 00:24:57.350 --> 00:25:07.990 align:middle line:84% One, definitely positive or specifically non-negative. 00:25:07.990 --> 00:25:12.260 align:middle line:84% I guess I should say that, non-negative. 00:25:12.260 --> 00:25:15.260 align:middle line:84% Now we just have to check positive definiteness. 00:25:15.260 --> 00:25:21.680 align:middle line:84% So if x is equal to y, then the distance from x to y is 0. 00:25:21.680 --> 00:25:27.050 align:middle line:84% And the distance from x to y is 0 only if x is equal to y. 00:25:27.050 --> 00:25:29.360 align:middle line:84% That's the definition of the metric. 00:25:29.360 --> 00:25:39.460 align:middle line:84% Two, it's definitely symmetric because equality is symmetric. 00:25:39.460 --> 00:25:43.000 align:middle line:84% And finally, three, the hardest one, triangle inequality. 00:25:43.000 --> 00:25:48.030 align:middle line:90% 00:25:48.030 --> 00:25:49.290 align:middle line:90% Why is this one the hardest? 00:25:49.290 --> 00:25:50.790 align:middle line:84% Because now we have three points, 00:25:50.790 --> 00:25:54.150 align:middle line:84% and we have a binary acting on them, right? 00:25:54.150 --> 00:26:02.670 align:middle line:90% So let x, y, and z be in x. 00:26:02.670 --> 00:26:04.560 align:middle line:84% Then we have three possibilities, 00:26:04.560 --> 00:26:08.340 align:middle line:84% or quite a few more, but I'll state it in generality. 00:26:08.340 --> 00:26:14.730 align:middle line:84% One, x is not equal to y, y is not equal to z, 00:26:14.730 --> 00:26:17.702 align:middle line:90% and z is not equal to x. 00:26:17.702 --> 00:26:23.810 align:middle line:84% Two, x is equal to y, but y is not equal to z. 00:26:23.810 --> 00:26:26.615 align:middle line:84% Or three, all of them are, in fact, equal. 00:26:26.615 --> 00:26:30.420 align:middle line:90% 00:26:30.420 --> 00:26:32.130 align:middle line:84% Why are these the only three cases? 00:26:32.130 --> 00:26:33.660 align:middle line:84% Well, because x, y, and z that we're 00:26:33.660 --> 00:26:35.620 align:middle line:84% choosing are just arbitrary, right? 00:26:35.620 --> 00:26:38.760 align:middle line:84% So if I wanted to decide the case where y is equal to z, 00:26:38.760 --> 00:26:43.250 align:middle line:84% but x is not equal to y, I would just relabel them. 00:26:43.250 --> 00:26:46.460 align:middle line:84% And so these are the three cases we have to check to make sure 00:26:46.460 --> 00:26:49.040 align:middle line:84% that our metric is actually a metric, which happens in cases 00:26:49.040 --> 00:26:50.900 align:middle line:84% like this, where our metric is defined 00:26:50.900 --> 00:26:55.257 align:middle line:84% via a binary or similarly simple cases. 00:26:55.257 --> 00:26:56.465 align:middle line:90% So let's check each of these. 00:26:56.465 --> 00:26:59.238 align:middle line:90% 00:26:59.238 --> 00:27:00.530 align:middle line:90% Actually, I'll just do it here. 00:27:00.530 --> 00:27:02.600 align:middle line:90% I can squeeze it in here. 00:27:02.600 --> 00:27:07.190 align:middle line:84% If none of them are equal, then the distance from x to z 00:27:07.190 --> 00:27:09.623 align:middle line:84% is just going to be 1, by definition. 00:27:09.623 --> 00:27:11.540 align:middle line:84% And this is certainly less than or equal to 2, 00:27:11.540 --> 00:27:14.370 align:middle line:84% which is the sum of the other two distances. 00:27:14.370 --> 00:27:16.190 align:middle line:90% So we're good there. 00:27:16.190 --> 00:27:19.790 align:middle line:90% Two, the distance from x to z-- 00:27:19.790 --> 00:27:21.020 align:middle line:90% what's this going to be? 00:27:21.020 --> 00:27:25.800 align:middle line:90% 00:27:25.800 --> 00:27:26.300 align:middle line:90% Anyone? 00:27:26.300 --> 00:27:29.440 align:middle line:90% 00:27:29.440 --> 00:27:31.750 align:middle line:84% If x is equal to y, and y is equal to z, 00:27:31.750 --> 00:27:33.430 align:middle line:90% what's the distance from x to z? 00:27:33.430 --> 00:27:33.930 align:middle line:90% Yeah? 00:27:33.930 --> 00:27:34.810 align:middle line:90% STUDENT: 1? 00:27:34.810 --> 00:27:38.320 align:middle line:84% PAIGE BRIGHT: 1, exactly, because x being equal to y 00:27:38.320 --> 00:27:41.110 align:middle line:90% means x is also not equal to z. 00:27:41.110 --> 00:27:51.570 align:middle line:84% And this is equal to 1, which is the distance from x 00:27:51.570 --> 00:27:56.640 align:middle line:84% to y plus the distance from y to z, because only one of them 00:27:56.640 --> 00:27:57.750 align:middle line:90% is 1. 00:27:57.750 --> 00:27:59.220 align:middle line:84% And then, of course, the last case, 00:27:59.220 --> 00:28:01.350 align:middle line:84% when you plug in the x equal to y is equal to 0, 00:28:01.350 --> 00:28:03.990 align:middle line:84% we're just going to get 0 is equal to 0 plus 0. 00:28:03.990 --> 00:28:05.515 align:middle line:90% So we're all good there. 00:28:05.515 --> 00:28:07.890 align:middle line:84% Now you're going to have an example on your homework that 00:28:07.890 --> 00:28:11.700 align:middle line:84% is similarly based off of a binary, where instead, 00:28:11.700 --> 00:28:18.870 align:middle line:84% the binary is going to be based off of if three points x, y, 00:28:18.870 --> 00:28:22.650 align:middle line:84% and z or specifically-- sorry, I'll just say x and y-- 00:28:22.650 --> 00:28:31.500 align:middle line:84% where if x and y are in R2, then the distance between x and y 00:28:31.500 --> 00:28:41.460 align:middle line:84% is just going to be the regular distance in R2 if x, y, and 0 00:28:41.460 --> 00:28:51.480 align:middle line:84% are colinear and the sum of the two otherwise. 00:28:51.480 --> 00:28:56.400 align:middle line:84% So to draw a picture of what I mean here, 00:28:56.400 --> 00:29:03.220 align:middle line:84% if x and y lie on the same line, then the distance 00:29:03.220 --> 00:29:04.570 align:middle line:90% is just as normal. 00:29:04.570 --> 00:29:09.820 align:middle line:84% If they're not, then I have to add up 00:29:09.820 --> 00:29:12.920 align:middle line:90% the two distances between them. 00:29:12.920 --> 00:29:14.290 align:middle line:90% This is similarly a binary. 00:29:14.290 --> 00:29:15.610 align:middle line:84% And when you're doing this on your homework, 00:29:15.610 --> 00:29:17.410 align:middle line:84% one thing I would suggest is breaking it 00:29:17.410 --> 00:29:19.570 align:middle line:90% up into similar casework. 00:29:19.570 --> 00:29:23.080 align:middle line:84% Either they all are colinear, or none of them are, or only one 00:29:23.080 --> 00:29:24.015 align:middle line:90% of them is. 00:29:24.015 --> 00:29:25.390 align:middle line:84% That's the sort of thing that you 00:29:25.390 --> 00:29:28.873 align:middle line:84% should do on a problem like this on the homework. 00:29:28.873 --> 00:29:30.790 align:middle line:84% This casework it can be a little bit annoying, 00:29:30.790 --> 00:29:32.470 align:middle line:90% but is useful to do. 00:29:32.470 --> 00:29:35.680 align:middle line:90% 00:29:35.680 --> 00:29:38.500 align:middle line:84% So even though this metric is vaguely interesting in that it 00:29:38.500 --> 00:29:41.440 align:middle line:84% makes any set a metric space, let's start 00:29:41.440 --> 00:29:45.760 align:middle line:84% studying a set that we care about a little bit more, 00:29:45.760 --> 00:29:49.460 align:middle line:90% continuous functions. 00:29:49.460 --> 00:29:53.740 align:middle line:84% And we can, in fact, define a metric on them. 00:29:53.740 --> 00:30:00.430 align:middle line:84% So I'm going to define c0 of ab to be 00:30:00.430 --> 00:30:09.470 align:middle line:84% the set of continuous functions from ab to R. 00:30:09.470 --> 00:30:12.050 align:middle line:84% And I'm going to define a metric on them. 00:30:12.050 --> 00:30:15.290 align:middle line:90% And the metric-- so, example-- 00:30:15.290 --> 00:30:18.440 align:middle line:84% or I guess I should say this is the definition. 00:30:18.440 --> 00:30:23.750 align:middle line:84% Example, if I'm considering two functions f 00:30:23.750 --> 00:30:28.460 align:middle line:84% and g in the set of continuous functions on the interval a 00:30:28.460 --> 00:30:33.110 align:middle line:84% to b, then the distance between these two functions 00:30:33.110 --> 00:30:35.240 align:middle line:84% is going to be the supremum over all 00:30:35.240 --> 00:30:44.640 align:middle line:84% x in a to b of the distances between the two points 00:30:44.640 --> 00:30:46.060 align:middle line:90% or between the two functions. 00:30:46.060 --> 00:30:49.318 align:middle line:84% This is what I'm claiming is a metric. 00:30:49.318 --> 00:30:51.110 align:middle line:84% And it's going to be a little bit difficult 00:30:51.110 --> 00:30:53.275 align:middle line:84% to prove in one of the steps, if you had to guess, 00:30:53.275 --> 00:30:54.650 align:middle line:84% probably the triangle inequality, 00:30:54.650 --> 00:30:57.095 align:middle line:84% because that's where everything messes up. 00:30:57.095 --> 00:30:59.220 align:middle line:84% But let's go ahead and check that this is a metric. 00:30:59.220 --> 00:31:02.440 align:middle line:90% 00:31:02.440 --> 00:31:04.740 align:middle line:84% Well, specifically, first, it's definitely 00:31:04.740 --> 00:31:13.970 align:middle line:84% going to be symmetric because the distance from f to g 00:31:13.970 --> 00:31:17.450 align:middle line:84% is equal to the supremum of the distances between the functions 00:31:17.450 --> 00:31:21.050 align:middle line:84% f and g, which is certainly equal to the distances 00:31:21.050 --> 00:31:26.520 align:middle line:84% of the supremum from g of x minus f of x. 00:31:26.520 --> 00:31:31.683 align:middle line:84% Now one thing to be careful about here, though, is-- 00:31:31.683 --> 00:31:33.350 align:middle line:84% you don't have to be as careful about it 00:31:33.350 --> 00:31:35.030 align:middle line:84% with symmetry, because symmetry is a little bit more 00:31:35.030 --> 00:31:37.490 align:middle line:84% clear that you can mess around with these operations. 00:31:37.490 --> 00:31:40.550 align:middle line:84% With the triangle inequality, it becomes a little bit harder. 00:31:40.550 --> 00:31:42.620 align:middle line:84% There is a technical thing I'll come back 00:31:42.620 --> 00:31:43.880 align:middle line:90% to you later about this step. 00:31:43.880 --> 00:31:46.100 align:middle line:84% But essentially, symmetry is done. 00:31:46.100 --> 00:31:49.130 align:middle line:90% 00:31:49.130 --> 00:31:51.530 align:middle line:90% Two, the positive definite. 00:31:51.530 --> 00:31:55.460 align:middle line:90% 00:31:55.460 --> 00:31:57.210 align:middle line:84% Well, it's definitely going to be positive 00:31:57.210 --> 00:32:00.334 align:middle line:84% because it's absolute value of zn or non-negative. 00:32:00.334 --> 00:32:04.070 align:middle line:90% 00:32:04.070 --> 00:32:07.250 align:middle line:84% And we have to check definiteness. 00:32:07.250 --> 00:32:11.350 align:middle line:84% Well, if f is equal to g, then by definition, f of x 00:32:11.350 --> 00:32:15.260 align:middle line:84% is equal to f of g of x everywhere 00:32:15.260 --> 00:32:22.880 align:middle line:84% for all x, which implies that the distance between f and g 00:32:22.880 --> 00:32:24.290 align:middle line:90% is 0. 00:32:24.290 --> 00:32:26.390 align:middle line:84% Now what if the distance between f and g is 0? 00:32:26.390 --> 00:32:34.260 align:middle line:90% 00:32:34.260 --> 00:32:38.620 align:middle line:84% How do we conclude that f and g are, in fact, equal? 00:32:38.620 --> 00:32:41.760 align:middle line:84% You know that because it's a supremum, everywhere, 00:32:41.760 --> 00:32:44.350 align:middle line:84% at every single point, they have to be equal. 00:32:44.350 --> 00:32:48.580 align:middle line:84% You can also make an argument via continuity, which is-- 00:32:48.580 --> 00:32:50.483 align:middle line:84% which makes use of the extreme value theorem. 00:32:50.483 --> 00:32:53.690 align:middle line:84% But we don't have to get into that much detail. 00:32:53.690 --> 00:32:54.940 align:middle line:90% The harder part, though-- 00:32:54.940 --> 00:32:57.140 align:middle line:90% so the positive definiteness. 00:32:57.140 --> 00:32:58.780 align:middle line:84% The hardest part, though, is going 00:32:58.780 --> 00:33:02.080 align:middle line:84% to be the triangle inequality, because here, we 00:33:02.080 --> 00:33:03.940 align:middle line:84% do have to use the extreme value theorem. 00:33:03.940 --> 00:33:15.985 align:middle line:90% 00:33:15.985 --> 00:33:25.536 align:middle line:84% A triangle inequality-- so let f, g, and h 00:33:25.536 --> 00:33:28.200 align:middle line:84% be a continuous functions on a to b. 00:33:28.200 --> 00:33:31.470 align:middle line:90% 00:33:31.470 --> 00:33:41.720 align:middle line:84% Then let's consider the distance from f to h minus h of x. 00:33:41.720 --> 00:33:44.660 align:middle line:84% We want to go from this to information about f and g and g 00:33:44.660 --> 00:33:48.000 align:middle line:84% and h so we can apply the triangle inequality. 00:33:48.000 --> 00:33:50.390 align:middle line:84% Well, to do so, this is the only part 00:33:50.390 --> 00:33:53.103 align:middle line:84% where we use the fact that the functions are continuous, 00:33:53.103 --> 00:33:54.770 align:middle line:84% because knowing that they're continuous, 00:33:54.770 --> 00:33:57.860 align:middle line:84% we can apply the extreme value theorem to them, right? 00:33:57.860 --> 00:34:00.560 align:middle line:84% So we know that this supremum has to exist somewhere. 00:34:00.560 --> 00:34:03.590 align:middle line:84% And I'm going to just let that point be y. 00:34:03.590 --> 00:34:06.980 align:middle line:84% So this is going to be equal to the distance from f of y 00:34:06.980 --> 00:34:08.060 align:middle line:90% to h of y. 00:34:08.060 --> 00:34:10.679 align:middle line:90% 00:34:10.679 --> 00:34:12.260 align:middle line:90% So what now? 00:34:12.260 --> 00:34:14.760 align:middle line:84% Well, now we can just directly apply the triangle inequality 00:34:14.760 --> 00:34:16.239 align:middle line:84% because these are absolute values. 00:34:16.239 --> 00:34:18.659 align:middle line:84% And we know that this is going to be less than 00:34:18.659 --> 00:34:21.000 align:middle line:84% or equal to the distance from f of y 00:34:21.000 --> 00:34:28.804 align:middle line:84% to g of y plus absolute value of g of y minus h of y. 00:34:28.804 --> 00:34:30.179 align:middle line:84% And then we can take the supremum 00:34:30.179 --> 00:34:33.989 align:middle line:84% of both of these individually, and we'll conclude the proof. 00:34:33.989 --> 00:34:36.270 align:middle line:84% But the thing we had to make sure of first 00:34:36.270 --> 00:34:39.300 align:middle line:84% was we couldn't just apply the triangle inequality right away. 00:34:39.300 --> 00:34:41.580 align:middle line:84% The supremum operator acting on it 00:34:41.580 --> 00:34:43.620 align:middle line:84% made it so that we had to go through these steps 00:34:43.620 --> 00:34:44.850 align:middle line:90% individually. 00:34:44.850 --> 00:34:47.010 align:middle line:84% And in fact, if we wanted to be really careful, 00:34:47.010 --> 00:34:49.788 align:middle line:84% it might have made sense to do it here as well, where 00:34:49.788 --> 00:34:51.330 align:middle line:84% we know the supremum exists, and it's 00:34:51.330 --> 00:34:54.900 align:middle line:84% going to be equal to if I swap the two orders. 00:34:54.900 --> 00:34:58.920 align:middle line:84% But really, it's important in the triangle inequality. 00:34:58.920 --> 00:35:04.490 align:middle line:84% And the thing to note here is that this is precisely 00:35:04.490 --> 00:35:06.530 align:middle line:84% the same sort of argument we had to do right 00:35:06.530 --> 00:35:08.480 align:middle line:90% here for the infinity metric. 00:35:08.480 --> 00:35:11.570 align:middle line:84% We knew that maximum existed, so we just went with it and ran. 00:35:11.570 --> 00:35:13.687 align:middle line:84% And that's, generally speaking, good advice. 00:35:13.687 --> 00:35:15.770 align:middle line:84% If we're dealing with something that's continuous, 00:35:15.770 --> 00:35:18.380 align:middle line:84% try narrowing your focus down as much as 00:35:18.380 --> 00:35:21.380 align:middle line:84% possible to a single point or to a single function, whatever 00:35:21.380 --> 00:35:24.065 align:middle line:84% you can do to finish off your proof. 00:35:24.065 --> 00:35:27.800 align:middle line:90% 00:35:27.800 --> 00:35:30.830 align:middle line:84% Now, depending on your background with analysis, 00:35:30.830 --> 00:35:32.930 align:middle line:84% you might be wondering to yourself, why do we only 00:35:32.930 --> 00:35:34.010 align:middle line:90% care about c0? 00:35:34.010 --> 00:35:37.445 align:middle line:90% Why don't we care about-- 00:35:37.445 --> 00:35:39.050 align:middle line:90% what is c, c0? 00:35:39.050 --> 00:35:41.030 align:middle line:90% Why is that 0 there? 00:35:41.030 --> 00:35:43.040 align:middle line:84% Well, in fact, the reason it's there 00:35:43.040 --> 00:35:45.920 align:middle line:84% is because we can study differentiable functions, 00:35:45.920 --> 00:35:49.980 align:middle line:84% specifically continuously differentiable functions. 00:35:49.980 --> 00:35:55.360 align:middle line:84% So, definition-- actually, I'll just define it for ck. 00:35:55.360 --> 00:36:05.490 align:middle line:84% This is the set of continuous functions on ab such 00:36:05.490 --> 00:36:22.260 align:middle line:84% that the first k derivatives of f1 exist, 00:36:22.260 --> 00:36:25.855 align:middle line:90% and two are continuous. 00:36:25.855 --> 00:36:27.230 align:middle line:84% This is the set of what are known 00:36:27.230 --> 00:36:30.050 align:middle line:84% as continuously differentiable functions, so ones in which 00:36:30.050 --> 00:36:33.200 align:middle line:84% when I take the derivative, this-- 00:36:33.200 --> 00:36:36.170 align:middle line:84% the derivative is going to be continuous. 00:36:36.170 --> 00:36:38.990 align:middle line:84% And once we have this new set defined, 00:36:38.990 --> 00:36:40.805 align:middle line:84% we can find even weirder metrics. 00:36:40.805 --> 00:36:44.203 align:middle line:90% 00:36:44.203 --> 00:36:45.870 align:middle line:84% Well, I guess not all that much weirder, 00:36:45.870 --> 00:36:47.760 align:middle line:84% but ones that are slightly more complicated, 00:36:47.760 --> 00:36:48.750 align:middle line:90% I guess I should say. 00:36:48.750 --> 00:36:55.070 align:middle line:90% 00:36:55.070 --> 00:36:59.900 align:middle line:84% So, example-- let's just consider the ones on c1 of ab. 00:36:59.900 --> 00:37:03.640 align:middle line:90% 00:37:03.640 --> 00:37:08.080 align:middle line:90% We want to show that-- 00:37:08.080 --> 00:37:13.255 align:middle line:90% the following, dc1 of f and g. 00:37:13.255 --> 00:37:15.130 align:middle line:84% I'm going to define what the metric is first. 00:37:15.130 --> 00:37:17.720 align:middle line:84% And what we want to show is that this is, in fact, a metric. 00:37:17.720 --> 00:37:19.990 align:middle line:84% This is going to be the supremum over x 00:37:19.990 --> 00:37:35.030 align:middle line:84% in a to b of f of x minus g of x plus the supremum of x and ab 00:37:35.030 --> 00:37:41.532 align:middle line:84% of f prime of x plus g prime-- or, sorry, minus g prime of x. 00:37:41.532 --> 00:37:42.990 align:middle line:84% This is going to be our new metric. 00:37:42.990 --> 00:37:45.800 align:middle line:90% 00:37:45.800 --> 00:37:48.940 align:middle line:84% Proof that this is, in fact, a metric-- 00:37:48.940 --> 00:37:52.274 align:middle line:84% proof-- one, definitely non-negative. 00:37:52.274 --> 00:37:55.260 align:middle line:90% 00:37:55.260 --> 00:37:57.355 align:middle line:84% But we have to check positive definiteness. 00:37:57.355 --> 00:38:03.540 align:middle line:90% 00:38:03.540 --> 00:38:07.940 align:middle line:84% Well, if f is equal to g, then f of x 00:38:07.940 --> 00:38:10.053 align:middle line:90% is equal to g of x everywhere. 00:38:10.053 --> 00:38:14.690 align:middle line:90% 00:38:14.690 --> 00:38:17.375 align:middle line:84% And in fact, f prime will be equal to a g prime, which 00:38:17.375 --> 00:38:19.250 align:middle line:84% implies the same result, because you can just 00:38:19.250 --> 00:38:21.300 align:middle line:90% take the derivative of this. 00:38:21.300 --> 00:38:24.230 align:middle line:90% So that includes one direction. 00:38:24.230 --> 00:38:30.660 align:middle line:84% If the distance on c1 of f to g is 0, what can we say? 00:38:30.660 --> 00:38:33.280 align:middle line:90% 00:38:33.280 --> 00:38:37.140 align:middle line:84% Well, again, we're summing over non-negative things. 00:38:37.140 --> 00:38:39.720 align:middle line:84% So if the sum of two non-negative things is 0, 00:38:39.720 --> 00:38:42.450 align:middle line:84% that implies that both terms must be 0. 00:38:42.450 --> 00:38:45.283 align:middle line:84% This fact, I cannot iterate it enough, is deeply important. 00:38:45.283 --> 00:38:46.950 align:middle line:84% When you're working on your problem set, 00:38:46.950 --> 00:38:49.140 align:middle line:84% you will have to use this fact repeatedly, 00:38:49.140 --> 00:38:51.180 align:middle line:84% that the sum of non-negative things being 0 00:38:51.180 --> 00:38:53.550 align:middle line:84% implies that the individual terms must 00:38:53.550 --> 00:38:55.560 align:middle line:84% be 0 as well, to prove positive definiteness. 00:38:55.560 --> 00:38:57.750 align:middle line:90% That's where it mostly comes at. 00:38:57.750 --> 00:39:01.080 align:middle line:84% So what this implies is that the supremum 00:39:01.080 --> 00:39:10.450 align:middle line:84% of x in ab of f of x minus g of x equals 0. 00:39:10.450 --> 00:39:12.100 align:middle line:84% And we've already explained above how 00:39:12.100 --> 00:39:15.100 align:middle line:84% this implies that this metric or that f equals g. 00:39:15.100 --> 00:39:19.217 align:middle line:90% 00:39:19.217 --> 00:39:20.800 align:middle line:84% So this is one of those examples where 00:39:20.800 --> 00:39:23.092 align:middle line:84% you want to boil it down to the examples you've already 00:39:23.092 --> 00:39:25.590 align:middle line:90% done before. 00:39:25.590 --> 00:39:29.136 align:middle line:84% So that proves positive definiteness. 00:39:29.136 --> 00:39:31.860 align:middle line:90% Symmetry is pretty immediate. 00:39:31.860 --> 00:39:36.270 align:middle line:84% And the triangle inequality follows by the same argument 00:39:36.270 --> 00:39:37.170 align:middle line:90% up here, right? 00:39:37.170 --> 00:39:39.510 align:middle line:84% We know that the-- we know that it-- the supremum 00:39:39.510 --> 00:39:42.960 align:middle line:84% exists for each of the terms, because it is-- because all 00:39:42.960 --> 00:39:44.890 align:middle line:90% of the terms are continuous. 00:39:44.890 --> 00:39:46.440 align:middle line:84% But the thing I want to note here 00:39:46.440 --> 00:39:49.200 align:middle line:84% is, what stops us from doing this at c1? 00:39:49.200 --> 00:39:51.450 align:middle line:90% Can we do this at ck? 00:39:51.450 --> 00:39:54.810 align:middle line:84% The answer is, yes, it's not too much more difficult. 00:39:54.810 --> 00:39:59.102 align:middle line:84% You instead sum over the 0th derivative, 00:39:59.102 --> 00:40:00.810 align:middle line:84% the first derivative, so on and so forth, 00:40:00.810 --> 00:40:02.340 align:middle line:90% up until the kth derivative. 00:40:02.340 --> 00:40:03.720 align:middle line:90% And that's fine. 00:40:03.720 --> 00:40:06.840 align:middle line:84% What if I want to study it on smooth functions, which 00:40:06.840 --> 00:40:10.100 align:middle line:90% I'll define right now? 00:40:10.100 --> 00:40:21.540 align:middle line:84% We define c infinity ab to be the set of smooth functions, 00:40:21.540 --> 00:40:24.210 align:middle line:90% i.e., infinitely differentiable. 00:40:24.210 --> 00:40:28.570 align:middle line:90% 00:40:28.570 --> 00:40:31.060 align:middle line:84% Notice that if it's infinitely differentiable, 00:40:31.060 --> 00:40:32.860 align:middle line:84% each of the derivatives must be continuous, 00:40:32.860 --> 00:40:35.980 align:middle line:84% because if it wasn't, then the next derivative wouldn't exist, 00:40:35.980 --> 00:40:40.010 align:middle line:84% or the next derivative wouldn't be continuous. 00:40:40.010 --> 00:40:41.680 align:middle line:84% So this is the set of smooth functions. 00:40:41.680 --> 00:40:44.560 align:middle line:84% What stops us from just taking the sum 00:40:44.560 --> 00:40:47.845 align:middle line:84% over all of these terms, over all infinitely many terms? 00:40:47.845 --> 00:40:51.970 align:middle line:90% 00:40:51.970 --> 00:40:55.510 align:middle line:84% The issue is that there are infinitely many, right? 00:40:55.510 --> 00:40:57.910 align:middle line:84% Again, a metric must be-- it must be the case that 00:40:57.910 --> 00:41:02.635 align:middle line:84% for a metric that the value that you get out is not infinity. 00:41:02.635 --> 00:41:05.260 align:middle line:84% I guess in theory you could mess around with this a little bit, 00:41:05.260 --> 00:41:07.660 align:middle line:84% and you'd get weirder types of metrics. 00:41:07.660 --> 00:41:10.340 align:middle line:84% But for our purposes, you don't want it to be infinity. 00:41:10.340 --> 00:41:16.600 align:middle line:84% But remark, there is a not-too-bad addendum to this. 00:41:16.600 --> 00:41:19.090 align:middle line:84% On the P set, there's an example that you 00:41:19.090 --> 00:41:22.000 align:middle line:84% can work through that's an optional one, where you define 00:41:22.000 --> 00:41:26.140 align:middle line:84% a metric on this, where what you do is you 00:41:26.140 --> 00:41:31.975 align:middle line:84% sum over this interesting fraction, dck(fg), 00:41:31.975 --> 00:41:33.625 align:middle line:90% 1 plus dck(fg). 00:41:33.625 --> 00:41:36.280 align:middle line:90% 00:41:36.280 --> 00:41:38.800 align:middle line:84% So it's summing over a bunch of metrics. 00:41:38.800 --> 00:41:40.300 align:middle line:84% You have to, in fact, show that this 00:41:40.300 --> 00:41:42.800 align:middle line:84% is a metric, which is one of the other problem set problems. 00:41:42.800 --> 00:41:44.710 align:middle line:84% But it's not that you can't define 00:41:44.710 --> 00:41:46.598 align:middle line:90% a metric on smooth functions. 00:41:46.598 --> 00:41:48.640 align:middle line:84% It's that you just have to be careful of the fact 00:41:48.640 --> 00:41:51.690 align:middle line:84% that there are infinitely many terms. 00:41:51.690 --> 00:41:53.010 align:middle line:90% And one small thing to note-- 00:41:53.010 --> 00:41:56.050 align:middle line:90% 00:41:56.050 --> 00:41:58.600 align:middle line:84% is it possible that we could have gotten away with not 00:41:58.600 --> 00:42:00.205 align:middle line:90% including this term in the sum? 00:42:00.205 --> 00:42:03.760 align:middle line:90% 00:42:03.760 --> 00:42:06.430 align:middle line:90% The answer is no. 00:42:06.430 --> 00:42:08.390 align:middle line:84% And I'll let you think about this some more. 00:42:08.390 --> 00:42:11.830 align:middle line:84% But the idea ends up coming from if we take this away, 00:42:11.830 --> 00:42:14.650 align:middle line:84% and the distances is 0, then what's 00:42:14.650 --> 00:42:17.980 align:middle line:84% the distances between f and g prime imply about f and g? 00:42:17.980 --> 00:42:20.480 align:middle line:84% So I'll let y'all think about that some more because that is 00:42:20.480 --> 00:42:21.397 align:middle line:90% a problem set problem. 00:42:21.397 --> 00:42:23.150 align:middle line:90% But the answer is no. 00:42:23.150 --> 00:42:26.960 align:middle line:84% And it's a bit interesting to think about why. 00:42:26.960 --> 00:42:28.430 align:middle line:84% And that reasoning is exactly why 00:42:28.430 --> 00:42:31.055 align:middle line:84% we have to be careful about the infinitely differentiable case. 00:42:31.055 --> 00:42:34.828 align:middle line:90% 00:42:34.828 --> 00:42:36.620 align:middle line:84% So now that we've done this, now that we've 00:42:36.620 --> 00:42:39.050 align:middle line:90% defined a metric on c1 and c0? 00:42:39.050 --> 00:42:40.190 align:middle line:90% Yeah? 00:42:40.190 --> 00:42:44.660 align:middle line:84% STUDENT: Can we use the very first metric as a metric 00:42:44.660 --> 00:42:45.920 align:middle line:90% on all of these spaces? 00:42:45.920 --> 00:42:48.750 align:middle line:90% Does it still read valid? 00:42:48.750 --> 00:42:50.150 align:middle line:84% PAIGE BRIGHT: Yeah, yeah, there's 00:42:50.150 --> 00:42:51.150 align:middle line:90% nothing wrong with that. 00:42:51.150 --> 00:42:52.817 align:middle line:84% The issue is that it's not encapsulating 00:42:52.817 --> 00:42:54.660 align:middle line:84% as much information as we want it to. 00:42:54.660 --> 00:42:56.690 align:middle line:84% So as we'll see in a moment, we want 00:42:56.690 --> 00:43:01.400 align:middle line:84% to understand differentiability and integration as functions 00:43:01.400 --> 00:43:03.360 align:middle line:90% that are continuous. 00:43:03.360 --> 00:43:04.850 align:middle line:90% So that's-- great question. 00:43:04.850 --> 00:43:06.850 align:middle line:84% Yes, we could have just considered the first one 00:43:06.850 --> 00:43:09.890 align:middle line:84% as a metric on all of them in the same way that we can 00:43:09.890 --> 00:43:11.918 align:middle line:84% consider the trivial metric, this one-- 00:43:11.918 --> 00:43:13.460 align:middle line:84% this is known as the trivial metric-- 00:43:13.460 --> 00:43:15.050 align:middle line:84% to be a metric on all of them, right? 00:43:15.050 --> 00:43:18.590 align:middle line:84% But we want more information, when possible. 00:43:18.590 --> 00:43:20.270 align:middle line:90% Great question. 00:43:20.270 --> 00:43:22.580 align:middle line:90% So I will come back over here. 00:43:22.580 --> 00:43:27.700 align:middle line:90% 00:43:27.700 --> 00:43:30.720 align:middle line:84% So now that we have c0 and c1 defined, 00:43:30.720 --> 00:43:34.020 align:middle line:84% we can, in fact, define differentiation 00:43:34.020 --> 00:43:35.880 align:middle line:90% and integration-- or not define. 00:43:35.880 --> 00:43:39.720 align:middle line:84% We can state that differentiation and integration 00:43:39.720 --> 00:43:42.270 align:middle line:90% are, in fact, continuous. 00:43:42.270 --> 00:43:43.420 align:middle line:90% And I'll do that right now. 00:43:43.420 --> 00:43:46.260 align:middle line:84% So I guess this is a proposition. 00:43:46.260 --> 00:43:51.040 align:middle line:90% 00:43:51.040 --> 00:43:58.340 align:middle line:84% If I consider differentiation as a map between c0 of ab, 00:43:58.340 --> 00:44:00.200 align:middle line:90% so continuous functions to-- 00:44:00.200 --> 00:44:07.650 align:middle line:84% or, sorry, c1-- to continuous functions, 00:44:07.650 --> 00:44:18.420 align:middle line:84% my claim is that differentiation is continuous as a function-- 00:44:18.420 --> 00:44:20.640 align:middle line:84% or I guess map is a better word-- but a map 00:44:20.640 --> 00:44:21.645 align:middle line:90% between metric spaces. 00:44:21.645 --> 00:44:30.260 align:middle line:90% 00:44:30.260 --> 00:44:32.010 align:middle line:84% Does this notation make sense to everyone, 00:44:32.010 --> 00:44:35.080 align:middle line:90% the differentiation as a map? 00:44:35.080 --> 00:44:35.580 align:middle line:90% Cool. 00:44:35.580 --> 00:44:40.690 align:middle line:84% So let's check that this is, in fact, a continuous function, 00:44:40.690 --> 00:44:42.960 align:middle line:90% which is pretty nice to do. 00:44:42.960 --> 00:44:45.750 align:middle line:84% By this setup, it's made to be nice, 00:44:45.750 --> 00:44:48.840 align:middle line:84% which addresses your question about why 00:44:48.840 --> 00:44:50.560 align:middle line:90% we sum over the two terms. 00:44:50.560 --> 00:44:54.840 align:middle line:84% So to prove continuity, what's often best to do 00:44:54.840 --> 00:44:58.830 align:middle line:84% is just to consider what the distances, in fact, are. 00:44:58.830 --> 00:45:01.770 align:middle line:84% Let's write out the left-hand side and the right-hand side 00:45:01.770 --> 00:45:05.820 align:middle line:90% of this implication. 00:45:05.820 --> 00:45:12.450 align:middle line:90% So let f and g be in c1 ab. 00:45:12.450 --> 00:45:15.060 align:middle line:90% 00:45:15.060 --> 00:45:21.300 align:middle line:84% And then we consider the distances on c1 of f and g. 00:45:21.300 --> 00:45:25.270 align:middle line:84% And we want to say that if this is less than delta, 00:45:25.270 --> 00:45:30.570 align:middle line:84% then the distances on c0 of the derivatives-- 00:45:30.570 --> 00:45:34.610 align:middle line:84% so I'll just say f prime and g prime-- 00:45:34.610 --> 00:45:35.900 align:middle line:90% is less than epsilon. 00:45:35.900 --> 00:45:38.297 align:middle line:84% We want to find for all epsilon bigger than 0, 00:45:38.297 --> 00:45:40.130 align:middle line:84% there exists a delta such that this is true. 00:45:40.130 --> 00:45:44.263 align:middle line:90% 00:45:44.263 --> 00:45:46.680 align:middle line:84% Well, let's state out what these two metrics actually are. 00:45:46.680 --> 00:45:50.350 align:middle line:84% And that'll make it clear that the reduction is not that bad. 00:45:50.350 --> 00:45:53.490 align:middle line:84% So again, this is equal to the supremum 00:45:53.490 --> 00:46:04.820 align:middle line:84% over x and ab of f of x minus g of x plus the supremum of x 00:46:04.820 --> 00:46:12.080 align:middle line:84% in a to b of f prime of x plus g prime-- or, sorry, minus g 00:46:12.080 --> 00:46:15.310 align:middle line:90% prime of x. 00:46:15.310 --> 00:46:18.820 align:middle line:84% And here, this is simply the supremum 00:46:18.820 --> 00:46:24.550 align:middle line:84% over the distances of f prime of x minus g prime of x. 00:46:24.550 --> 00:46:26.590 align:middle line:84% The reason why this step is nice to do, 00:46:26.590 --> 00:46:29.470 align:middle line:84% first, is because we notice that this term is 00:46:29.470 --> 00:46:32.440 align:middle line:84% precisely the same as this one on the left-hand side. 00:46:32.440 --> 00:46:35.530 align:middle line:84% And all the terms are non-negative. 00:46:35.530 --> 00:46:42.430 align:middle line:90% So let delta equal epsilon. 00:46:42.430 --> 00:46:46.270 align:middle line:84% If you let delta equal epsilon, then this term 00:46:46.270 --> 00:46:53.260 align:middle line:84% being dc1 of f to g being less than delta implies 00:46:53.260 --> 00:46:56.950 align:middle line:84% that this term must be less than delta, or, in fact, less than 00:46:56.950 --> 00:46:58.510 align:middle line:90% epsilon by its construction. 00:46:58.510 --> 00:47:01.423 align:middle line:90% 00:47:01.423 --> 00:47:03.840 align:middle line:84% So that implies that this term is less than epsilon, which 00:47:03.840 --> 00:47:05.690 align:middle line:90% is what we wanted to show. 00:47:05.690 --> 00:47:08.920 align:middle line:84% So this shows-- or that's the end of the proof. 00:47:08.920 --> 00:47:12.510 align:middle line:84% This shows that differentiation is a continuous map. 00:47:12.510 --> 00:47:14.220 align:middle line:84% And here, we use the fact that we're 00:47:14.220 --> 00:47:15.910 align:middle line:90% summing over the two terms. 00:47:15.910 --> 00:47:18.030 align:middle line:84% Now on your problem set what you'll do 00:47:18.030 --> 00:47:21.900 align:middle line:84% is you'll show that integration from a to a point t from a to b 00:47:21.900 --> 00:47:24.190 align:middle line:84% is, in fact, also a continuous operator. 00:47:24.190 --> 00:47:26.647 align:middle line:84% So that, I think, is pretty interesting. 00:47:26.647 --> 00:47:28.730 align:middle line:84% It's going to be a little bit harder than this one 00:47:28.730 --> 00:47:31.380 align:middle line:84% because this direction doesn't include both terms. 00:47:31.380 --> 00:47:33.733 align:middle line:84% But I think it will be a worthwhile exercise 00:47:33.733 --> 00:47:34.400 align:middle line:90% to work through. 00:47:34.400 --> 00:47:37.458 align:middle line:90% 00:47:37.458 --> 00:47:38.375 align:middle line:90% Let's see what's next. 00:47:38.375 --> 00:47:46.013 align:middle line:90% 00:47:46.013 --> 00:47:47.930 align:middle line:84% So while you're going to show that integration 00:47:47.930 --> 00:47:49.880 align:middle line:84% is a continuous operator, this doesn't stop us 00:47:49.880 --> 00:47:52.400 align:middle line:84% from studying integration as a worthwhile metric right 00:47:52.400 --> 00:47:57.600 align:middle line:84% now, because we can define integration not too bad. 00:47:57.600 --> 00:48:07.760 align:middle line:84% So we define i1 to be a metric on c0 ab times c0-- 00:48:07.760 --> 00:48:11.150 align:middle line:90% 00:48:11.150 --> 00:48:13.190 align:middle line:84% actually I'll change this a little bit-- 00:48:13.190 --> 00:48:22.850 align:middle line:84% from ab 0, 1, 0, 1, to 0 to infinity 00:48:22.850 --> 00:48:28.160 align:middle line:90% given by i1 of f and g. 00:48:28.160 --> 00:48:34.790 align:middle line:84% It's simply the integral from 0 to 1 of f of x minus g of x dx. 00:48:34.790 --> 00:48:38.490 align:middle line:90% 00:48:38.490 --> 00:48:41.250 align:middle line:84% All right, we're going to show that this 00:48:41.250 --> 00:48:42.450 align:middle line:90% is, in fact, a metric. 00:48:42.450 --> 00:48:45.550 align:middle line:84% Now what are the three components? 00:48:45.550 --> 00:48:46.260 align:middle line:90% One, symmetry. 00:48:46.260 --> 00:48:52.450 align:middle line:90% 00:48:52.450 --> 00:48:57.220 align:middle line:84% Here we have to use the fact from Riemann integration 00:48:57.220 --> 00:49:02.380 align:middle line:84% that because f of x minus g of x is 00:49:02.380 --> 00:49:09.780 align:middle line:84% equal to g of x minus f of x under absolute values, 00:49:09.780 --> 00:49:11.670 align:middle line:84% then we get that the integrals are, 00:49:11.670 --> 00:49:15.160 align:middle line:90% in fact, the same from 0 to 1. 00:49:15.160 --> 00:49:16.840 align:middle line:90% You can just apply-- 00:49:16.840 --> 00:49:20.830 align:middle line:84% has everyone seen this fact about integration? 00:49:20.830 --> 00:49:22.300 align:middle line:90% Cool. 00:49:22.300 --> 00:49:24.297 align:middle line:84% One way you can prove it is that if they're 00:49:24.297 --> 00:49:25.880 align:middle line:84% less than or equal to each other, then 00:49:25.880 --> 00:49:27.910 align:middle line:84% the integration still follows, and you can 00:49:27.910 --> 00:49:29.780 align:middle line:90% apply the inequality both ways. 00:49:29.780 --> 00:49:31.960 align:middle line:90% So I just wanted to say that. 00:49:31.960 --> 00:49:34.930 align:middle line:84% But symmetry works out pretty nicely. 00:49:34.930 --> 00:49:37.572 align:middle line:90% Two, we know it's non-negative. 00:49:37.572 --> 00:49:41.340 align:middle line:90% 00:49:41.340 --> 00:49:44.250 align:middle line:84% But positive definiteness is going to be a little bit harder 00:49:44.250 --> 00:49:45.900 align:middle line:90% this time. 00:49:45.900 --> 00:49:54.690 align:middle line:84% If f is equal to g, then clearly i1 of f to g is 0. 00:49:54.690 --> 00:49:57.210 align:middle line:84% This comes from the fact that the distances between the two 00:49:57.210 --> 00:49:59.400 align:middle line:90% points is just literally 0. 00:49:59.400 --> 00:50:00.817 align:middle line:84% But how to do the other direction? 00:50:00.817 --> 00:50:03.400 align:middle line:84% Because this is where it can be a little bit more complicated. 00:50:03.400 --> 00:50:05.970 align:middle line:84% What if, for instance, the two functions differ by a point? 00:50:05.970 --> 00:50:08.160 align:middle line:84% Then their integrals are still the same, 00:50:08.160 --> 00:50:10.510 align:middle line:84% but what does this actually tell us? 00:50:10.510 --> 00:50:12.300 align:middle line:90% Let me write this out. 00:50:12.300 --> 00:50:21.035 align:middle line:90% What if i1 fg equals to 0? 00:50:21.035 --> 00:50:24.270 align:middle line:90% 00:50:24.270 --> 00:50:25.520 align:middle line:90% Well here's-- yeah, go for it. 00:50:25.520 --> 00:50:27.312 align:middle line:84% STUDENT: Well, it would continuous, though. 00:50:27.312 --> 00:50:30.965 align:middle line:84% So it would have a little ball where it all is, 00:50:30.965 --> 00:50:33.440 align:middle line:90% like where the [INAUDIBLE] is. 00:50:33.440 --> 00:50:35.000 align:middle line:90% PAIGE BRIGHT: Exactly, right. 00:50:35.000 --> 00:50:36.980 align:middle line:90% So because it's continuous-- 00:50:36.980 --> 00:50:40.850 align:middle line:84% suppose that this was not 0, proof by contradiction. 00:50:40.850 --> 00:50:44.650 align:middle line:84% And I'll draw a nice picture of what's happening here. 00:50:44.650 --> 00:50:49.500 align:middle line:84% So assuming that the integral of them is non-zero, 00:50:49.500 --> 00:50:51.550 align:middle line:84% then there must at least be one point. 00:50:51.550 --> 00:50:55.340 align:middle line:84% Let's say this is f, and this is g. 00:50:55.340 --> 00:50:56.285 align:middle line:90% It's pretty close by. 00:50:56.285 --> 00:51:00.280 align:middle line:90% 00:51:00.280 --> 00:51:03.820 align:middle line:84% Then there must exist at least one point such 00:51:03.820 --> 00:51:06.200 align:middle line:90% that they're different. 00:51:06.200 --> 00:51:08.218 align:middle line:84% Otherwise, if there was no singular point such 00:51:08.218 --> 00:51:10.760 align:middle line:84% that they were different, then they must be equal everywhere. 00:51:10.760 --> 00:51:13.872 align:middle line:84% And we reduce to the first case, right? 00:51:13.872 --> 00:51:15.830 align:middle line:84% Or, then, in fact, that's what we want to show. 00:51:15.830 --> 00:51:18.100 align:middle line:84% But it not being equal to 0 means 00:51:18.100 --> 00:51:21.160 align:middle line:84% that there must exist a point where they're not the same. 00:51:21.160 --> 00:51:24.310 align:middle line:84% And what this tells you is there must exist a little ball 00:51:24.310 --> 00:51:27.220 align:middle line:84% around x, ball of radius epsilon, 00:51:27.220 --> 00:51:30.640 align:middle line:84% let's say, such that they're, in fact, not 00:51:30.640 --> 00:51:33.710 align:middle line:90% equal on that entire ball. 00:51:33.710 --> 00:51:36.610 align:middle line:84% But we can use this to show that the integrals over that, 00:51:36.610 --> 00:51:39.080 align:middle line:84% the integral over the entire interval 0 to 1, 00:51:39.080 --> 00:51:43.230 align:middle line:84% must therefore not be 0, and reach a contradiction. 00:51:43.230 --> 00:51:48.710 align:middle line:84% So specifically we would reach the conclusion 00:51:48.710 --> 00:51:49.880 align:middle line:90% that f is not equal to g. 00:51:49.880 --> 00:51:53.060 align:middle line:84% And I'll let you all work through the statements of that. 00:51:53.060 --> 00:51:54.620 align:middle line:84% But, yeah, it is precisely the fact 00:51:54.620 --> 00:51:57.590 align:middle line:84% that if there's a point where the integral is non-zero, 00:51:57.590 --> 00:52:00.230 align:middle line:84% then there must be a ball around that point such 00:52:00.230 --> 00:52:02.780 align:middle line:90% that the distances is not 0. 00:52:02.780 --> 00:52:04.130 align:middle line:90% Cool. 00:52:04.130 --> 00:52:06.770 align:middle line:84% And this just highlights why the fact-- why continuity 00:52:06.770 --> 00:52:10.580 align:middle line:84% is important here yet again, not just 00:52:10.580 --> 00:52:14.300 align:middle line:84% to show that nice things that are continuous. 00:52:14.300 --> 00:52:15.890 align:middle line:84% Lastly, the triangle inequality-- 00:52:15.890 --> 00:52:20.240 align:middle line:90% 00:52:20.240 --> 00:52:23.300 align:middle line:84% I just want to state this, because it's not too bad. 00:52:23.300 --> 00:52:25.780 align:middle line:84% We just note before even integrating 00:52:25.780 --> 00:52:28.840 align:middle line:84% that the distances between f of x and h of x is less than 00:52:28.840 --> 00:52:37.840 align:middle line:84% or equal to f of x minus g of x plus g of x minus h of x. 00:52:37.840 --> 00:52:41.570 align:middle line:90% 00:52:41.570 --> 00:52:45.770 align:middle line:84% And then integrate on both sides of the inequality. 00:52:45.770 --> 00:52:47.820 align:middle line:84% And that allows us to reach our conclusion. 00:52:47.820 --> 00:52:50.300 align:middle line:84% So in statements like this, either choosing a point 00:52:50.300 --> 00:52:51.650 align:middle line:90% such that the supremum exists. 00:52:51.650 --> 00:52:54.660 align:middle line:84% If it's based off the supremum, that's a good way to do it. 00:52:54.660 --> 00:52:57.440 align:middle line:84% The other way to do it is try to utilize facts 00:52:57.440 --> 00:53:00.110 align:middle line:84% about the absolute values before you even apply the metric. 00:53:00.110 --> 00:53:02.420 align:middle line:84% So before you even apply integration, 00:53:02.420 --> 00:53:03.890 align:middle line:90% see what you can say. 00:53:03.890 --> 00:53:06.750 align:middle line:84% These are the few techniques that are deeply helpful. 00:53:06.750 --> 00:53:09.500 align:middle line:84% And there's one thing I want to note here. 00:53:09.500 --> 00:53:10.740 align:middle line:90% Do I still have it up? 00:53:10.740 --> 00:53:15.940 align:middle line:84% I do not because it was a while ago. 00:53:15.940 --> 00:53:18.430 align:middle line:84% We can view integration as a sum. 00:53:18.430 --> 00:53:19.540 align:middle line:90% It's an infinite sum. 00:53:19.540 --> 00:53:25.090 align:middle line:84% Sure, it's Riemann integration, but a sum nonetheless. 00:53:25.090 --> 00:53:26.680 align:middle line:90% What this is called-- 00:53:26.680 --> 00:53:27.700 align:middle line:90% I'll note it here-- 00:53:27.700 --> 00:53:36.980 align:middle line:84% this is known as the capital L1 metric on c0 ab. 00:53:36.980 --> 00:53:41.600 align:middle line:84% Or specifically, the notation would be it's L1 ab. 00:53:41.600 --> 00:53:45.260 align:middle line:90% This is symmetric. 00:53:45.260 --> 00:53:47.690 align:middle line:84% The reason it's called L1 might remind you of the fact 00:53:47.690 --> 00:53:49.903 align:middle line:84% that we had a little l1 that we'd find earlier, 00:53:49.903 --> 00:53:51.320 align:middle line:84% where instead there, we're summing 00:53:51.320 --> 00:53:54.230 align:middle line:90% over finitely many terms. 00:53:54.230 --> 00:53:55.423 align:middle line:90% It's the same exact notion. 00:53:55.423 --> 00:53:56.840 align:middle line:84% It's just the only difference here 00:53:56.840 --> 00:53:58.850 align:middle line:84% is that we're summing over infinitely many terms. 00:53:58.850 --> 00:54:00.530 align:middle line:84% And that's why the notation is the same. 00:54:00.530 --> 00:54:04.520 align:middle line:84% Now this notion is deeply important for 18.102 00:54:04.520 --> 00:54:07.250 align:middle line:84% because this is the definition-- this leads to definitions 00:54:07.250 --> 00:54:08.780 align:middle line:84% of Lebesque integration, which we'll 00:54:08.780 --> 00:54:11.630 align:middle line:84% talk a little bit about in this class. 00:54:11.630 --> 00:54:14.480 align:middle line:84% But really, it's a question of integration 00:54:14.480 --> 00:54:17.090 align:middle line:84% as we already understand it via Riemann integration. 00:54:17.090 --> 00:54:18.620 align:middle line:90% It's really, really bad, right? 00:54:18.620 --> 00:54:21.350 align:middle line:84% What if we don't want to study things on continuous functions 00:54:21.350 --> 00:54:24.860 align:middle line:84% but instead ones with finitely many discontinuities? 00:54:24.860 --> 00:54:27.230 align:middle line:84% The Riemann integral just becomes so much more 00:54:27.230 --> 00:54:28.380 align:middle line:90% annoying to deal with. 00:54:28.380 --> 00:54:31.790 align:middle line:84% So the Lebesque integral is the way around that. 00:54:31.790 --> 00:54:35.660 align:middle line:84% And it's called capital L1 because of Lebesque, 00:54:35.660 --> 00:54:37.040 align:middle line:90% the person who invented it. 00:54:37.040 --> 00:54:40.330 align:middle line:90% 00:54:40.330 --> 00:54:44.030 align:middle line:84% But one thing I want to note is that all 00:54:44.030 --> 00:54:46.550 align:middle line:84% of the spaces that we've considered so far 00:54:46.550 --> 00:54:47.967 align:middle line:90% are vector spaces. 00:54:47.967 --> 00:54:50.300 align:middle line:84% So for those of you all who have studied linear algebra, 00:54:50.300 --> 00:54:52.800 align:middle line:84% this is-- this will be slight review. 00:54:52.800 --> 00:54:54.590 align:middle line:84% But I just want to note it here because it 00:54:54.590 --> 00:54:56.990 align:middle line:84% is somewhat important and will come up a little bit later 00:54:56.990 --> 00:54:59.000 align:middle line:90% in the class. 00:54:59.000 --> 00:55:01.400 align:middle line:84% A vector space is simply a space in which 00:55:01.400 --> 00:55:03.350 align:middle line:84% you can add two terms together, and it'll 00:55:03.350 --> 00:55:05.360 align:middle line:90% stay in the same space. 00:55:05.360 --> 00:55:07.288 align:middle line:84% This isn't a class on linear algebra, 00:55:07.288 --> 00:55:09.830 align:middle line:84% so there's not going to be too many times when the definition 00:55:09.830 --> 00:55:10.955 align:middle line:90% of a vector space comes up. 00:55:10.955 --> 00:55:13.880 align:middle line:84% But it's a useful thing to keep in mind and to know about. 00:55:13.880 --> 00:55:14.825 align:middle line:90% So, vector space. 00:55:14.825 --> 00:55:18.190 align:middle line:90% 00:55:18.190 --> 00:55:18.930 align:middle line:90% This is the TLDR. 00:55:18.930 --> 00:55:21.207 align:middle line:90% 00:55:21.207 --> 00:55:23.790 align:middle line:84% I'm not going to write out the entire definition because it's, 00:55:23.790 --> 00:55:25.510 align:middle line:90% in fact, quite lengthy. 00:55:25.510 --> 00:55:28.320 align:middle line:84% But it's a space V such that we have 00:55:28.320 --> 00:55:34.090 align:middle line:84% addition, which maps from V and V to V, 00:55:34.090 --> 00:55:37.570 align:middle line:84% and multiplication, which maps from real numbers 00:55:37.570 --> 00:55:41.050 align:middle line:84% across V to V. And ways to think about this 00:55:41.050 --> 00:55:47.650 align:middle line:84% is, for instance, the set of continuous functions, c0 ab. 00:55:47.650 --> 00:55:52.390 align:middle line:84% We have addition defined on this such that we define f plus g 00:55:52.390 --> 00:55:56.050 align:middle line:84% to just be f of x plus g of x everywhere. 00:55:56.050 --> 00:56:00.450 align:middle line:90% 00:56:00.450 --> 00:56:03.190 align:middle line:84% And we define a constant times a function, 00:56:03.190 --> 00:56:06.240 align:middle line:84% which is what R here is doing, as just 00:56:06.240 --> 00:56:09.240 align:middle line:84% being the constant times f of x everywhere. 00:56:09.240 --> 00:56:12.210 align:middle line:90% 00:56:12.210 --> 00:56:16.480 align:middle line:84% The basic idea for vector space is that addition 00:56:16.480 --> 00:56:17.730 align:middle line:90% acts how you would want it to. 00:56:17.730 --> 00:56:19.860 align:middle line:84% It maps from two points in the vector space 00:56:19.860 --> 00:56:21.690 align:middle line:90% to a point in the vector space. 00:56:21.690 --> 00:56:24.090 align:middle line:84% And scalar multiplication acts the same way. 00:56:24.090 --> 00:56:26.610 align:middle line:84% It takes a scalar and a point in your vector space 00:56:26.610 --> 00:56:29.520 align:middle line:84% and maps it to another point in your vector space. 00:56:29.520 --> 00:56:30.720 align:middle line:90% Why do I note this? 00:56:30.720 --> 00:56:32.520 align:middle line:84% Because metric spaces do not only 00:56:32.520 --> 00:56:35.670 align:middle line:90% have to be on vector spaces. 00:56:35.670 --> 00:56:37.740 align:middle line:90% We can have it be much weirder. 00:56:37.740 --> 00:56:40.020 align:middle line:90% A good example of this-- 00:56:40.020 --> 00:56:42.900 align:middle line:84% so if you did not catch the basic idea of what a vector 00:56:42.900 --> 00:56:46.050 align:middle line:84% space is, I'll-- this example might highlight it. 00:56:46.050 --> 00:56:51.700 align:middle line:84% Example-- consider the sphere S1 to be 00:56:51.700 --> 00:56:56.228 align:middle line:84% the set of x in, let's say, R3 such 00:56:56.228 --> 00:56:57.520 align:middle line:90% that-- actually, you know what? 00:56:57.520 --> 00:57:04.690 align:middle line:84% I'll just do a circle, the circle of radius 1 in R2. 00:57:04.690 --> 00:57:08.080 align:middle line:84% This is not a vector space under usual pointwise addition, 00:57:08.080 --> 00:57:11.080 align:middle line:84% because if I take a point on my vector space-- 00:57:11.080 --> 00:57:12.910 align:middle line:90% let's say that one-- 00:57:12.910 --> 00:57:14.110 align:middle line:90% let's call it x. 00:57:14.110 --> 00:57:16.840 align:middle line:84% And I do x plus x, I'm going to end up 00:57:16.840 --> 00:57:20.740 align:middle line:84% with a point that's not on the circle. 00:57:20.740 --> 00:57:22.900 align:middle line:84% This is not a vector space in the usual sense 00:57:22.900 --> 00:57:26.960 align:middle line:84% because we want addition to be contained in the vector space. 00:57:26.960 --> 00:57:28.092 align:middle line:90% Why is this important? 00:57:28.092 --> 00:57:30.550 align:middle line:84% Because we can still define distances on the circle, right? 00:57:30.550 --> 00:57:32.740 align:middle line:84% We can define distances on the circle 00:57:32.740 --> 00:57:35.770 align:middle line:84% via just the regular distances, so the distance from x to y 00:57:35.770 --> 00:57:38.920 align:middle line:90% being the usual distance on R2. 00:57:38.920 --> 00:57:44.150 align:middle line:84% You can also define it via the shortest distance between them 00:57:44.150 --> 00:57:45.580 align:middle line:90% which is known as a geodesic. 00:57:45.580 --> 00:57:48.535 align:middle line:90% 00:57:48.535 --> 00:57:52.340 align:middle line:84% I don't like this chalk, but anyways. 00:57:52.340 --> 00:57:54.350 align:middle line:84% So the point that I'm trying to highlight here 00:57:54.350 --> 00:57:56.663 align:middle line:84% is you don't always have addition defined, 00:57:56.663 --> 00:57:58.580 align:middle line:84% which is going to limit the number of theorems 00:57:58.580 --> 00:58:01.250 align:middle line:84% we can actually say about metrics, right? 00:58:01.250 --> 00:58:03.920 align:middle line:84% Nowhere in our definition of a metric does it tell us how 00:58:03.920 --> 00:58:06.590 align:middle line:84% to add two functions together because sometimes we just 00:58:06.590 --> 00:58:08.180 align:middle line:90% simply can't. 00:58:08.180 --> 00:58:10.700 align:middle line:84% The set of-- or when we can, it starts 00:58:10.700 --> 00:58:14.120 align:middle line:84% becoming more like functional analysis, which is 18.102. 00:58:14.120 --> 00:58:17.610 align:middle line:84% We'll talk briefly about that in two lectures from now. 00:58:17.610 --> 00:58:19.790 align:middle line:84% But this notion of it not being a vector space 00:58:19.790 --> 00:58:22.382 align:middle line:90% is pretty important. 00:58:22.382 --> 00:58:24.090 align:middle line:84% Was there anything else I wanted to note? 00:58:24.090 --> 00:58:28.530 align:middle line:84% We might end today a little bit early, which will be nice. 00:58:28.530 --> 00:58:30.750 align:middle line:84% The main other thing I wanted to note here 00:58:30.750 --> 00:58:33.030 align:middle line:84% was something that I noted earlier for the little l1 00:58:33.030 --> 00:58:38.820 align:middle line:84% spaces, is you can also define lp spaces 00:58:38.820 --> 00:58:44.580 align:middle line:84% f of x minus h of x to the p-- to the 1 over p. 00:58:44.580 --> 00:58:48.210 align:middle line:90% This is known as lp spaces. 00:58:48.210 --> 00:58:54.220 align:middle line:84% So functions in which this term is finite is an lp space. 00:58:54.220 --> 00:58:56.910 align:middle line:84% All right, so I actually went through this 00:58:56.910 --> 00:58:58.270 align:middle line:90% 20 minutes earlier-- 00:58:58.270 --> 00:58:59.200 align:middle line:90% faster than last time. 00:58:59.200 --> 00:59:02.813 align:middle line:84% So if y'all have any questions, I'm happy to talk about more 00:59:02.813 --> 00:59:03.480 align:middle line:90% of the material. 00:59:03.480 --> 00:59:05.438 align:middle line:84% But we might just end today a little bit early. 00:59:05.438 --> 00:59:08.430 align:middle line:84% I don't want to get into the general theory quite yet. 00:59:08.430 --> 00:59:10.470 align:middle line:84% What I would highly suggest is trying 00:59:10.470 --> 00:59:12.830 align:middle line:84% to sit with this notion a little bit, 00:59:12.830 --> 00:59:17.170 align:middle line:84% because though it seems like a simple notion, what we've 00:59:17.170 --> 00:59:21.100 align:middle line:84% really done here today is gone from understanding, 00:59:21.100 --> 00:59:23.470 align:middle line:84% for instance, functions as things that take in points 00:59:23.470 --> 00:59:26.350 align:middle line:84% and spit out points as things that can be manipulated, 00:59:26.350 --> 00:59:28.730 align:middle line:84% as things that have a distance between them. 00:59:28.730 --> 00:59:30.710 align:middle line:84% And once we study sequences of functions, 00:59:30.710 --> 00:59:33.010 align:middle line:84% which is deeply important in analysis, 00:59:33.010 --> 00:59:36.900 align:middle line:84% it's going to be-- become more and more important. 00:59:36.900 --> 00:59:43.000 align:middle line:90%