WEBVTT 00:00:00.000 --> 00:00:21.980 align:middle line:84% [SQUEAKING] [RUSTLING] [CLICKING] 00:00:21.980 --> 00:00:24.840 align:middle line:84% PROFESSOR: OK, so let's get started. 00:00:24.840 --> 00:00:28.390 align:middle line:84% This is the first lecture for the course-- 00:00:28.390 --> 00:00:36.690 align:middle line:90% 00:00:36.690 --> 00:00:39.120 align:middle line:84% Introduction to Functional Analysis. 00:00:39.120 --> 00:00:47.100 align:middle line:90% 00:00:47.100 --> 00:00:53.480 align:middle line:84% OK, so let me give you a brief preview or a few words 00:00:53.480 --> 00:00:58.460 align:middle line:84% about what functional analysis is or initially aimed to do. 00:00:58.460 --> 00:01:04.240 align:middle line:84% So by this point, you've taken-- or at least 00:01:04.240 --> 00:01:05.990 align:middle line:84% what the prerequisites are supposed to be, 00:01:05.990 --> 00:01:07.460 align:middle line:90% you've taken linear algebra. 00:01:07.460 --> 00:01:09.110 align:middle line:90% You've taken calculus. 00:01:09.110 --> 00:01:14.330 align:middle line:84% And what these subjects allow you to do 00:01:14.330 --> 00:01:21.690 align:middle line:84% is solve problems where you have finitely 00:01:21.690 --> 00:01:24.330 align:middle line:84% many independent variables, if you like. 00:01:24.330 --> 00:01:26.280 align:middle line:84% You're working in finite dimensions. 00:01:26.280 --> 00:01:27.265 align:middle line:90% So you always have-- 00:01:27.265 --> 00:01:28.890 align:middle line:84% for example, in calculus, you're trying 00:01:28.890 --> 00:01:33.560 align:middle line:84% to find the min or max of a function of 1, 2, 00:01:33.560 --> 00:01:36.180 align:middle line:90% 3, or 4 variables, right? 00:01:36.180 --> 00:01:40.080 align:middle line:84% Or in linear algebra, you're trying to solve a linear-- 00:01:40.080 --> 00:01:43.650 align:middle line:84% set of linear equations, but there's five equations 00:01:43.650 --> 00:01:47.380 align:middle line:90% and five unknowns. 00:01:47.380 --> 00:01:53.390 align:middle line:84% So there's always finitely many independent variables. 00:01:53.390 --> 00:01:58.080 align:middle line:84% Now, that allows you to solve a lot 00:01:58.080 --> 00:02:02.667 align:middle line:84% of fun problems of how fast water is leaking from a cone. 00:02:02.667 --> 00:02:04.500 align:middle line:84% I think that's one of the problems you solve 00:02:04.500 --> 00:02:09.030 align:middle line:90% or something of that sort. 00:02:09.030 --> 00:02:11.640 align:middle line:84% But then when you move on in life, 00:02:11.640 --> 00:02:16.710 align:middle line:84% you come across ODEs and PDEs and other types 00:02:16.710 --> 00:02:19.890 align:middle line:84% of minimization, maximization problems where 00:02:19.890 --> 00:02:25.910 align:middle line:84% now, if you'd like, the set of independent variables 00:02:25.910 --> 00:02:29.290 align:middle line:84% is no longer a finite dimensional, OK? 00:02:29.290 --> 00:02:33.450 align:middle line:84% So let's think of the number of-- 00:02:33.450 --> 00:02:36.600 align:middle line:84% or your independent variable as being a member of some vector 00:02:36.600 --> 00:02:38.350 align:middle line:90% space. 00:02:38.350 --> 00:02:41.880 align:middle line:84% So when we were talking about calculus, functions of 1, 00:02:41.880 --> 00:02:46.440 align:middle line:84% 2, or 3 variables, these are functions of R, R2, R3, 00:02:46.440 --> 00:02:49.620 align:middle line:90% so and so on. 00:02:49.620 --> 00:02:54.410 align:middle line:84% But it turns out that if you want to consider, 00:02:54.410 --> 00:02:58.560 align:middle line:84% for example, what's the shortest curve between two points, 00:02:58.560 --> 00:03:03.980 align:middle line:84% this is a natural functional, meaning-- 00:03:03.980 --> 00:03:06.080 align:middle line:84% well, this is terminology that means 00:03:06.080 --> 00:03:09.680 align:middle line:84% now the argument is a function, that the points, 00:03:09.680 --> 00:03:14.150 align:middle line:84% your independent variables, are curves, are functions. 00:03:14.150 --> 00:03:18.380 align:middle line:90% 00:03:18.380 --> 00:03:21.170 align:middle line:84% For a vector in R3, say, it takes three numbers 00:03:21.170 --> 00:03:26.420 align:middle line:84% to specify a vector, the three coordinates. 00:03:26.420 --> 00:03:29.270 align:middle line:84% How many numbers do you need to specify, say, 00:03:29.270 --> 00:03:33.500 align:middle line:84% a continuous curve on the interval, say, 0, 1? 00:03:33.500 --> 00:03:35.750 align:middle line:84% Well, you need infinitely many numbers. 00:03:35.750 --> 00:03:39.000 align:middle line:84% You need to know the graph of that curve. 00:03:39.000 --> 00:03:43.100 align:middle line:84% So functional analysis, in short, 00:03:43.100 --> 00:03:45.380 align:middle line:84% was built to be able to start solving problems 00:03:45.380 --> 00:03:51.180 align:middle line:84% where the vector spaces are not necessarily finite dimensional. 00:03:51.180 --> 00:03:56.340 align:middle line:84% And as we'll see in the problems that we work through 00:03:56.340 --> 00:03:59.940 align:middle line:84% and the situations that pop up, this 00:03:59.940 --> 00:04:04.080 align:middle line:84% arises quite naturally for very particular-- 00:04:04.080 --> 00:04:07.470 align:middle line:84% I mean, not particular, but for concrete problems. 00:04:07.470 --> 00:04:12.540 align:middle line:84% This is not just some sort of academic exercise. 00:04:12.540 --> 00:04:15.840 align:middle line:84% This whole subject grew out of trying 00:04:15.840 --> 00:04:22.380 align:middle line:84% to understand particular concrete problems involving 00:04:22.380 --> 00:04:26.730 align:middle line:84% PDEs, and minimization, and optimization 00:04:26.730 --> 00:04:32.250 align:middle line:90% of now functions of functions. 00:04:32.250 --> 00:04:33.690 align:middle line:84% That was the original terminology, 00:04:33.690 --> 00:04:36.330 align:middle line:84% functions of functions, where now the vector space 00:04:36.330 --> 00:04:38.460 align:middle line:84% is some space of functions, not just spaces 00:04:38.460 --> 00:04:40.950 align:middle line:84% of, say, three-dimensional vectors 00:04:40.950 --> 00:04:45.370 align:middle line:84% or two-dimensional vectors, and thus the name functional. 00:04:45.370 --> 00:04:54.420 align:middle line:84% OK, so that's a few words about how this subject grew, 00:04:54.420 --> 00:04:56.100 align:middle line:90% what's the point. 00:04:56.100 --> 00:04:58.260 align:middle line:84% So let's start getting into specifics. 00:04:58.260 --> 00:05:01.320 align:middle line:84% So again, I'm going to be using a lot of terminology that comes 00:05:01.320 --> 00:05:08.650 align:middle line:84% from linear algebra and the Real Analysis course, 18.100B. 00:05:08.650 --> 00:05:10.420 align:middle line:84% But at the start, I'll be reminding you 00:05:10.420 --> 00:05:15.430 align:middle line:84% of what some of these terms that I'm using mean. 00:05:15.430 --> 00:05:21.460 align:middle line:84% But as the course goes on, I will stop redefining terms 00:05:21.460 --> 00:05:25.360 align:middle line:84% that you should have seen in Real Analysis or Linear Algebra 00:05:25.360 --> 00:05:26.835 align:middle line:90% and just use them. 00:05:26.835 --> 00:05:35.620 align:middle line:84% OK, so the first topic that we're going to deal with-- 00:05:35.620 --> 00:05:42.060 align:middle line:84% so our norm spaces, these are the central objects 00:05:42.060 --> 00:05:46.360 align:middle line:84% and/or starting point in functional analysis, which 00:05:46.360 --> 00:05:52.430 align:middle line:84% are the analog of our R2, R3, and so on. 00:05:52.430 --> 00:05:54.590 align:middle line:90% So what's the setup? 00:05:54.590 --> 00:06:04.290 align:middle line:84% Let V be vector space over R or even C. 00:06:04.290 --> 00:06:07.440 align:middle line:84% So it could be a complex vector space. 00:06:07.440 --> 00:06:18.810 align:middle line:84% And either of these spaces we'll usually denote by a boldface K. 00:06:18.810 --> 00:06:20.170 align:middle line:90% So what does this mean? 00:06:20.170 --> 00:06:21.840 align:middle line:84% So again, this is one of those points 00:06:21.840 --> 00:06:24.660 align:middle line:84% where I'll just quickly remind you what a vector space is. 00:06:24.660 --> 00:06:29.560 align:middle line:90% 00:06:29.560 --> 00:06:37.060 align:middle line:84% Then V comes with two operations-- 00:06:37.060 --> 00:06:41.840 align:middle line:90% plus and scalar multiplication. 00:06:41.840 --> 00:06:49.020 align:middle line:84% So plus, going from V cross V into V, which we denote-- 00:06:49.020 --> 00:06:53.280 align:middle line:84% if I have two vectors in V, they get 00:06:53.280 --> 00:06:58.560 align:middle line:84% mapped to the new vector, which I denote V1 plus V2. 00:06:58.560 --> 00:07:05.210 align:middle line:84% And then I have scalar multiplication 00:07:05.210 --> 00:07:13.040 align:middle line:84% from the set of scalars cross V into V. And this goes alpha-- 00:07:13.040 --> 00:07:18.330 align:middle line:90% V gets mapped to alpha times V. 00:07:18.330 --> 00:07:21.240 align:middle line:84% All right, so you have these two operations. 00:07:21.240 --> 00:07:26.880 align:middle line:84% And they satisfy certain conditions, assumptions, 00:07:26.880 --> 00:07:28.620 align:middle line:84% relations between them, which are 00:07:28.620 --> 00:07:32.760 align:middle line:84% part of the axioms of a vector space, which you can read 00:07:32.760 --> 00:07:35.040 align:middle line:84% in the last section of the notes, 00:07:35.040 --> 00:07:38.110 align:middle line:84% if you want to refresh your memory. 00:07:38.110 --> 00:07:43.930 align:middle line:84% And so V as a vector space has these two operations satisfying 00:07:43.930 --> 00:07:45.610 align:middle line:90% a certain set of axioms. 00:07:45.610 --> 00:07:46.765 align:middle line:90% And so for example-- 00:07:46.765 --> 00:07:50.080 align:middle line:90% 00:07:50.080 --> 00:07:52.510 align:middle line:90% familiar old examples-- 00:07:52.510 --> 00:08:04.120 align:middle line:84% R2, Rn, and then, of course, C, the set of n tuples of R 00:08:04.120 --> 00:08:12.030 align:middle line:84% or C. But here's another simple example-- 00:08:12.030 --> 00:08:19.590 align:middle line:84% C 0, 1, which I'll remind you this notation here means 00:08:19.590 --> 00:08:23.790 align:middle line:84% the set of functions from 0, 1 into-- 00:08:23.790 --> 00:08:29.220 align:middle line:84% let's say their complex value, just to fix a field of scalars 00:08:29.220 --> 00:08:35.240 align:middle line:84% to work with, such that f is continuous, 00:08:35.240 --> 00:08:37.059 align:middle line:84% meaning it's continuous at every point. 00:08:37.059 --> 00:08:40.030 align:middle line:84% This is also a vector field because the sum 00:08:40.030 --> 00:08:43.909 align:middle line:84% of two continuous functions is continuous. 00:08:43.909 --> 00:08:47.420 align:middle line:84% And if I take a scalar multiple of a continuous function, 00:08:47.420 --> 00:08:48.730 align:middle line:90% then that's also continuous. 00:08:48.730 --> 00:08:51.310 align:middle line:84% And then these operations satisfy the axioms 00:08:51.310 --> 00:08:54.950 align:middle line:90% that you need for vector space. 00:08:54.950 --> 00:08:59.030 align:middle line:90% But there really is a big-- 00:08:59.030 --> 00:09:03.410 align:middle line:84% pun intended-- difference between these spaces here 00:09:03.410 --> 00:09:05.000 align:middle line:90% and these spaces here-- 00:09:05.000 --> 00:09:06.320 align:middle line:90% or that space there. 00:09:06.320 --> 00:09:09.380 align:middle line:90% And what is the difference? 00:09:09.380 --> 00:09:11.600 align:middle line:90% The size, OK? 00:09:11.600 --> 00:09:14.720 align:middle line:90% Now, in analysis, size was-- 00:09:14.720 --> 00:09:18.340 align:middle line:90% 00:09:18.340 --> 00:09:21.475 align:middle line:84% you had maybe one or two different notions of size 00:09:21.475 --> 00:09:22.850 align:middle line:84% you were introduced to, depending 00:09:22.850 --> 00:09:25.910 align:middle line:84% on how much analysis you've seen, but one was cardinality. 00:09:25.910 --> 00:09:29.600 align:middle line:84% That's not what I'm talking about when I mean size. 00:09:29.600 --> 00:09:30.950 align:middle line:90% What I mean is the following-- 00:09:30.950 --> 00:09:32.930 align:middle line:84% I mean the dimension of these spaces. 00:09:32.930 --> 00:09:37.680 align:middle line:84% So let me recall the following definition, 00:09:37.680 --> 00:09:44.980 align:middle line:84% that we say a vector space V is finite dimensional-- 00:09:44.980 --> 00:09:47.380 align:middle line:84% and these were a lot of the vector spaces you were first 00:09:47.380 --> 00:09:49.570 align:middle line:90% introduced to-- 00:09:49.570 --> 00:10:09.340 align:middle line:84% if every linearly independent set is, in fact, a finite set. 00:10:09.340 --> 00:10:13.240 align:middle line:90% 00:10:13.240 --> 00:10:14.900 align:middle line:90% And so let me again-- 00:10:14.900 --> 00:10:17.830 align:middle line:84% since I'm using some of these words to help you recall, 00:10:17.830 --> 00:10:18.790 align:middle line:90% what does this mean? 00:10:18.790 --> 00:10:22.790 align:middle line:84% In math, this means for every set E 00:10:22.790 --> 00:10:27.680 align:middle line:84% that is linearly independent, meaning 00:10:27.680 --> 00:10:29.810 align:middle line:84% which have the following properties so that if I 00:10:29.810 --> 00:10:38.530 align:middle line:90% take any elements in E that-- 00:10:38.530 --> 00:10:45.450 align:middle line:84% so E satisfies that if I take any finitely many elements 00:10:45.450 --> 00:10:48.480 align:middle line:84% of E, the assumption that there's 00:10:48.480 --> 00:10:57.180 align:middle line:84% a linear combination of them giving 0 00:10:57.180 --> 00:11:02.405 align:middle line:84% implies that all of these scalars must be 0. 00:11:02.405 --> 00:11:08.778 align:middle line:90% 00:11:08.778 --> 00:11:13.520 align:middle line:84% OK, so this here is the definition 00:11:13.520 --> 00:11:15.410 align:middle line:90% of being linearly independent. 00:11:15.410 --> 00:11:19.194 align:middle line:90% 00:11:19.194 --> 00:11:21.800 align:middle line:84% For all E satisfying this condition-- so 00:11:21.800 --> 00:11:24.350 align:middle line:84% this is poorly written, but I hope you follow. 00:11:24.350 --> 00:11:27.020 align:middle line:84% This is the definition of linearly independence. 00:11:27.020 --> 00:11:30.230 align:middle line:84% Then for every set that's linearly independent, 00:11:30.230 --> 00:11:36.290 align:middle line:84% then the set E is finite in the sense of cardinality, 00:11:36.290 --> 00:11:42.340 align:middle line:84% in the sense there's only 100 elements in there or not. 00:11:42.340 --> 00:11:45.960 align:middle line:90% So this is finite dimensional. 00:11:45.960 --> 00:11:56.980 align:middle line:84% And say V is infinite dimensional 00:11:56.980 --> 00:12:08.060 align:middle line:90% if V is not finite dimensional. 00:12:08.060 --> 00:12:11.240 align:middle line:84% So some of you I saw have taken a course from me before. 00:12:11.240 --> 00:12:18.030 align:middle line:84% I tend to use a lot of abbreviations when I write. 00:12:18.030 --> 00:12:20.010 align:middle line:84% But typically, these abbreviations 00:12:20.010 --> 00:12:23.100 align:middle line:84% are pretty clear what they mean, if you just sound it out. 00:12:23.100 --> 00:12:30.060 align:middle line:90% 00:12:30.060 --> 00:12:31.590 align:middle line:90% So vector space V-- 00:12:31.590 --> 00:12:33.120 align:middle line:84% and these infinite dimensional guys, 00:12:33.120 --> 00:12:35.400 align:middle line:84% these are the guys we're going to be dealing 00:12:35.400 --> 00:12:37.290 align:middle line:90% with a lot in this course. 00:12:37.290 --> 00:12:41.520 align:middle line:84% The finite dimensional ones you dealt with in linear algebra. 00:12:41.520 --> 00:12:45.480 align:middle line:84% Maybe you used a few-- you had a few infinite dimensional 00:12:45.480 --> 00:12:49.380 align:middle line:84% examples if you were looking at examples of vector spaces. 00:12:49.380 --> 00:12:52.470 align:middle line:84% But these infinite dimensional vector spaces, 00:12:52.470 --> 00:12:55.260 align:middle line:84% these are the type of guys or the type of vector spaces 00:12:55.260 --> 00:13:02.790 align:middle line:84% we're now going to solve linear equations on 00:13:02.790 --> 00:13:05.767 align:middle line:84% and, in some sense, do calculus on. 00:13:05.767 --> 00:13:07.350 align:middle line:84% That's not quite true, but we're going 00:13:07.350 --> 00:13:10.200 align:middle line:84% to use calculus and some tools to be 00:13:10.200 --> 00:13:14.670 align:middle line:84% able to say some things about linear equations 00:13:14.670 --> 00:13:18.730 align:middle line:84% on these infinite dimensional spaces. 00:13:18.730 --> 00:13:21.470 align:middle line:84% But they won't just be any type of infinite dimensional space. 00:13:21.470 --> 00:13:24.075 align:middle line:84% And I'll say what type we're looking at in a minute. 00:13:24.075 --> 00:13:27.140 align:middle line:90% 00:13:27.140 --> 00:13:29.450 align:middle line:84% OK, so finite dimensional, infinite dimensional-- 00:13:29.450 --> 00:13:32.420 align:middle line:90% so the first set of examples-- 00:13:32.420 --> 00:13:39.320 align:middle line:84% R1, R2, Rn, Cn, and so on, those are finite dimensional spaces. 00:13:39.320 --> 00:13:41.480 align:middle line:84% And the dimension is n, if I had to define 00:13:41.480 --> 00:13:42.720 align:middle line:90% what the dimension is. 00:13:42.720 --> 00:13:45.470 align:middle line:84% What is an example of infinite dimensional? 00:13:45.470 --> 00:13:51.130 align:middle line:84% Well, you can probably guess since I led up 00:13:51.130 --> 00:13:53.680 align:middle line:84% to it with saying there's a big difference between this one 00:13:53.680 --> 00:13:55.030 align:middle line:90% and-- 00:13:55.030 --> 00:13:58.330 align:middle line:90% Rn and Cn. 00:13:58.330 --> 00:14:02.590 align:middle line:84% The space of continues functions on the interval 0, 1, 00:14:02.590 --> 00:14:04.270 align:middle line:90% this is infinite dimensional. 00:14:04.270 --> 00:14:08.560 align:middle line:90% 00:14:08.560 --> 00:14:09.860 align:middle line:90% And why is that? 00:14:09.860 --> 00:14:14.810 align:middle line:84% This is because the set by like E given by the functions 00:14:14.810 --> 00:14:20.670 align:middle line:90% fn of x equals x to the n-- 00:14:20.670 --> 00:14:26.280 align:middle line:84% here, n is a natural number or 0-- 00:14:26.280 --> 00:14:28.080 align:middle line:84% this is a linearly independent set. 00:14:28.080 --> 00:14:29.865 align:middle line:84% I'll let you think about why that is. 00:14:29.865 --> 00:14:33.260 align:middle line:90% 00:14:33.260 --> 00:14:36.400 align:middle line:84% And you see that it's not a finite set. 00:14:36.400 --> 00:14:40.040 align:middle line:84% And it contains infinitely many different functions. 00:14:40.040 --> 00:14:48.620 align:middle line:84% So this is a linearly independent set, OK? 00:14:48.620 --> 00:14:51.980 align:middle line:84% So like I said, what we're going to be dealing with 00:14:51.980 --> 00:14:57.160 align:middle line:84% is how to handle solving linear equations or questions 00:14:57.160 --> 00:15:00.190 align:middle line:84% about analysis that we'll need to solve 00:15:00.190 --> 00:15:05.620 align:middle line:84% other problems on these infinite dimensional spaces, 00:15:05.620 --> 00:15:07.900 align:middle line:84% unlike in the past where we did analysis 00:15:07.900 --> 00:15:10.120 align:middle line:90% on finite dimensional spaces. 00:15:10.120 --> 00:15:10.990 align:middle line:90% And the types of-- 00:15:10.990 --> 00:15:14.120 align:middle line:90% 00:15:14.120 --> 00:15:17.600 align:middle line:84% the types of things you proved in the analysis class 00:15:17.600 --> 00:15:22.370 align:middle line:84% were something like the Heine-Borel theorem 00:15:22.370 --> 00:15:26.780 align:middle line:84% that closed and bounded subsets of Rn 00:15:26.780 --> 00:15:32.580 align:middle line:84% are compact, meaning every bounded sequence has 00:15:32.580 --> 00:15:34.338 align:middle line:90% a convergent subsequence. 00:15:34.338 --> 00:15:35.880 align:middle line:84% This is, in fact, something you prove 00:15:35.880 --> 00:15:38.670 align:middle line:84% to show that every continuous function on a closed-- 00:15:38.670 --> 00:15:45.760 align:middle line:84% on a closed and bounded set has a min in a max in that set. 00:15:45.760 --> 00:15:47.560 align:middle line:84% But that statement that I just said 00:15:47.560 --> 00:15:50.890 align:middle line:84% about the Heine-Borel theorem, that is not true once we 00:15:50.890 --> 00:15:52.630 align:middle line:90% get to infinite dimensions. 00:15:52.630 --> 00:15:54.520 align:middle line:90% And so we have to-- 00:15:54.520 --> 00:15:56.500 align:middle line:84% if we want to be able to solve problems, 00:15:56.500 --> 00:15:59.800 align:middle line:84% we'll have to develop some machinery to get around that. 00:15:59.800 --> 00:16:03.850 align:middle line:84% It is one of the main issues that 00:16:03.850 --> 00:16:08.770 align:middle line:84% arise when you move from doing analysis on finite dimensions 00:16:08.770 --> 00:16:10.780 align:middle line:90% to infinite dimensions. 00:16:10.780 --> 00:16:14.860 align:middle line:84% So OK, mathematically, I haven't said much, 00:16:14.860 --> 00:16:18.920 align:middle line:84% but I have been trying to gin up the subject 00:16:18.920 --> 00:16:20.170 align:middle line:90% to make sure you stay engaged. 00:16:20.170 --> 00:16:25.850 align:middle line:90% 00:16:25.850 --> 00:16:29.030 align:middle line:84% All right, so we have vector spaces. 00:16:29.030 --> 00:16:33.515 align:middle line:84% But to do analysis, we need some notion of how close things are. 00:16:33.515 --> 00:16:36.550 align:middle line:90% 00:16:36.550 --> 00:16:42.960 align:middle line:84% And to do that, we introduced the notion of a norm. 00:16:42.960 --> 00:16:48.210 align:middle line:90% So a norm on a vector space V-- 00:16:48.210 --> 00:16:53.980 align:middle line:84% if I don't write vector space V, you should say to yourself, 00:16:53.980 --> 00:16:55.890 align:middle line:84% this capital V is a vector space. 00:16:55.890 --> 00:17:00.750 align:middle line:84% A norm on a vector space V, this is a function. 00:17:00.750 --> 00:17:04.420 align:middle line:84% This is going to be an object that generalizes length. 00:17:04.420 --> 00:17:10.020 align:middle line:84% So it's a function from V to the set of non-negative numbers 00:17:10.020 --> 00:17:16.500 align:middle line:84% with three properties that we associate to length, 00:17:16.500 --> 00:17:21.550 align:middle line:84% namely that the norm of V is 0 implies-- 00:17:21.550 --> 00:17:24.420 align:middle line:90% 00:17:24.420 --> 00:17:32.595 align:middle line:84% well, I should say if and only if V equals 0. 00:17:32.595 --> 00:17:37.890 align:middle line:90% 00:17:37.890 --> 00:17:43.950 align:middle line:84% 2, if I take V and I multiply it by a scalar 00:17:43.950 --> 00:17:46.290 align:middle line:84% and then I take its length, I should get something 00:17:46.290 --> 00:17:50.790 align:middle line:84% like that scalar times the length of the vector. 00:17:50.790 --> 00:17:53.910 align:middle line:84% If I take a vector, multiply it by 2, 00:17:53.910 --> 00:17:56.200 align:middle line:84% I should get twice times the length of the vector. 00:17:56.200 --> 00:18:00.090 align:middle line:90% So this is expressed by this. 00:18:00.090 --> 00:18:06.010 align:middle line:84% And this is for all lambda in my field of scalars 00:18:06.010 --> 00:18:10.510 align:middle line:90% and for all V in V. 00:18:10.510 --> 00:18:16.350 align:middle line:84% So this property here is referred to as homogeneity. 00:18:16.350 --> 00:18:18.890 align:middle line:90% 00:18:18.890 --> 00:18:23.482 align:middle line:84% This property here is referred to as definiteness. 00:18:23.482 --> 00:18:27.820 align:middle line:84% Definite-- I don't know if I'm spelling that right, 00:18:27.820 --> 00:18:29.130 align:middle line:90% but definiteness. 00:18:29.130 --> 00:18:33.630 align:middle line:90% 00:18:33.630 --> 00:18:38.940 align:middle line:84% And then the third property is that it satisfies the triangle 00:18:38.940 --> 00:18:40.480 align:middle line:90% inequality. 00:18:40.480 --> 00:18:52.290 align:middle line:84% This is that for all V1, V2 in V, the norm of V1 plus V2, 00:18:52.290 --> 00:18:58.890 align:middle line:84% this is less than or equal to the norm of V1 plus V2. 00:18:58.890 --> 00:19:01.260 align:middle line:84% And any vector space that has a norm on it 00:19:01.260 --> 00:19:02.385 align:middle line:90% we call a norm space. 00:19:02.385 --> 00:19:11.640 align:middle line:90% 00:19:11.640 --> 00:19:15.950 align:middle line:84% With norm is what we call a norm space. 00:19:15.950 --> 00:19:20.280 align:middle line:90% 00:19:20.280 --> 00:19:23.520 align:middle line:90% And this thing here, I usually-- 00:19:23.520 --> 00:19:26.220 align:middle line:84% so it's referred to as a triangle inequality. 00:19:26.220 --> 00:19:29.310 align:middle line:84% All right, so a vector space with a function on it 00:19:29.310 --> 00:19:31.140 align:middle line:84% that satisfies these three properties 00:19:31.140 --> 00:19:33.255 align:middle line:90% we call a norm space. 00:19:33.255 --> 00:19:37.810 align:middle line:90% 00:19:37.810 --> 00:19:41.110 align:middle line:84% And then so like I said in my past classes, 00:19:41.110 --> 00:19:44.290 align:middle line:84% whenever I see a-- or whenever you see a decent definition 00:19:44.290 --> 00:19:47.890 align:middle line:84% or something with substance, you should do examples. 00:19:47.890 --> 00:19:49.480 align:middle line:84% We'll do that in just a minute after I 00:19:49.480 --> 00:19:51.100 align:middle line:90% give a few more definitions. 00:19:51.100 --> 00:19:54.100 align:middle line:90% 00:19:54.100 --> 00:19:57.500 align:middle line:90% So let me just add to this one. 00:19:57.500 --> 00:20:00.440 align:middle line:84% This was the definition of a norm. 00:20:00.440 --> 00:20:08.880 align:middle line:84% A semi-norm wants to be a norm, but it's not quite. 00:20:08.880 --> 00:20:11.550 align:middle line:84% This is a function, which I'll also 00:20:11.550 --> 00:20:18.380 align:middle line:84% denote with these two parallel lines on each side. 00:20:18.380 --> 00:20:24.040 align:middle line:90% 00:20:24.040 --> 00:20:31.720 align:middle line:84% It's a function satisfying homogeneity and the triangle 00:20:31.720 --> 00:20:36.100 align:middle line:84% inequality, but not necessarily positive-- or definiteness-- 00:20:36.100 --> 00:20:39.220 align:middle line:84% or positive definiteness is also what that goes by, 00:20:39.220 --> 00:20:44.625 align:middle line:84% satisfying 2 and 3, but maybe not necessarily 1. 00:20:44.625 --> 00:20:49.450 align:middle line:90% 00:20:49.450 --> 00:20:53.470 align:middle line:84% Again, semi-norms pop up in a natural way. 00:20:53.470 --> 00:20:57.100 align:middle line:84% And I'll give you an example in just a second. 00:20:57.100 --> 00:21:02.250 align:middle line:84% So first off, we have this notion of length 00:21:02.250 --> 00:21:06.300 align:middle line:84% in a vector space, which is a norm. 00:21:06.300 --> 00:21:09.740 align:middle line:84% And if we're given a norm on a vector space, 00:21:09.740 --> 00:21:11.330 align:middle line:90% we can associate a metric. 00:21:11.330 --> 00:21:12.515 align:middle line:90% So remember, a metric-- 00:21:12.515 --> 00:21:15.130 align:middle line:90% 00:21:15.130 --> 00:21:17.140 align:middle line:84% so from real analysis, I want you to recall 00:21:17.140 --> 00:21:24.310 align:middle line:90% that a function d on a set-- 00:21:24.310 --> 00:21:32.080 align:middle line:90% 00:21:32.080 --> 00:21:39.580 align:middle line:84% if x is a set, a function d from x cross x into 0, 00:21:39.580 --> 00:21:52.980 align:middle line:84% infinity is a metric if you have three conditions satisfied-- 00:21:52.980 --> 00:22:01.580 align:middle line:84% a, so the distance between a point in itself-- 00:22:01.580 --> 00:22:04.580 align:middle line:84% or let me write it this way, the distance between two points 00:22:04.580 --> 00:22:10.980 align:middle line:90% is 0 if and only if x equals y. 00:22:10.980 --> 00:22:20.880 align:middle line:84% b, for all x, y in the set X, dxy equals dyx-- 00:22:20.880 --> 00:22:25.590 align:middle line:84% this is symmetry of the distance. 00:22:25.590 --> 00:22:27.240 align:middle line:84% And you have the triangle inequality 00:22:27.240 --> 00:22:32.650 align:middle line:84% for the distance for all xy in X, d. 00:22:32.650 --> 00:22:43.210 align:middle line:84% Or for all x, y, and z in x, dxy xy 00:22:43.210 --> 00:22:47.260 align:middle line:84% is less than or equal to the distance from x to z 00:22:47.260 --> 00:22:51.340 align:middle line:90% plus the distance from z to y. 00:22:51.340 --> 00:22:54.550 align:middle line:90% 00:22:54.550 --> 00:22:57.980 align:middle line:84% So if I have a vector space and I have a norm on it, 00:22:57.980 --> 00:23:01.060 align:middle line:84% I can turn that space now into a metric space. 00:23:01.060 --> 00:23:04.330 align:middle line:84% And this metric that I'm about to write down 00:23:04.330 --> 00:23:07.780 align:middle line:84% is usually referred to as the metric induced by the norm. 00:23:07.780 --> 00:23:10.675 align:middle line:84% So our first little kind of mini-theorem here-- 00:23:10.675 --> 00:23:15.590 align:middle line:90% 00:23:15.590 --> 00:23:31.188 align:middle line:84% if I have a norm on a vector space, then d of-- 00:23:31.188 --> 00:23:33.640 align:middle line:90% let me use the same-- 00:23:33.640 --> 00:23:40.420 align:middle line:84% if I define a function dvw to be simply the length 00:23:40.420 --> 00:23:41.455 align:middle line:90% of v minus w-- 00:23:41.455 --> 00:23:44.030 align:middle line:90% 00:23:44.030 --> 00:23:53.230 align:middle line:84% so for elements of v and w in the vector space, 00:23:53.230 --> 00:24:02.480 align:middle line:84% defines metric on V. And this metric 00:24:02.480 --> 00:24:07.760 align:middle line:84% we refer to as the metric induced by the norm, all right? 00:24:07.760 --> 00:24:13.100 align:middle line:84% And this is not difficult to prove. 00:24:13.100 --> 00:24:20.240 align:middle line:84% Basically, 1, 2, and 3 imply respectively a, b, and c. 00:24:20.240 --> 00:24:24.335 align:middle line:84% 1, the first property of a norm, implies a. 00:24:24.335 --> 00:24:25.295 align:middle line:90% This is pretty clear. 00:24:25.295 --> 00:24:28.630 align:middle line:90% 00:24:28.630 --> 00:24:32.780 align:middle line:84% The distance between vw is 0 if and only 00:24:32.780 --> 00:24:36.290 align:middle line:84% if v minus w equals 0, which is if and only if v equals w. 00:24:36.290 --> 00:24:38.570 align:middle line:90% That gives us part a. 00:24:38.570 --> 00:24:51.680 align:middle line:84% b we get from 2, since by 2 v minus w 00:24:51.680 --> 00:24:59.000 align:middle line:84% is equal to minus 1 times w minus v equals-- 00:24:59.000 --> 00:25:01.505 align:middle line:84% now, we pull the scalar out and take its absolute value. 00:25:01.505 --> 00:25:09.350 align:middle line:90% 00:25:09.350 --> 00:25:13.520 align:middle line:84% So since by 2, we have the length of v minus w 00:25:13.520 --> 00:25:18.000 align:middle line:84% equals the length of w minus w, this implies b. 00:25:18.000 --> 00:25:20.550 align:middle line:84% And then c just follows again immediately 00:25:20.550 --> 00:25:24.750 align:middle line:84% from the triangle inequality 3 by just adding and subtracting 00:25:24.750 --> 00:25:26.880 align:middle line:90% a third element. 00:25:26.880 --> 00:25:36.930 align:middle line:84% So let me just write 3 implies c essentially immediately, 00:25:36.930 --> 00:25:37.430 align:middle line:90% all right? 00:25:37.430 --> 00:25:40.710 align:middle line:90% 00:25:40.710 --> 00:25:45.980 align:middle line:84% OK, so when we have a norm, we get a notion 00:25:45.980 --> 00:25:47.720 align:middle line:84% of a metric, a notion of distance 00:25:47.720 --> 00:25:50.450 align:middle line:84% between two vectors in our vector space 00:25:50.450 --> 00:25:52.980 align:middle line:84% by just taking the length of the difference of them. 00:25:52.980 --> 00:25:56.947 align:middle line:90% So again, this should sort of-- 00:25:56.947 --> 00:26:06.695 align:middle line:84% this is supposed to be an analog of what we see in R and Rn 00:26:06.695 --> 00:26:07.195 align:middle line:90% in general. 00:26:07.195 --> 00:26:10.750 align:middle line:90% 00:26:10.750 --> 00:26:14.435 align:middle line:84% So for example, let me just recall this. 00:26:14.435 --> 00:26:17.728 align:middle line:90% 00:26:17.728 --> 00:26:19.145 align:middle line:84% So now, let's look at a few norms. 00:26:19.145 --> 00:26:22.460 align:middle line:90% 00:26:22.460 --> 00:26:27.230 align:middle line:84% Instead of n tuples of Rn or Cn where 00:26:27.230 --> 00:26:28.580 align:middle line:90% we have the Euclidean norm-- 00:26:28.580 --> 00:26:39.210 align:middle line:90% 00:26:39.210 --> 00:26:46.620 align:middle line:84% Euclidean norm, here, if I have an n-tuple given by x-- 00:26:46.620 --> 00:26:48.960 align:middle line:84% so let me put a 2 here to denote this norm-- 00:26:48.960 --> 00:26:54.040 align:middle line:84% then the Euclidean norm of this vector-- 00:26:54.040 --> 00:27:00.300 align:middle line:90% so this guy is in Rn or Cn-- 00:27:00.300 --> 00:27:06.480 align:middle line:84% then its Euclidean norm is given by the sum i equals 1 00:27:06.480 --> 00:27:12.630 align:middle line:90% to n xi length squared 1/2. 00:27:12.630 --> 00:27:19.160 align:middle line:84% And that gives you the standard notion of length and distance 00:27:19.160 --> 00:27:24.140 align:middle line:84% between points that you've dealt with in Euclidean space. 00:27:24.140 --> 00:27:25.820 align:middle line:84% But that's not the only norm you could 00:27:25.820 --> 00:27:31.580 align:middle line:84% have on these spaces that you've dealt with before. 00:27:31.580 --> 00:27:38.210 align:middle line:84% Another one-- put an infinity here. 00:27:38.210 --> 00:27:48.800 align:middle line:84% This norm is the max between 1 and n of xi. 00:27:48.800 --> 00:27:52.190 align:middle line:84% So to measure the length of a vector, 00:27:52.190 --> 00:27:58.080 align:middle line:84% I take this to be the magnitude of the largest 00:27:58.080 --> 00:28:00.025 align:middle line:90% entry in that vector, OK? 00:28:00.025 --> 00:28:06.960 align:middle line:90% 00:28:06.960 --> 00:28:12.180 align:middle line:84% And in general, there's a whole family 00:28:12.180 --> 00:28:14.310 align:middle line:84% of norms I could put on Rn or Cn. 00:28:14.310 --> 00:28:20.850 align:middle line:90% 00:28:20.850 --> 00:28:22.440 align:middle line:90% Let me put a p here. 00:28:22.440 --> 00:28:31.370 align:middle line:84% This is norm, the lp norm, little lp norm, 00:28:31.370 --> 00:28:37.040 align:middle line:84% is the sum of the p-th power of the xi's raised to the 1 00:28:37.040 --> 00:28:37.580 align:middle line:90% over p. 00:28:37.580 --> 00:28:40.220 align:middle line:84% Of course, you need this 1 over p for homogeneity 00:28:40.220 --> 00:28:43.070 align:middle line:84% to work out and also the triangle inequality. 00:28:43.070 --> 00:28:49.077 align:middle line:84% And this is for 1 less than or equal to p less than infinity. 00:28:49.077 --> 00:28:51.160 align:middle line:84% I didn't write Infinity there because that doesn't 00:28:51.160 --> 00:28:52.840 align:middle line:84% make sense, although you can actually 00:28:52.840 --> 00:28:55.300 align:middle line:84% prove that if I take a fixed vector 00:28:55.300 --> 00:28:59.170 align:middle line:84% and I let p go to infinity, then this quantity here 00:28:59.170 --> 00:29:01.470 align:middle line:90% converges to this quantity. 00:29:01.470 --> 00:29:02.470 align:middle line:90% That's not hard to show. 00:29:02.470 --> 00:29:06.250 align:middle line:90% 00:29:06.250 --> 00:29:11.370 align:middle line:84% So let me quickly draw you a quick picture 00:29:11.370 --> 00:29:14.370 align:middle line:84% of what the unit balls look like-- 00:29:14.370 --> 00:29:18.700 align:middle line:90% 00:29:18.700 --> 00:29:23.000 align:middle line:84% R2, say, with the different norms. 00:29:23.000 --> 00:29:30.730 align:middle line:84% So let me recall that if I have, in a metric space, a metric, 00:29:30.730 --> 00:29:34.450 align:middle line:84% then B x, r this is a set of all y in x 00:29:34.450 --> 00:29:42.665 align:middle line:84% so that the distance from x to y is less than or equal to r. 00:29:42.665 --> 00:29:45.710 align:middle line:90% 00:29:45.710 --> 00:29:48.230 align:middle line:84% Now, you know what the ball looks 00:29:48.230 --> 00:29:49.730 align:middle line:90% like for the Euclidean norm. 00:29:49.730 --> 00:29:59.430 align:middle line:84% This is just a circle and filled in, of course. 00:29:59.430 --> 00:30:00.680 align:middle line:90% I'm not going to fill this in. 00:30:00.680 --> 00:30:02.380 align:middle line:90% So now, what I'm looking at is-- 00:30:02.380 --> 00:30:05.350 align:middle line:90% and I'll put a 2. 00:30:05.350 --> 00:30:10.260 align:middle line:84% This is the ball centered at 0 of radius 1, 00:30:10.260 --> 00:30:12.300 align:middle line:90% so everything inside. 00:30:12.300 --> 00:30:19.490 align:middle line:84% What about for the infinity norm? 00:30:19.490 --> 00:30:26.170 align:middle line:84% Let's say I want to look at what does this look like. 00:30:26.170 --> 00:30:33.730 align:middle line:90% This, in fact, is square. 00:30:33.730 --> 00:30:38.540 align:middle line:90% 00:30:38.540 --> 00:30:45.900 align:middle line:84% And what about-- let's do one more about the little l1 norm, 00:30:45.900 --> 00:30:50.240 align:middle line:84% which is just the length of the magnitude 00:30:50.240 --> 00:30:52.760 align:middle line:84% of the entries, the length of the absolute value 00:30:52.760 --> 00:30:54.740 align:middle line:90% of the entries. 00:30:54.740 --> 00:31:02.120 align:middle line:84% So all these go through these points here on the axis. 00:31:02.120 --> 00:31:06.680 align:middle line:84% This ball-- so first off, everything inside the blue 00:31:06.680 --> 00:31:08.420 align:middle line:90% is the l infinity ball. 00:31:08.420 --> 00:31:11.210 align:middle line:84% Everything inside the white is that little l2 ball. 00:31:11.210 --> 00:31:13.700 align:middle line:84% The little l1 ball is everything inside 00:31:13.700 --> 00:31:17.515 align:middle line:84% of this square, which is now tilted. 00:31:17.515 --> 00:31:21.070 align:middle line:90% 00:31:21.070 --> 00:31:27.190 align:middle line:84% And every other little lp ball is in between the yellow 00:31:27.190 --> 00:31:28.060 align:middle line:90% and the blue. 00:31:28.060 --> 00:31:30.160 align:middle line:84% And if you take p goes to infinity, 00:31:30.160 --> 00:31:32.800 align:middle line:84% then that ball converges, in a certain sense, 00:31:32.800 --> 00:31:35.380 align:middle line:84% to this l infinity ball, which is in blue. 00:31:35.380 --> 00:31:38.580 align:middle line:90% 00:31:38.580 --> 00:31:41.610 align:middle line:84% So you see, changing the norm, even 00:31:41.610 --> 00:31:43.650 align:middle line:84% on these finite dimensional spaces, 00:31:43.650 --> 00:31:49.860 align:middle line:84% changes the geometry of the balls, if you like, all right? 00:31:49.860 --> 00:31:52.930 align:middle line:90% 00:31:52.930 --> 00:31:56.560 align:middle line:84% But not in two drastic of a way, meaning 00:31:56.560 --> 00:31:59.290 align:middle line:90% if I take a large enough B-- 00:31:59.290 --> 00:32:01.930 align:middle line:84% if I take a large enough l1 ball, 00:32:01.930 --> 00:32:06.190 align:middle line:84% that will swallow up an l infinity ball. 00:32:06.190 --> 00:32:10.920 align:middle line:84% And therefore, the two balls are kind of an-- 00:32:10.920 --> 00:32:20.500 align:middle line:84% I can take the size of that ball to be maybe of size 3. 00:32:20.500 --> 00:32:23.620 align:middle line:84% And so the balls are kind of the same. 00:32:23.620 --> 00:32:25.520 align:middle line:84% One can always be sandwiching the other. 00:32:25.520 --> 00:32:27.820 align:middle line:84% I'll talk a little more about that 00:32:27.820 --> 00:32:31.420 align:middle line:84% when we talk about equivalent norms, at least 00:32:31.420 --> 00:32:32.900 align:middle line:90% in the problem sets. 00:32:32.900 --> 00:32:37.000 align:middle line:84% So OK, so that's an example of norms 00:32:37.000 --> 00:32:42.330 align:middle line:84% on a finite dimensional vector space. 00:32:42.330 --> 00:32:44.000 align:middle line:90% Let's look at another norm. 00:32:44.000 --> 00:32:52.890 align:middle line:90% 00:32:52.890 --> 00:32:56.670 align:middle line:84% Let's take a metric space so I remember we have a-- 00:32:56.670 --> 00:32:58.320 align:middle line:84% so this is just any old metric space. 00:32:58.320 --> 00:33:04.410 align:middle line:90% 00:33:04.410 --> 00:33:10.440 align:middle line:84% And now, I'm going to define a vector space, C lower infinity 00:33:10.440 --> 00:33:11.070 align:middle line:90% x. 00:33:11.070 --> 00:33:20.170 align:middle line:84% This is defined to be the set of all continuous functions 00:33:20.170 --> 00:33:21.860 align:middle line:90% on this metric space-- 00:33:21.860 --> 00:33:23.360 align:middle line:84% remember, if we have a metric space, 00:33:23.360 --> 00:33:26.830 align:middle line:84% we have a notion of continuous functions on it-- 00:33:26.830 --> 00:33:28.630 align:middle line:90% such that-- what? 00:33:28.630 --> 00:33:32.440 align:middle line:84% So these are-- so again, to fix this, 00:33:32.440 --> 00:33:36.670 align:middle line:84% let me just clarify where it's going. 00:33:36.670 --> 00:33:49.840 align:middle line:84% f goes from x to, let's say, C such that f is continuous 00:33:49.840 --> 00:34:00.400 align:middle line:84% and f is bounded, meaning the image of x under f 00:34:00.400 --> 00:34:02.980 align:middle line:84% is a bounded subset of C, all right? 00:34:02.980 --> 00:34:10.850 align:middle line:90% 00:34:10.850 --> 00:34:14.000 align:middle line:84% Just so you can connect this with something I just 00:34:14.000 --> 00:34:16.670 align:middle line:84% wrote down a minute ago, the set of all bounded functions, 00:34:16.670 --> 00:34:18.920 align:middle line:84% continuous functions on 0, 1, this 00:34:18.920 --> 00:34:20.929 align:middle line:84% is just a set of continuous functions on 0, 00:34:20.929 --> 00:34:23.960 align:middle line:84% 1 because we know continuous functions 00:34:23.960 --> 00:34:28.820 align:middle line:84% on a closed and bounded interval are bounded. 00:34:28.820 --> 00:34:34.110 align:middle line:84% So I don't have to say bounded whenever I write this. 00:34:34.110 --> 00:34:36.550 align:middle line:84% OK, so I have this space of continuous functions 00:34:36.550 --> 00:34:39.940 align:middle line:84% on a metric space, which are bounded. 00:34:39.940 --> 00:34:42.590 align:middle line:84% And I'm going to define a norm on this. 00:34:42.590 --> 00:34:43.855 align:middle line:90% So this is a vector space. 00:34:43.855 --> 00:34:51.250 align:middle line:90% 00:34:51.250 --> 00:34:54.489 align:middle line:84% Because any sum of two continuous functions 00:34:54.489 --> 00:34:57.940 align:middle line:84% is continuous, a scalar multiple times a continuous function 00:34:57.940 --> 00:34:59.170 align:middle line:90% is continuous. 00:34:59.170 --> 00:35:02.110 align:middle line:84% And again, those two operations satisfy 00:35:02.110 --> 00:35:05.870 align:middle line:90% the axioms of a vector space. 00:35:05.870 --> 00:35:08.435 align:middle line:84% And I can define a norm on this space. 00:35:08.435 --> 00:35:12.020 align:middle line:90% 00:35:12.020 --> 00:35:15.020 align:middle line:90% Then I claim that-- 00:35:15.020 --> 00:35:17.990 align:middle line:90% what do I use here? 00:35:17.990 --> 00:35:22.788 align:middle line:84% Then if I define an infinity norm again, 00:35:22.788 --> 00:35:24.830 align:middle line:84% which will look similar to what I wrote up there, 00:35:24.830 --> 00:35:35.510 align:middle line:84% as this is the sup of all x in capital X of u of x, 00:35:35.510 --> 00:35:42.880 align:middle line:84% then this is a norm on this space 00:35:42.880 --> 00:35:45.280 align:middle line:84% of continuous-- bounded continuous functions. 00:35:45.280 --> 00:35:49.640 align:middle line:90% 00:35:49.640 --> 00:35:56.370 align:middle line:84% So 1 and 2 are pretty easy to see from the definition. 00:35:56.370 --> 00:36:05.770 align:middle line:84% So properties 1 and 2 are easy to see. 00:36:05.770 --> 00:36:07.870 align:middle line:84% For the triangle inequality, well, that 00:36:07.870 --> 00:36:11.230 align:middle line:84% follows essentially from the triangle inequality for C. 00:36:11.230 --> 00:36:15.510 align:middle line:84% These are C-valued, complex valued functions. 00:36:15.510 --> 00:36:20.585 align:middle line:84% If you like, replace it with real valued functions 00:36:20.585 --> 00:36:22.710 align:middle line:84% if that makes you feel better in the first lecture, 00:36:22.710 --> 00:36:26.660 align:middle line:84% although we'll need complex numbers eventually. 00:36:26.660 --> 00:36:31.070 align:middle line:84% So let's check that this function satisfies the triangle 00:36:31.070 --> 00:36:32.585 align:middle line:84% inequality and therefore is a norm. 00:36:32.585 --> 00:36:35.450 align:middle line:90% 00:36:35.450 --> 00:36:43.730 align:middle line:84% If u and v are two bounded continuous functions on x, then 00:36:43.730 --> 00:36:50.150 align:middle line:84% for all x in X, if I take ux plus vx, 00:36:50.150 --> 00:36:55.400 align:middle line:84% this is less than or equal to, by the triangle inequality 00:36:55.400 --> 00:37:02.050 align:middle line:84% for C, the absolute value of u of x plus v of x. 00:37:02.050 --> 00:37:04.240 align:middle line:84% Again, u of x is a complex number and so 00:37:04.240 --> 00:37:07.390 align:middle line:84% is v of x So this is by triangle inequality. 00:37:07.390 --> 00:37:14.370 align:middle line:90% 00:37:14.370 --> 00:37:18.290 align:middle line:84% But now, this, for any old x, is always 00:37:18.290 --> 00:37:24.600 align:middle line:84% bounded by the supremum of that over all values. 00:37:24.600 --> 00:37:25.640 align:middle line:90% So let me write-- 00:37:25.640 --> 00:37:31.610 align:middle line:90% 00:37:31.610 --> 00:37:32.300 align:middle line:90% OK? 00:37:32.300 --> 00:37:34.490 align:middle line:84% So u of x for any fixed value of x 00:37:34.490 --> 00:37:37.280 align:middle line:84% is always bounded by the supremum over all 00:37:37.280 --> 00:37:40.400 align:middle line:84% x's is in the capital X, which is the norm of u. 00:37:40.400 --> 00:37:44.660 align:middle line:84% So what I've shown is that for all x in capital X, 00:37:44.660 --> 00:37:46.970 align:middle line:84% the absolute value of u of x plus v 00:37:46.970 --> 00:37:50.690 align:middle line:90% of x is bounded by this number. 00:37:50.690 --> 00:37:53.120 align:middle line:84% And therefore, this number is an upper bound 00:37:53.120 --> 00:37:58.280 align:middle line:84% for the set of absolute values of these guys and therefore 00:37:58.280 --> 00:38:01.490 align:middle line:84% the supremum, which implies that-- 00:38:01.490 --> 00:38:12.530 align:middle line:90% 00:38:12.530 --> 00:38:14.860 align:middle line:84% which is the least upper bound of all these quantities 00:38:14.860 --> 00:38:19.825 align:middle line:84% as x ranges over x is less than or equal to this number here. 00:38:19.825 --> 00:38:24.270 align:middle line:90% 00:38:24.270 --> 00:38:26.610 align:middle line:84% And therefore, we have the triangle inequality 00:38:26.610 --> 00:38:27.630 align:middle line:90% for this function. 00:38:27.630 --> 00:38:29.890 align:middle line:84% And therefore, this defines a norm, 00:38:29.890 --> 00:38:35.040 align:middle line:84% which I'll often refer to as the uniform norm or l infinity norm 00:38:35.040 --> 00:38:36.480 align:middle line:84% just because there's an infinity. 00:38:36.480 --> 00:38:45.140 align:middle line:84% But the content should be clear on what I'm talking about. 00:38:45.140 --> 00:38:47.240 align:middle line:84% I mean, this is not inconsistent with what 00:38:47.240 --> 00:38:51.335 align:middle line:84% I wrote down because Rn, that's just a set of-- 00:38:51.335 --> 00:38:55.550 align:middle line:90% 00:38:55.550 --> 00:38:59.190 align:middle line:84% well, OK, never mind, let me stop 00:38:59.190 --> 00:39:02.830 align:middle line:84% talking before I get myself twisted in a knot. 00:39:02.830 --> 00:39:06.240 align:middle line:84% So what does convergence mean in this norm here? 00:39:06.240 --> 00:39:09.690 align:middle line:84% We know what convergence means in the Euclidean or any 00:39:09.690 --> 00:39:10.380 align:middle line:90% of these norms. 00:39:10.380 --> 00:39:12.300 align:middle line:84% It means the points are getting together. 00:39:12.300 --> 00:39:14.040 align:middle line:84% The fixed points in the plane are 00:39:14.040 --> 00:39:16.210 align:middle line:84% starting to get closer and closer together, 00:39:16.210 --> 00:39:18.030 align:middle line:90% at least in R2 or C2. 00:39:18.030 --> 00:39:18.885 align:middle line:90% What about here? 00:39:18.885 --> 00:39:23.790 align:middle line:90% 00:39:23.790 --> 00:39:28.500 align:middle line:90% So let me just note that-- 00:39:28.500 --> 00:39:33.980 align:middle line:84% so remember, un converges to u in this space 00:39:33.980 --> 00:39:35.390 align:middle line:90% of bounded continuous functions. 00:39:35.390 --> 00:39:36.223 align:middle line:90% What does this mean. 00:39:36.223 --> 00:39:38.300 align:middle line:84% This means the distance between un and u 00:39:38.300 --> 00:39:40.610 align:middle line:90% goes to 0 as n goes to infinity. 00:39:40.610 --> 00:39:43.410 align:middle line:90% 00:39:43.410 --> 00:39:45.740 align:middle line:84% So here, I'm talking about what does 00:39:45.740 --> 00:39:52.680 align:middle line:84% convergence of a sequence of elements of this space mean. 00:39:52.680 --> 00:39:58.780 align:middle line:84% So that means that the distance between un and u 00:39:58.780 --> 00:40:02.700 align:middle line:90% goes to 0 as n goes to infinity. 00:40:02.700 --> 00:40:05.200 align:middle line:84% And the distance, remember, is defined in terms of the norm. 00:40:05.200 --> 00:40:11.780 align:middle line:90% 00:40:11.780 --> 00:40:16.910 align:middle line:84% So convergence in the space of bounded continuous functions 00:40:16.910 --> 00:40:19.340 align:middle line:90% just means this. 00:40:19.340 --> 00:40:23.990 align:middle line:84% But in terms of something covered in past analysis class, 00:40:23.990 --> 00:40:29.870 align:middle line:90% what is that equivalent to? 00:40:29.870 --> 00:40:31.010 align:middle line:90% I'll write it out. 00:40:31.010 --> 00:40:33.570 align:middle line:84% This means for all epsilon positive, 00:40:33.570 --> 00:40:39.530 align:middle line:84% there exists a natural number n such that-- 00:40:39.530 --> 00:40:43.310 align:middle line:84% now, this is the sup of u of x for all-- 00:40:43.310 --> 00:40:46.610 align:middle line:84% or sup un of x minus u of x for all x in X 00:40:46.610 --> 00:40:52.310 align:middle line:84% So I can write such that for all n bigger than or equal to n 00:40:52.310 --> 00:41:02.040 align:middle line:84% for all x in X, un of x minus u of x is less than epsilon. 00:41:02.040 --> 00:41:13.030 align:middle line:84% But that's just the definition of uniform convergence 00:41:13.030 --> 00:41:18.680 align:middle line:90% uniformly on x. 00:41:18.680 --> 00:41:23.840 align:middle line:84% So the point of this little note is that convergence 00:41:23.840 --> 00:41:26.420 align:middle line:90% in this norm or in this metric-- 00:41:26.420 --> 00:41:29.570 align:middle line:84% I'm going to use norm and metric interchangeably 00:41:29.570 --> 00:41:32.270 align:middle line:84% because this metric is induced by this norm. 00:41:32.270 --> 00:41:34.730 align:middle line:84% So convergence in norm in this space 00:41:34.730 --> 00:41:38.060 align:middle line:84% of bounded continuous functions is the same 00:41:38.060 --> 00:41:41.270 align:middle line:84% as saying that this sequence of functions 00:41:41.270 --> 00:41:45.700 align:middle line:84% converges to this function uniformly on x 00:41:45.700 --> 00:41:47.695 align:middle line:90% So I hope that's clear. 00:41:47.695 --> 00:41:50.590 align:middle line:90% 00:41:50.590 --> 00:41:52.450 align:middle line:84% So maybe I should say this now instead 00:41:52.450 --> 00:41:55.330 align:middle line:90% of in the eighth lecture. 00:41:55.330 --> 00:41:57.130 align:middle line:84% But really, take the time to actually watch 00:41:57.130 --> 00:41:59.960 align:middle line:90% these lectures for two reasons. 00:41:59.960 --> 00:42:03.940 align:middle line:84% One, I'm actually here recording them. 00:42:03.940 --> 00:42:08.700 align:middle line:84% So instead of just saying read this and ask me 00:42:08.700 --> 00:42:11.790 align:middle line:84% if you have any questions, I'm giving you some more insight 00:42:11.790 --> 00:42:14.040 align:middle line:90% than just what's in the notes. 00:42:14.040 --> 00:42:17.010 align:middle line:84% And two, it keeps you engaged, OK? 00:42:17.010 --> 00:42:21.330 align:middle line:84% So you should treat these as you actually being there-- 00:42:21.330 --> 00:42:25.440 align:middle line:84% have your notebook, take notes as I lecture. 00:42:25.440 --> 00:42:30.360 align:middle line:84% The great thing is that you can pause and rewind. 00:42:30.360 --> 00:42:32.280 align:middle line:84% This space of continuous functions-- 00:42:32.280 --> 00:42:36.930 align:middle line:90% 00:42:36.930 --> 00:42:37.755 align:middle line:90% we have that norm. 00:42:37.755 --> 00:42:42.140 align:middle line:90% 00:42:42.140 --> 00:42:49.850 align:middle line:84% OK, so some more examples of norm spaces is-- 00:42:49.850 --> 00:42:54.780 align:middle line:84% so I was calling that little lp norm. 00:42:54.780 --> 00:42:57.170 align:middle line:84% But the little is not necessarily little. 00:42:57.170 --> 00:43:04.030 align:middle line:84% So the actual little lp space that I will typically 00:43:04.030 --> 00:43:05.290 align:middle line:90% refer to-- 00:43:05.290 --> 00:43:08.230 align:middle line:84% this is going to be the space of all-- 00:43:08.230 --> 00:43:12.640 align:middle line:90% 00:43:12.640 --> 00:43:18.340 align:middle line:84% this is the space of all sequences now. 00:43:18.340 --> 00:43:23.880 align:middle line:84% So the vectors in this space are sequences. 00:43:23.880 --> 00:43:26.940 align:middle line:84% The points in this space are sequences, such that-- 00:43:26.940 --> 00:43:30.500 align:middle line:90% 00:43:30.500 --> 00:43:33.920 align:middle line:90% so call this thing element a. 00:43:33.920 --> 00:43:35.555 align:middle line:90% So a is a sequence. 00:43:35.555 --> 00:43:39.730 align:middle line:90% 00:43:39.730 --> 00:43:41.490 align:middle line:84% So this is the set of all sequences that 00:43:41.490 --> 00:43:44.220 align:middle line:90% have lp norm that's finite. 00:43:44.220 --> 00:43:46.950 align:middle line:84% And what is the lp norm in this case? 00:43:46.950 --> 00:43:48.750 align:middle line:84% Well, it's the natural generalization 00:43:48.750 --> 00:43:56.445 align:middle line:84% of that guy, where the lp norm is equal to the sum 00:43:56.445 --> 00:44:03.985 align:middle line:90% j equals 1 to infinity for 1-- 00:44:03.985 --> 00:44:06.775 align:middle line:90% 00:44:06.775 --> 00:44:08.650 align:middle line:90% p between 1 and infinity. 00:44:08.650 --> 00:44:17.350 align:middle line:84% And then the l infinity norm is just the sup over . 00:44:17.350 --> 00:44:32.580 align:middle line:90% J 00:44:32.580 --> 00:44:36.930 align:middle line:84% So little lp stands for the space of sequences 00:44:36.930 --> 00:44:39.000 align:middle line:90% with finite lp norm. 00:44:39.000 --> 00:44:41.700 align:middle line:84% Or you could say their lp is summable. 00:44:41.700 --> 00:44:44.640 align:middle line:90% Their p-th power is summable. 00:44:44.640 --> 00:44:52.390 align:middle line:84% So let me just state a silly example. 00:44:52.390 --> 00:44:56.730 align:middle line:84% The sequence 1 over j, j equals 1 00:44:56.730 --> 00:45:01.620 align:middle line:84% to infinity, this is in little lp for all p bigger than 1, 00:45:01.620 --> 00:45:03.600 align:middle line:84% but not for p equals 1 because then we 00:45:03.600 --> 00:45:04.950 align:middle line:90% get the harmonic series. 00:45:04.950 --> 00:45:12.250 align:middle line:90% 00:45:12.250 --> 00:45:14.290 align:middle line:90% Now, why? 00:45:14.290 --> 00:45:16.980 align:middle line:90% 00:45:16.980 --> 00:45:20.930 align:middle line:84% First off, why the triangle inequality holds even 00:45:20.930 --> 00:45:26.090 align:middle line:84% in the finitely many, in the finite dimensional case 00:45:26.090 --> 00:45:30.830 align:middle line:84% and why this is a natural vector space is non-trivial. 00:45:30.830 --> 00:45:32.540 align:middle line:84% So you shouldn't be able to actually 00:45:32.540 --> 00:45:33.950 align:middle line:90% say, oh OK, that makes sense. 00:45:33.950 --> 00:45:35.510 align:middle line:84% It satisfies the triangle inequality. 00:45:35.510 --> 00:45:36.920 align:middle line:90% It's not clear. 00:45:36.920 --> 00:45:40.640 align:middle line:84% It's non-trivial that this is, in fact, a norm. 00:45:40.640 --> 00:45:42.470 align:middle line:90% This one's easier to see. 00:45:42.470 --> 00:45:46.250 align:middle line:84% But it's not clear for p bigger than 1 00:45:46.250 --> 00:45:52.340 align:middle line:84% that this is a norm and that therefore little lp, 00:45:52.340 --> 00:45:55.070 align:middle line:84% and that the sum of two guys in little lp 00:45:55.070 --> 00:45:57.695 align:middle line:84% are in lp so that this is an actual vector space, 00:45:57.695 --> 00:45:59.570 align:middle line:84% and then we have the triangle inequalities so 00:45:59.570 --> 00:46:00.770 align:middle line:90% that this is a norm-- 00:46:00.770 --> 00:46:02.900 align:middle line:90% that will be in the exercises. 00:46:02.900 --> 00:46:04.400 align:middle line:90% But just take it as-- 00:46:04.400 --> 00:46:09.500 align:middle line:84% take me at my word that this is indeed a vector space. 00:46:09.500 --> 00:46:13.610 align:middle line:84% If I take two sequences that are in little lp, 00:46:13.610 --> 00:46:18.020 align:middle line:84% meaning their p-th power is summable, then 00:46:18.020 --> 00:46:23.240 align:middle line:84% their sum, entry by entry, is also in little lp and so on, 00:46:23.240 --> 00:46:26.750 align:middle line:84% and that this function here defines 00:46:26.750 --> 00:46:29.510 align:middle line:90% a norm on this space little lp. 00:46:29.510 --> 00:46:31.430 align:middle line:90% So just accept that. 00:46:31.430 --> 00:46:38.360 align:middle line:90% 00:46:38.360 --> 00:46:42.800 align:middle line:84% And so now, we're not interested-- 00:46:42.800 --> 00:46:46.390 align:middle line:90% so we've gone a little bit-- 00:46:46.390 --> 00:46:49.750 align:middle line:84% narrowed down our spaces we're interested 00:46:49.750 --> 00:46:50.810 align:middle line:90% in a little bit more. 00:46:50.810 --> 00:46:54.790 align:middle line:84% So we've gone from general infinite dimensional vector 00:46:54.790 --> 00:46:57.970 align:middle line:84% spaces now to norm spaces being our interest, 00:46:57.970 --> 00:46:59.110 align:middle line:90% what we're interested in. 00:46:59.110 --> 00:47:05.224 align:middle line:84% But we're not exactly interested in just any norm vector space. 00:47:05.224 --> 00:47:09.980 align:middle line:84% The central objects and functional analysis 00:47:09.980 --> 00:47:18.400 align:middle line:84% that we're interested in are the analogs of Rn. 00:47:18.400 --> 00:47:20.740 align:middle line:84% Now, what property that are in and Cn 00:47:20.740 --> 00:47:23.800 align:middle line:90% have is that their metric-- 00:47:23.800 --> 00:47:30.100 align:middle line:84% this metric that you have on these sets is complete. 00:47:30.100 --> 00:47:32.330 align:middle line:84% The Cauchy sequences always converge. 00:47:32.330 --> 00:47:36.040 align:middle line:84% And this is our next narrowing down of the spaces 00:47:36.040 --> 00:47:37.160 align:middle line:90% we're interested in. 00:47:37.160 --> 00:47:44.960 align:middle line:84% And these have a special name, so so-called Banach spaces 00:47:44.960 --> 00:47:46.100 align:middle line:90% after Stefan Banach. 00:47:46.100 --> 00:47:50.320 align:middle line:90% 00:47:50.320 --> 00:47:58.490 align:middle line:84% So norm space, so a vector space with a norm, 00:47:58.490 --> 00:48:16.430 align:middle line:84% is a Banach space if is complete with respect 00:48:16.430 --> 00:48:24.980 align:middle line:84% to the metric induced by the norm. 00:48:24.980 --> 00:48:32.080 align:middle line:90% 00:48:32.080 --> 00:48:36.370 align:middle line:84% So if we have a norm space, we have a metric 00:48:36.370 --> 00:48:39.160 align:middle line:84% that's associated to that by defining 00:48:39.160 --> 00:48:47.000 align:middle line:84% the distance between two guys by that equation up there. 00:48:47.000 --> 00:48:49.240 align:middle line:84% And so we say it's a Banach space 00:48:49.240 --> 00:48:55.570 align:middle line:84% if that metric is complete, meaning Cauchy sequences 00:48:55.570 --> 00:49:00.190 align:middle line:84% in that space with respect to this metric converge 00:49:00.190 --> 00:49:00.880 align:middle line:90% in the space. 00:49:00.880 --> 00:49:07.990 align:middle line:84% Now, in first year analysis, you learn 00:49:07.990 --> 00:49:10.775 align:middle line:84% that instead of rational numbers, they are not complete. 00:49:10.775 --> 00:49:12.400 align:middle line:84% There are Cauchy sequences which do not 00:49:12.400 --> 00:49:15.650 align:middle line:90% converge to a rational number. 00:49:15.650 --> 00:49:18.730 align:middle line:84% Every real number you can write as a limit 00:49:18.730 --> 00:49:22.000 align:middle line:90% as finitely many decimals. 00:49:22.000 --> 00:49:27.010 align:middle line:84% So square root of 2 will be a number 00:49:27.010 --> 00:49:29.890 align:middle line:84% where you can form a Cauchy sequence of rational numbers 00:49:29.890 --> 00:49:31.070 align:middle line:90% converging to it. 00:49:31.070 --> 00:49:35.110 align:middle line:84% But it's not a rational number, so the rational numbers 00:49:35.110 --> 00:49:36.640 align:middle line:90% are not complete. 00:49:36.640 --> 00:49:39.720 align:middle line:90% But R is complete. 00:49:39.720 --> 00:49:42.570 align:middle line:84% And in general, we give a name to those norm linear spaces-- 00:49:42.570 --> 00:49:48.810 align:middle line:84% those norm spaces such that this metric is complete. 00:49:48.810 --> 00:49:50.205 align:middle line:90% We call those Banach spaces. 00:49:50.205 --> 00:49:57.710 align:middle line:90% 00:49:57.710 --> 00:50:01.910 align:middle line:84% So examples which you saw, at least for the Euclidean norm 00:50:01.910 --> 00:50:06.150 align:middle line:84% and should be able to prove on your own, 00:50:06.150 --> 00:50:11.820 align:middle line:84% assuming the triangle inequality for the little lp norms, 00:50:11.820 --> 00:50:28.425 align:middle line:84% these are complete with respect to any of the little lp norms. 00:50:28.425 --> 00:50:38.600 align:middle line:90% 00:50:38.600 --> 00:50:43.380 align:middle line:84% All right, now, let's do a non-trivial one. 00:50:43.380 --> 00:50:50.360 align:middle line:84% Let's show that the space of bounded continuous functions 00:50:50.360 --> 00:50:53.345 align:middle line:90% on a metric space is complete. 00:50:53.345 --> 00:51:09.190 align:middle line:90% 00:51:09.190 --> 00:51:11.470 align:middle line:90% So let's make this a theorem. 00:51:11.470 --> 00:51:19.370 align:middle line:90% 00:51:19.370 --> 00:51:48.120 align:middle line:84% If x is a complete geometric space, 00:51:48.120 --> 00:51:56.770 align:middle line:90% then it is a Banach space. 00:51:56.770 --> 00:51:59.530 align:middle line:84% BSP, that's my abbreviation for Banach space. 00:51:59.530 --> 00:52:06.340 align:middle line:90% 00:52:06.340 --> 00:52:10.700 align:middle line:84% All right, so for a complete metric space, 00:52:10.700 --> 00:52:13.820 align:middle line:84% the space of bounded continuous functions on x 00:52:13.820 --> 00:52:15.920 align:middle line:90% is a Banach space. 00:52:15.920 --> 00:52:18.710 align:middle line:84% We've already shown that that function over there 00:52:18.710 --> 00:52:19.760 align:middle line:90% defines a norm on it. 00:52:19.760 --> 00:52:23.600 align:middle line:84% So I'm saying it's complete with respect to that norm. 00:52:23.600 --> 00:52:28.620 align:middle line:84% Cauchy sequences always converge in this space. 00:52:28.620 --> 00:52:29.870 align:middle line:90% So let me just write this out. 00:52:29.870 --> 00:52:50.360 align:middle line:84% We want to show that every Cauchy sequence un in the space 00:52:50.360 --> 00:52:54.740 align:middle line:84% C bounded continuous functions has 00:52:54.740 --> 00:53:04.080 align:middle line:84% a limit in the space of bounded continuous functions. 00:53:04.080 --> 00:53:07.600 align:middle line:90% 00:53:07.600 --> 00:53:11.300 align:middle line:84% All right, so let's take a Cauchy sequence. 00:53:11.300 --> 00:53:14.570 align:middle line:84% And the way this proof is going to work 00:53:14.570 --> 00:53:18.530 align:middle line:84% is the way essentially really all of the proofs of showing 00:53:18.530 --> 00:53:22.430 align:middle line:84% something is a Banach space works, 00:53:22.430 --> 00:53:25.740 align:middle line:90% is you take a Cauchy sequence. 00:53:25.740 --> 00:53:29.730 align:middle line:84% Then you'll be able to come up with a candidate for the limit. 00:53:29.730 --> 00:53:32.160 align:middle line:84% And then your job is to show two things-- 00:53:32.160 --> 00:53:36.800 align:middle line:84% that that candidate is in the space itself, 00:53:36.800 --> 00:53:40.670 align:middle line:84% and then that the convergence happens. 00:53:40.670 --> 00:53:42.980 align:middle line:84% And sometimes, those two come together 00:53:42.980 --> 00:53:44.455 align:middle line:90% or can be done at the same time. 00:53:44.455 --> 00:53:48.955 align:middle line:90% 00:53:48.955 --> 00:53:50.080 align:middle line:90% And you'll see what I mean. 00:53:50.080 --> 00:54:01.520 align:middle line:84% Let un be a Cauchy sequence in C infinity, 00:54:01.520 --> 00:54:05.090 align:middle line:84% in the space of bounded continuous functions. 00:54:05.090 --> 00:54:08.720 align:middle line:84% First off, I claim that this forms a bounded sequence 00:54:08.720 --> 00:54:11.190 align:middle line:90% in this space. 00:54:11.190 --> 00:54:15.743 align:middle line:84% This is a fact from metric space theory, 00:54:15.743 --> 00:54:16.910 align:middle line:90% but I'll write it out again. 00:54:16.910 --> 00:54:20.360 align:middle line:90% 00:54:20.360 --> 00:54:27.740 align:middle line:84% Then there exists natural number N0 00:54:27.740 --> 00:54:30.980 align:middle line:84% such that for all n bigger than or equal to 00:54:30.980 --> 00:54:38.470 align:middle line:84% N0, the difference between u N0 and u-- 00:54:38.470 --> 00:54:47.360 align:middle line:84% so let me say for all nm bigger than or equal to N0, un 00:54:47.360 --> 00:54:51.995 align:middle line:84% minus um in the infinity norm is less than 1. 00:54:51.995 --> 00:54:55.310 align:middle line:90% 00:54:55.310 --> 00:54:56.940 align:middle line:90% So what I'm first going to do-- 00:54:56.940 --> 00:54:59.900 align:middle line:84% I'm first going to show that this sequence of functions 00:54:59.900 --> 00:55:02.840 align:middle line:90% are bounded in this space. 00:55:02.840 --> 00:55:05.000 align:middle line:84% Each of them are, of course, bounded. 00:55:05.000 --> 00:55:07.925 align:middle line:84% But I'm saying that they form a bounded sequence in the space. 00:55:07.925 --> 00:55:14.100 align:middle line:90% 00:55:14.100 --> 00:55:19.860 align:middle line:90% Then let's see, rewrite this. 00:55:19.860 --> 00:55:23.660 align:middle line:90% 00:55:23.660 --> 00:55:28.280 align:middle line:84% For all n bigger than or equal to N0, un, the norm, 00:55:28.280 --> 00:55:36.260 align:middle line:84% this is less than or equal to un minus u N0 plus u N0. 00:55:36.260 --> 00:55:43.330 align:middle line:84% And this is less than 1 plus u N0. 00:55:43.330 --> 00:55:47.320 align:middle line:84% So for all n bigger than or equal to this fixed number N0, 00:55:47.320 --> 00:55:53.370 align:middle line:84% un and l infinity norm are bounded 00:55:53.370 --> 00:55:57.330 align:middle line:84% by 1 plus the norm of this fixed guy. 00:55:57.330 --> 00:56:04.850 align:middle line:84% Then for all n, a natural number, 00:56:04.850 --> 00:56:09.440 align:middle line:84% the norm of un is less than or equal to-- if I put u1-- 00:56:09.440 --> 00:56:21.330 align:middle line:90% 00:56:21.330 --> 00:56:24.000 align:middle line:84% now, if n is bigger than or equal to N0, 00:56:24.000 --> 00:56:26.340 align:middle line:84% then this is less than or equal to this guy. 00:56:26.340 --> 00:56:27.840 align:middle line:84% So it's certainly less than or equal 00:56:27.840 --> 00:56:31.680 align:middle line:84% to some non-negative things plus this guy. 00:56:31.680 --> 00:56:35.670 align:middle line:84% Meanwhile, if n is less than N0, than the norm of it 00:56:35.670 --> 00:56:38.590 align:middle line:84% is less than or equal to one of these norms that appear here, 00:56:38.590 --> 00:56:41.430 align:middle line:84% which is also, again, less than or equal to this entire number 00:56:41.430 --> 00:56:43.650 align:middle line:90% that I wrote down here. 00:56:43.650 --> 00:56:47.340 align:middle line:84% And let me define that to be this number B. 00:56:47.340 --> 00:56:53.330 align:middle line:84% So I've shown for all natural numbers n, the norm of un 00:56:53.330 --> 00:56:57.560 align:middle line:84% is bounded by B. So this forms a bounded sequence 00:56:57.560 --> 00:57:01.340 align:middle line:84% in the space of bounded continuous functions. 00:57:01.340 --> 00:57:06.240 align:middle line:84% You have to keep track of where this bounded-- 00:57:06.240 --> 00:57:08.880 align:middle line:84% where I'm saying this bounded is taking place. 00:57:08.880 --> 00:57:14.160 align:middle line:84% Each of these functions is bounded, is a bounded function. 00:57:14.160 --> 00:57:17.580 align:middle line:84% What I'm saying here is that they form a bounded sequence 00:57:17.580 --> 00:57:18.645 align:middle line:90% in this space. 00:57:18.645 --> 00:57:27.210 align:middle line:90% 00:57:27.210 --> 00:57:36.046 align:middle line:84% OK, so let me make a note of this. 00:57:36.046 --> 00:57:37.620 align:middle line:90% This is bounded by this. 00:57:37.620 --> 00:57:41.990 align:middle line:90% 00:57:41.990 --> 00:57:51.780 align:middle line:90% All right, now, what do we know? 00:57:51.780 --> 00:57:54.330 align:middle line:84% And let me-- in fact, I should have written this out 00:57:54.330 --> 00:57:56.742 align:middle line:84% over here, but let me write it out again. 00:57:56.742 --> 00:57:58.200 align:middle line:84% What does it mean for this sequence 00:57:58.200 --> 00:58:00.150 align:middle line:90% to be Cauchy in this space? 00:58:00.150 --> 00:58:03.550 align:middle line:84% This means for all epsilon positive, 00:58:03.550 --> 00:58:07.650 align:middle line:84% there exist a natural number n such 00:58:07.650 --> 00:58:13.980 align:middle line:84% that for all nm bigger than or equal to N un 00:58:13.980 --> 00:58:19.815 align:middle line:84% minus um in this uniform norm is less than epsilon. 00:58:19.815 --> 00:58:27.860 align:middle line:90% 00:58:27.860 --> 00:58:37.810 align:middle line:84% Now, since for all x I have that un of x 00:58:37.810 --> 00:58:41.260 align:middle line:84% minus um of x in absolute value is 00:58:41.260 --> 00:58:44.890 align:middle line:84% less than or equal to the sup over all x's in capital X, 00:58:44.890 --> 00:58:54.920 align:middle line:84% which is just the uniform norm, the l infinity 00:58:54.920 --> 00:59:01.220 align:middle line:84% norm on this space, and I am assuming it's Cauchy, 00:59:01.220 --> 00:59:05.240 align:middle line:84% so it satisfies this property, I get 00:59:05.240 --> 00:59:13.250 align:middle line:84% that for all x in X, the sequence now of complex numbers 00:59:13.250 --> 00:59:14.300 align:middle line:90% un of x-- 00:59:14.300 --> 00:59:18.890 align:middle line:90% 00:59:18.890 --> 00:59:21.080 align:middle line:84% so this is now just a sequence of complex numbers. 00:59:21.080 --> 00:59:22.310 align:middle line:90% It's the value. 00:59:22.310 --> 00:59:23.090 align:middle line:90% I take x. 00:59:23.090 --> 00:59:26.070 align:middle line:90% I stick it into u sub n. 00:59:26.070 --> 00:59:27.870 align:middle line:84% I have now a sequence of complex numbers. 00:59:27.870 --> 00:59:36.040 align:middle line:90% This is the Cauchy sequence. 00:59:36.040 --> 00:59:39.950 align:middle line:84% All of this may seem a little bit weird at first. 00:59:39.950 --> 00:59:45.050 align:middle line:84% But after I finish this proof, do it again now for little l 00:59:45.050 --> 00:59:47.235 align:middle line:90% infinity. 00:59:47.235 --> 00:59:48.610 align:middle line:84% And you'll start getting the hang 00:59:48.610 --> 00:59:50.740 align:middle line:90% and seeing how the arguments go. 00:59:50.740 --> 00:59:53.470 align:middle line:90% 00:59:53.470 --> 01:00:05.750 align:middle line:84% All right, so for every x in capital X, un of x-- 01:00:05.750 --> 01:00:08.980 align:middle line:90% 01:00:08.980 --> 01:00:11.830 align:middle line:84% so that's now a sequence of complex numbers. 01:00:11.830 --> 01:00:15.770 align:middle line:90% 01:00:15.770 --> 01:00:19.020 align:middle line:84% And yeah, I thought I didn't need it to be complete. 01:00:19.020 --> 01:00:21.630 align:middle line:84% I don't know why I wrote it down. 01:00:21.630 --> 01:00:25.090 align:middle line:84% Any metric space, if x is a metric space then, OK? 01:00:25.090 --> 01:00:36.880 align:middle line:90% 01:00:36.880 --> 01:00:40.510 align:middle line:84% All right, so for each x in the metric space, 01:00:40.510 --> 01:00:44.980 align:middle line:84% un of x, which is now forming a sequence of complex numbers, 01:00:44.980 --> 01:00:47.140 align:middle line:90% this is a Cauchy sequence. 01:00:47.140 --> 01:00:49.090 align:middle line:84% But the space of complex numbers, this 01:00:49.090 --> 01:00:53.830 align:middle line:90% is a complete metric space. 01:00:53.830 --> 01:00:58.510 align:middle line:84% And therefore, for each x, this sequence has a limit. 01:00:58.510 --> 01:01:03.150 align:middle line:90% 01:01:03.150 --> 01:01:15.900 align:middle line:84% So by completeness of C for all x 01:01:15.900 --> 01:01:32.100 align:middle line:84% in capital X, this sequence, un of x, has a limit in C. 01:01:32.100 --> 01:01:40.020 align:middle line:84% And now, I define what will be my candidate function, 01:01:40.020 --> 01:01:48.190 align:middle line:84% u of x, to be this pointwise limit, 01:01:48.190 --> 01:01:51.250 align:middle line:84% limit as n goes to infinity of u sub n of x. 01:01:51.250 --> 01:01:54.530 align:middle line:90% 01:01:54.530 --> 01:01:57.750 align:middle line:84% So in fact, in a few words, if you 01:01:57.750 --> 01:01:59.940 align:middle line:84% remember what these words mean, what we've shown 01:01:59.940 --> 01:02:02.490 align:middle line:84% is that every Cauchy sequence in the space 01:02:02.490 --> 01:02:04.980 align:middle line:84% of bounded continuous functions has a pointwise limit. 01:02:04.980 --> 01:02:16.120 align:middle line:90% 01:02:16.120 --> 01:02:19.000 align:middle line:84% OK, then now what we're going to show 01:02:19.000 --> 01:02:22.060 align:middle line:84% is that this guy is, in fact, in the space 01:02:22.060 --> 01:02:25.450 align:middle line:84% of bounded continuous functions and that we have convergence 01:02:25.450 --> 01:02:31.960 align:middle line:84% of this sequence of functions to u in the space C infinity x, 01:02:31.960 --> 01:02:34.030 align:middle line:84% in the space of bounded continuous functions. 01:02:34.030 --> 01:02:38.065 align:middle line:84% We've only defined this guy as the pointwise limit-- 01:02:38.065 --> 01:02:40.730 align:middle line:90% 01:02:40.730 --> 01:02:46.678 align:middle line:84% so as the point wise limit of these guys, OK? 01:02:46.678 --> 01:02:53.510 align:middle line:90% 01:02:53.510 --> 01:02:57.740 align:middle line:84% Then for all x in capital X, the absolute value of u of x 01:02:57.740 --> 01:03:00.932 align:middle line:84% is equal to the limit as n goes to infinity, 01:03:00.932 --> 01:03:02.390 align:middle line:84% so the absolute value of the limit. 01:03:02.390 --> 01:03:03.765 align:middle line:84% But since this converges, this is 01:03:03.765 --> 01:03:06.800 align:middle line:84% equal to the limit of the absolute values. 01:03:06.800 --> 01:03:10.370 align:middle line:84% And each of these guys are bounded 01:03:10.370 --> 01:03:17.640 align:middle line:84% by B. They're bounded by the infinity norm, which 01:03:17.640 --> 01:03:23.940 align:middle line:84% is bounded by B. So this is less than or equal to B. 01:03:23.940 --> 01:03:34.070 align:middle line:84% And thus, so this is a bounded function at least 01:03:34.070 --> 01:03:35.480 align:middle line:90% on this metric space. 01:03:35.480 --> 01:03:40.980 align:middle line:90% 01:03:40.980 --> 01:03:44.460 align:middle line:84% So now, we're going to achieve two things at once. 01:03:44.460 --> 01:03:46.260 align:middle line:84% We're going to show that it's continuous 01:03:46.260 --> 01:03:51.240 align:middle line:84% and that we have convergence in this space by showing it is-- 01:03:51.240 --> 01:03:55.930 align:middle line:90% 01:03:55.930 --> 01:03:59.560 align:middle line:84% by showing this l infinity norm of the difference 01:03:59.560 --> 01:04:02.050 align:middle line:90% goes to 0 of u minus un. 01:04:02.050 --> 01:04:19.290 align:middle line:90% 01:04:19.290 --> 01:04:29.980 align:middle line:84% OK, so first, we're going to show that-- 01:04:29.980 --> 01:04:34.840 align:middle line:84% so think of this just now as a function, 01:04:34.840 --> 01:04:37.910 align:middle line:90% not necessarily a norm. 01:04:37.910 --> 01:04:39.698 align:middle line:90% OK, maybe I'm being too careful. 01:04:39.698 --> 01:04:41.240 align:middle line:84% I'm just going to say now we're going 01:04:41.240 --> 01:04:43.010 align:middle line:84% to show that this quantity here goes 01:04:43.010 --> 01:04:52.070 align:middle line:90% to 0 as n goes to infinity. 01:04:52.070 --> 01:04:53.030 align:middle line:90% And how do we do this? 01:04:53.030 --> 01:05:00.170 align:middle line:84% Well, since we're just going to do this 01:05:00.170 --> 01:05:04.910 align:middle line:84% the old fashioned way of let epsilon be positive and show 01:05:04.910 --> 01:05:07.820 align:middle line:84% that we can choose an n such that this is less than 01:05:07.820 --> 01:05:10.190 align:middle line:84% or equal to epsilon, I know in the definition 01:05:10.190 --> 01:05:11.940 align:middle line:84% you're supposed to have less than epsilon. 01:05:11.940 --> 01:05:15.050 align:middle line:84% But less than or equal to is good enough. 01:05:15.050 --> 01:05:20.470 align:middle line:90% OK, so let epsilon be positive. 01:05:20.470 --> 01:05:28.860 align:middle line:84% Since the sequence is Cauchy in this space, 01:05:28.860 --> 01:05:35.820 align:middle line:84% this implies there exists a natural number N such that 01:05:35.820 --> 01:05:39.120 align:middle line:84% for all nm bigger than or equal to N, 01:05:39.120 --> 01:05:44.818 align:middle line:84% I have that un minus um l infinity is less than-- 01:05:44.818 --> 01:05:46.485 align:middle line:84% all right, let's make it epsilon over 2. 01:05:46.485 --> 01:05:54.200 align:middle line:90% 01:05:54.200 --> 01:05:58.580 align:middle line:84% Then for all nm bigger than or equal to N, 01:05:58.580 --> 01:06:00.150 align:middle line:90% This means un of x-- 01:06:00.150 --> 01:06:03.490 align:middle line:90% 01:06:03.490 --> 01:06:10.760 align:middle line:90% so let x be in capital X. 01:06:10.760 --> 01:06:12.950 align:middle line:84% We now want to show that for all little n bigger 01:06:12.950 --> 01:06:18.800 align:middle line:84% than or equal to capital N, in fact, un minus u in this norm 01:06:18.800 --> 01:06:20.780 align:middle line:84% is less than or equal to epsilon over 2. 01:06:20.780 --> 01:06:24.380 align:middle line:90% 01:06:24.380 --> 01:06:28.130 align:middle line:84% For all nm bigger than or equal to N, 01:06:28.130 --> 01:06:38.166 align:middle line:84% I have that u, which is less than or equal to un minus um-- 01:06:38.166 --> 01:06:41.638 align:middle line:90% 01:06:41.638 --> 01:06:47.570 align:middle line:84% and therefore, if I take the limit as m goes to infinity, 01:06:47.570 --> 01:06:50.030 align:middle line:84% remember, u of x is the pointwise limit. 01:06:50.030 --> 01:06:55.190 align:middle line:90% I fixed x in capital X. 01:06:55.190 --> 01:07:00.970 align:middle line:84% I get that for all n bigger than or equal to m, 01:07:00.970 --> 01:07:07.120 align:middle line:84% un of x minus u of x is less than or equal 01:07:07.120 --> 01:07:09.920 align:middle line:90% to epsilon over 2. 01:07:09.920 --> 01:07:10.955 align:middle line:90% So what have I shown? 01:07:10.955 --> 01:07:15.610 align:middle line:84% I've shown for all n bigger than or equal to capital 01:07:15.610 --> 01:07:18.520 align:middle line:84% N-- and this capital N came from this condition, not anything 01:07:18.520 --> 01:07:19.840 align:middle line:90% having to do with x. 01:07:19.840 --> 01:07:21.430 align:middle line:90% It came from this condition-- 01:07:21.430 --> 01:07:24.050 align:middle line:90% 01:07:24.050 --> 01:07:27.230 align:middle line:84% I have un of x minus u of x is less than 01:07:27.230 --> 01:07:29.240 align:middle line:90% or equal to epsilon over 2. 01:07:29.240 --> 01:07:32.750 align:middle line:84% And therefore, the sup over all x's is less than or equal 01:07:32.750 --> 01:07:40.040 align:middle line:84% to epsilon over 2, which is less than epsilon. 01:07:40.040 --> 01:07:42.770 align:middle line:90% 01:07:42.770 --> 01:07:46.640 align:middle line:84% Thus, un converges to u-- or I should 01:07:46.640 --> 01:07:57.430 align:middle line:90% say as n goes to infinity. 01:07:57.430 --> 01:07:59.580 align:middle line:90% Now, what's the last step? 01:07:59.580 --> 01:08:02.370 align:middle line:84% I have this candidate, u. un's converged 01:08:02.370 --> 01:08:08.140 align:middle line:90% to u with respect to this sense. 01:08:08.140 --> 01:08:11.830 align:middle line:84% I need to conclude also that u is 01:08:11.830 --> 01:08:15.760 align:middle line:84% an element of the space of bounded continuous functions. 01:08:15.760 --> 01:08:17.020 align:middle line:90% I know it's bounded. 01:08:17.020 --> 01:08:18.790 align:middle line:90% Why is it continuous? 01:08:18.790 --> 01:08:34.979 align:middle line:84% Well, since un minus u goes to 0, this implies that un-- 01:08:34.979 --> 01:08:37.694 align:middle line:84% by what we remarked a little bit earlier, 01:08:37.694 --> 01:08:45.700 align:middle line:84% it means un converges to u uniformly on x. 01:08:45.700 --> 01:08:48.770 align:middle line:90% 01:08:48.770 --> 01:08:53.359 align:middle line:84% And since u is the uniform limit of a sequence 01:08:53.359 --> 01:09:03.770 align:middle line:84% of continuous functions, that implies that u is continuous. 01:09:03.770 --> 01:09:06.859 align:middle line:90% 01:09:06.859 --> 01:09:12.970 align:middle line:84% So let me just reiterate what we've done. 01:09:12.970 --> 01:09:18.529 align:middle line:84% In sequence, we've shown that ux is bounded. 01:09:18.529 --> 01:09:26.120 align:middle line:84% We've then shown that the sup over all x in X-- 01:09:26.120 --> 01:09:30.109 align:middle line:84% so I should have put that in yellow. 01:09:30.109 --> 01:09:34.899 align:middle line:90% 01:09:34.899 --> 01:09:39.710 align:middle line:84% We've shown convergence to u with respect to this norm. 01:09:39.710 --> 01:09:43.240 align:middle line:84% And we've shown u is, in fact, in the space. 01:09:43.240 --> 01:09:48.110 align:middle line:90% So therefore, u is in-- 01:09:48.110 --> 01:10:06.660 align:middle line:90% 01:10:06.660 --> 01:10:16.205 align:middle line:84% i.e., the space is complete, and therefore a Banach space. 01:10:16.205 --> 01:10:29.140 align:middle line:90% 01:10:29.140 --> 01:10:32.380 align:middle line:84% So the first time you see that proof-- 01:10:32.380 --> 01:10:36.430 align:middle line:84% and again, this is kind of how all the proofs of something 01:10:36.430 --> 01:10:39.640 align:middle line:90% being a Banach space goes. 01:10:39.640 --> 01:10:42.685 align:middle line:84% When you see that for the first time, it's a little weird. 01:10:42.685 --> 01:10:47.530 align:middle line:90% 01:10:47.530 --> 01:10:49.510 align:middle line:84% And this will be in the exercise so 01:10:49.510 --> 01:10:51.010 align:middle line:84% that you can get a jumpstart on that 01:10:51.010 --> 01:10:52.990 align:middle line:84% by looking at maybe the simplest one. 01:10:52.990 --> 01:10:56.840 align:middle line:90% 01:10:56.840 --> 01:10:58.400 align:middle line:84% Well, I still have space over here. 01:10:58.400 --> 01:11:06.820 align:middle line:90% 01:11:06.820 --> 01:11:18.530 align:middle line:84% So little lp, this is a Banach space 01:11:18.530 --> 01:11:25.880 align:middle line:84% for all p between 1 and infinity. 01:11:25.880 --> 01:11:29.050 align:middle line:90% Another space, which-- 01:11:29.050 --> 01:11:33.465 align:middle line:84% OK, maybe try your hand at this one 01:11:33.465 --> 01:11:35.840 align:middle line:84% instead of little l infinity because at least something's 01:11:35.840 --> 01:11:38.550 align:middle line:90% a little different. 01:11:38.550 --> 01:11:44.730 align:middle line:84% Little c0, which is the set of sequences in little l 01:11:44.730 --> 01:11:47.730 align:middle line:84% infinity-- so each element of c0 is 01:11:47.730 --> 01:11:57.910 align:middle line:84% a sequence, a bounded sequence such that limit j goes 01:11:57.910 --> 01:12:00.430 align:middle line:90% to infinity of aj equals 0. 01:12:00.430 --> 01:12:03.290 align:middle line:90% 01:12:03.290 --> 01:12:04.855 align:middle line:90% This is also a Banach space. 01:12:04.855 --> 01:12:08.300 align:middle line:90% 01:12:08.300 --> 01:12:15.210 align:middle line:84% So I encourage you to take this example of bounded sequences 01:12:15.210 --> 01:12:16.950 align:middle line:90% that converge to 0. 01:12:16.950 --> 01:12:22.500 align:middle line:84% First off, it's pretty clear it's a vector space. 01:12:22.500 --> 01:12:24.030 align:middle line:84% It's actually a subspace-- and I'll 01:12:24.030 --> 01:12:28.515 align:middle line:84% get to subspaces of Banach spaces next lecture. 01:12:28.515 --> 01:12:31.850 align:middle line:90% 01:12:31.850 --> 01:12:37.330 align:middle line:84% So it's a Banach space with the l infinity norm. 01:12:37.330 --> 01:12:52.790 align:middle line:90% 01:12:52.790 --> 01:12:57.260 align:middle line:84% So again, how would you prove that this is a Banach space? 01:12:57.260 --> 01:12:59.360 align:middle line:90% You would take a Cauchy. 01:12:59.360 --> 01:13:00.815 align:middle line:84% So you have to start thinking of-- 01:13:00.815 --> 01:13:07.160 align:middle line:90% 01:13:07.160 --> 01:13:12.430 align:middle line:84% again, these Banach spaces, these spaces you're looking at 01:13:12.430 --> 01:13:16.930 align:middle line:84% can be made up of complicated things. 01:13:16.930 --> 01:13:19.300 align:middle line:84% Little lp, this is a space of sequences. 01:13:19.300 --> 01:13:23.155 align:middle line:84% So each point in the space is a sequence. 01:13:23.155 --> 01:13:24.280 align:middle line:90% It's a sequence of numbers. 01:13:24.280 --> 01:13:27.800 align:middle line:90% 01:13:27.800 --> 01:13:32.120 align:middle line:84% And here, so your sequence of points 01:13:32.120 --> 01:13:37.310 align:middle line:84% is a sequence of sequences, just like in the example 01:13:37.310 --> 01:13:39.530 align:middle line:84% we did here, your sequence of points 01:13:39.530 --> 01:13:42.350 align:middle line:84% in your space of bounded continuous functions 01:13:42.350 --> 01:13:43.790 align:middle line:90% is a sequence of functions. 01:13:43.790 --> 01:13:46.100 align:middle line:84% Here, we have a sequence of sequences, 01:13:46.100 --> 01:13:48.360 align:middle line:84% which again a sequence is just a function, 01:13:48.360 --> 01:13:49.760 align:middle line:90% so you shouldn't be too scared. 01:13:49.760 --> 01:13:53.870 align:middle line:84% But try your hand at showing that this is a bonus space 01:13:53.870 --> 01:13:59.930 align:middle line:84% just from what we've done so far, following 01:13:59.930 --> 01:14:01.430 align:middle line:90% this kind of blueprint. 01:14:01.430 --> 01:14:03.725 align:middle line:84% And again, it'll be kind of the same. 01:14:03.725 --> 01:14:07.665 align:middle line:84% You first show that a Cauchy sequence in here is, in fact, 01:14:07.665 --> 01:14:08.165 align:middle line:90% bounded. 01:14:08.165 --> 01:14:11.610 align:middle line:90% 01:14:11.610 --> 01:14:18.100 align:middle line:84% Then show that pointwise each of the-- 01:14:18.100 --> 01:14:21.760 align:middle line:84% and here, pointwise should mean each of the entries 01:14:21.760 --> 01:14:24.270 align:middle line:90% of your sequence of sequences-- 01:14:24.270 --> 01:14:27.190 align:middle line:90% 01:14:27.190 --> 01:14:29.620 align:middle line:90% forms a Cauchy sequence. 01:14:29.620 --> 01:14:32.170 align:middle line:84% Then that allows you to obtain a candidate 01:14:32.170 --> 01:14:35.770 align:middle line:84% sequence as the limit of your sequence of sequences, 01:14:35.770 --> 01:14:36.950 align:middle line:90% all right? 01:14:36.950 --> 01:14:40.070 align:middle line:84% And then you have to show, again through this argument, 01:14:40.070 --> 01:14:48.636 align:middle line:84% that you do have convergence with respect to this norm, 01:14:48.636 --> 01:14:53.770 align:middle line:84% that that sequence is bound it, and, in fact, 01:14:53.770 --> 01:14:56.800 align:middle line:84% that sequence satisfies this condition in order 01:14:56.800 --> 01:14:57.655 align:middle line:90% to be in this space. 01:14:57.655 --> 01:15:08.900 align:middle line:90% 01:15:08.900 --> 01:15:11.950 align:middle line:90% OK, so I think I'll stop there. 01:15:11.950 --> 01:15:19.000 align:middle line:90%