WEBVTT 00:00:00.000 --> 00:00:18.740 align:middle line:84% [SQUEAKING] [RUSTLING] [CLICKING] 00:00:18.740 --> 00:00:29.020 align:middle line:84% PROFESSOR: OK, so let's continue our discussion of normed spaces 00:00:29.020 --> 00:00:30.010 align:middle line:90% and Banach spaces. 00:00:30.010 --> 00:00:33.940 align:middle line:84% So last time, I introduced the space 00:00:33.940 --> 00:00:38.500 align:middle line:84% of bounded linear operators from one normed space from another. 00:00:38.500 --> 00:00:42.200 align:middle line:84% And we saw that when the target is a Banach space, 00:00:42.200 --> 00:00:44.680 align:middle line:84% then this space of bounded linear operators 00:00:44.680 --> 00:00:45.865 align:middle line:90% is also a Banach space. 00:00:45.865 --> 00:00:49.420 align:middle line:90% 00:00:49.420 --> 00:00:55.290 align:middle line:84% So now, we'll look at another example of a way 00:00:55.290 --> 00:00:59.430 align:middle line:84% to obtain a normed space from a given normed space. 00:00:59.430 --> 00:01:17.265 align:middle line:84% So I'll be talking about subspaces and quotients, 00:01:17.265 --> 00:01:17.765 align:middle line:90% all right? 00:01:17.765 --> 00:01:20.350 align:middle line:90% 00:01:20.350 --> 00:01:25.720 align:middle line:84% So let me just recall briefly what a subspace is. 00:01:25.720 --> 00:01:27.610 align:middle line:84% You should remember this from linear algebra. 00:01:27.610 --> 00:01:31.510 align:middle line:84% V will always be at least a vector 00:01:31.510 --> 00:01:36.380 align:middle line:90% space, usually a normed space. 00:01:36.380 --> 00:01:39.050 align:middle line:90% So we say a subset-- 00:01:39.050 --> 00:01:40.730 align:middle line:84% and I've been using this notation. 00:01:40.730 --> 00:01:41.750 align:middle line:90% It doesn't mean strict-- 00:01:41.750 --> 00:01:44.380 align:middle line:90% 00:01:44.380 --> 00:01:45.490 align:middle line:90% a strict subset. 00:01:45.490 --> 00:01:49.510 align:middle line:84% I'm just kind of used to using this notation from 18.100A. 00:01:49.510 --> 00:01:51.910 align:middle line:84% And the textbook use this notation 00:01:51.910 --> 00:01:56.080 align:middle line:84% for just being a subset, not necessarily a strict subset. 00:01:56.080 --> 00:01:59.820 align:middle line:90% So quick note about that. 00:01:59.820 --> 00:02:14.760 align:middle line:84% So a subset of V is a subspace if for all elements, two 00:02:14.760 --> 00:02:23.420 align:middle line:84% elements in the field of scalars and elements in W, 00:02:23.420 --> 00:02:30.280 align:middle line:84% their linear combination is also in W. So W is closed, 00:02:30.280 --> 00:02:32.620 align:middle line:90% undertaking linear combinations. 00:02:32.620 --> 00:02:38.650 align:middle line:90% 00:02:38.650 --> 00:02:43.020 align:middle line:84% And so it's quite easy to prove-- 00:02:43.020 --> 00:02:53.940 align:middle line:84% and I'll leave this to you-- that the subspace of a normed 00:02:53.940 --> 00:02:54.870 align:middle line:90% space-- 00:02:54.870 --> 00:03:02.460 align:middle line:84% of a Banach space, sorry, is itself 00:03:02.460 --> 00:03:06.480 align:middle line:84% a Banach space, meaning it's complete with respect 00:03:06.480 --> 00:03:13.830 align:middle line:84% to the norm it inherits from V. It's a Banach space 00:03:13.830 --> 00:03:26.080 align:middle line:84% if and only if it's a closed subset of V with respect 00:03:26.080 --> 00:03:29.395 align:middle line:84% to this metric we have induced by the norm. 00:03:29.395 --> 00:03:35.970 align:middle line:90% 00:03:35.970 --> 00:03:40.590 align:middle line:84% I mean, you can let me talk our way through it real quick. 00:03:40.590 --> 00:03:43.470 align:middle line:84% Assuming W is a Banach space, we want 00:03:43.470 --> 00:03:47.400 align:middle line:84% to show W is also a closed subset of V. So 00:03:47.400 --> 00:03:51.980 align:middle line:84% that means you need to show, for example, that-- 00:03:51.980 --> 00:03:53.570 align:middle line:84% one equivalent way of showing this 00:03:53.570 --> 00:03:56.600 align:middle line:84% is that every sequence that converges-- 00:03:56.600 --> 00:03:58.370 align:middle line:84% every sequence of elements from W 00:03:58.370 --> 00:04:02.330 align:middle line:84% that converges converges to an element of W. 00:04:02.330 --> 00:04:08.600 align:middle line:84% Now, take a sequence in W that converges to some element in V. 00:04:08.600 --> 00:04:11.540 align:middle line:84% We now want to show that element's in W. Then 00:04:11.540 --> 00:04:14.330 align:middle line:90% that sequence converges in W-- 00:04:14.330 --> 00:04:18.980 align:middle line:84% or is Cauchy in W. And since W is a Banach space, 00:04:18.980 --> 00:04:21.560 align:middle line:84% there must be an element in W which it converges to. 00:04:21.560 --> 00:04:23.870 align:middle line:84% And therefore, by uniqueness of limits, 00:04:23.870 --> 00:04:31.430 align:middle line:84% that limit of that original sequence must lie in W. 00:04:31.430 --> 00:04:34.310 align:middle line:84% And now, going the opposite direction, 00:04:34.310 --> 00:04:38.810 align:middle line:84% assuming the subspace is closed, we 00:04:38.810 --> 00:04:43.050 align:middle line:84% want to show that now this subspace is a Banach 00:04:43.050 --> 00:04:44.870 align:middle line:90% space, meaning it's complete. 00:04:44.870 --> 00:04:47.690 align:middle line:84% Well, let's take a Cauchy sequence in W. 00:04:47.690 --> 00:04:50.630 align:middle line:84% This is also a Cauchy sequence in V. Therefore, 00:04:50.630 --> 00:04:52.130 align:middle line:90% it has a limit. 00:04:52.130 --> 00:04:55.220 align:middle line:84% This limit has to be in W, since W is closed. 00:04:55.220 --> 00:04:58.580 align:middle line:84% And therefore, every Cauchy sequence in W has a limit in W. 00:04:58.580 --> 00:05:01.050 align:middle line:84% And therefore, W is a Banach space. 00:05:01.050 --> 00:05:03.332 align:middle line:84% OK, so I didn't write out what the-- 00:05:03.332 --> 00:05:05.540 align:middle line:84% I didn't write out the details, but we talked our way 00:05:05.540 --> 00:05:06.140 align:middle line:90% through it. 00:05:06.140 --> 00:05:09.830 align:middle line:90% 00:05:09.830 --> 00:05:12.290 align:middle line:84% Since we're taking a trip down memory lane, 00:05:12.290 --> 00:05:14.270 align:middle line:84% I just recalled what a subspace is. 00:05:14.270 --> 00:05:19.310 align:middle line:84% Now, given a subspace, we can obtain another vector space 00:05:19.310 --> 00:05:24.230 align:middle line:84% with the data of the space and the subspace 00:05:24.230 --> 00:05:25.500 align:middle line:90% called the quotient. 00:05:25.500 --> 00:05:27.320 align:middle line:90% So let me recall the quotient. 00:05:27.320 --> 00:05:31.790 align:middle line:90% 00:05:31.790 --> 00:05:38.750 align:middle line:84% So this is about briefly recalling what the quotient is. 00:05:38.750 --> 00:05:58.040 align:middle line:84% So let's take a subspace of V. We 00:05:58.040 --> 00:06:05.090 align:middle line:84% define an equivalence relation on V 00:06:05.090 --> 00:06:14.260 align:middle line:84% by saying that v is related to v-prime if 00:06:14.260 --> 00:06:20.670 align:middle line:84% and only if v minus v-prime is in W. 00:06:20.670 --> 00:06:24.815 align:middle line:84% Now, what do I mean by equivalence relation? 00:06:24.815 --> 00:06:29.050 align:middle line:90% 00:06:29.050 --> 00:06:33.040 align:middle line:84% v is always related to v because the difference of v 00:06:33.040 --> 00:06:35.410 align:middle line:84% is equal to 0, which is an element of W 00:06:35.410 --> 00:06:40.940 align:middle line:84% because W is the subspace and therefore a vector 00:06:40.940 --> 00:06:42.140 align:middle line:90% space in its own right. 00:06:42.140 --> 00:06:46.820 align:middle line:90% 00:06:46.820 --> 00:06:49.810 align:middle line:84% If v is related to v-prime, then this 00:06:49.810 --> 00:06:54.640 align:middle line:84% implies v-prime is related to v, simply because we-- again, 00:06:54.640 --> 00:06:58.120 align:middle line:84% if this is in W, then it's minus is also in W. 00:06:58.120 --> 00:07:04.250 align:middle line:90% And so this is reflexivity. 00:07:04.250 --> 00:07:07.670 align:middle line:84% And then we also have transitivity, 00:07:07.670 --> 00:07:12.170 align:middle line:84% that if v is related to v-prime and v is related-- 00:07:12.170 --> 00:07:15.350 align:middle line:84% and v-prime is related to v-double prime, 00:07:15.350 --> 00:07:19.340 align:middle line:84% then v is related to v-double prime. 00:07:19.340 --> 00:07:22.110 align:middle line:90% 00:07:22.110 --> 00:07:24.960 align:middle line:84% Just simply take v minus v-double prime 00:07:24.960 --> 00:07:26.910 align:middle line:90% and add and subtract v-prime. 00:07:26.910 --> 00:07:29.880 align:middle line:84% Then you obtain the sum of two elements in W 00:07:29.880 --> 00:07:34.510 align:middle line:84% which must be in W. So this is an equivalence relation. 00:07:34.510 --> 00:07:47.590 align:middle line:84% And so then we define the equivalence class 00:07:47.590 --> 00:07:51.880 align:middle line:84% of v. This is the set of all v-prime in V 00:07:51.880 --> 00:08:00.590 align:middle line:84% such that v-prime is related to v. 00:08:00.590 --> 00:08:07.390 align:middle line:84% And then we define a new set, v-mod W, 00:08:07.390 --> 00:08:09.565 align:middle line:84% to be the set of all equivalence classes. 00:08:09.565 --> 00:08:20.280 align:middle line:90% 00:08:20.280 --> 00:08:23.490 align:middle line:84% And instead of writing the equivalence class of v, 00:08:23.490 --> 00:08:38.760 align:middle line:84% we typically write v plus W instead 00:08:38.760 --> 00:08:42.330 align:middle line:84% of the equivalence class of v. So you 00:08:42.330 --> 00:08:45.150 align:middle line:84% think of the equivalence class of v 00:08:45.150 --> 00:08:50.460 align:middle line:84% as being v plus all elements of W. 00:08:50.460 --> 00:08:59.670 align:middle line:84% Or two elements are the same if they differ by an element of W. 00:08:59.670 --> 00:09:02.690 align:middle line:90% So this is a set. 00:09:02.690 --> 00:09:19.170 align:middle line:84% But it becomes a vector space with the addition 00:09:19.170 --> 00:09:21.300 align:middle line:84% and scalar multiplication operations defined 00:09:21.300 --> 00:09:24.165 align:middle line:84% in a kind of natural way with v1. 00:09:24.165 --> 00:09:35.840 align:middle line:90% 00:09:35.840 --> 00:10:01.320 align:middle line:84% And scalar multiplication is just the equivalence class 00:10:01.320 --> 00:10:07.780 align:middle line:84% given by lambda times v. Now, you have to take a second 00:10:07.780 --> 00:10:10.220 align:middle line:84% and make sure that these operations are well-defined, 00:10:10.220 --> 00:10:12.340 align:middle line:84% which I'm sure you did in linear algebra. 00:10:12.340 --> 00:10:13.990 align:middle line:90% What do I mean by well-defined? 00:10:13.990 --> 00:10:16.370 align:middle line:84% Well, these are two equivalent classes. 00:10:16.370 --> 00:10:18.490 align:middle line:84% So if I take two other representatives-- 00:10:18.490 --> 00:10:20.770 align:middle line:90% say v1-prime and v2-prime-- 00:10:20.770 --> 00:10:24.790 align:middle line:84% do I get the same equivalence class 00:10:24.790 --> 00:10:27.940 align:middle line:84% as I did with the unprime, meaning is v1-prime 00:10:27.940 --> 00:10:32.470 align:middle line:84% plus v2-prime plus W the same equivalence class as v1 00:10:32.470 --> 00:10:34.420 align:middle line:90% plus v2 plus W? 00:10:34.420 --> 00:10:38.065 align:middle line:84% And this is not too hard to say-- or to see. 00:10:38.065 --> 00:10:48.330 align:middle line:90% 00:10:48.330 --> 00:10:52.245 align:middle line:84% So v-mod W, which is typically how you pronounce this v 00:10:52.245 --> 00:10:55.120 align:middle line:84% slash W-- v-mod W is a vector space. 00:10:55.120 --> 00:11:07.780 align:middle line:84% And so note, we can identify W, so just the set W, 00:11:07.780 --> 00:11:11.690 align:middle line:84% with the equivalence class 0 plus W, 00:11:11.690 --> 00:11:29.970 align:middle line:84% which is also equal to little w plus capital W. 00:11:29.970 --> 00:11:39.415 align:middle line:84% OK, so when we first started talking about norms, 00:11:39.415 --> 00:11:42.380 align:middle line:84% I also introduced what was called a seminorm. 00:11:42.380 --> 00:11:46.970 align:middle line:84% Now, a seminorm, recall, satisfied not 00:11:46.970 --> 00:11:48.830 align:middle line:84% all three properties a norm satisfied. 00:11:48.830 --> 00:11:53.150 align:middle line:84% It satisfies homogeneity, meaning how scholars pull out, 00:11:53.150 --> 00:11:56.020 align:middle line:84% and also the triangle inequality. 00:11:56.020 --> 00:11:58.980 align:middle line:84% But it didn't satisfy positive definiteness, 00:11:58.980 --> 00:12:05.906 align:middle line:84% meaning the seminorm could be 0 for certain non-zero vectors, 00:12:05.906 --> 00:12:06.406 align:middle line:90% OK? 00:12:06.406 --> 00:12:11.800 align:middle line:90% 00:12:11.800 --> 00:12:12.790 align:middle line:90% How could that arise? 00:12:12.790 --> 00:12:14.080 align:middle line:90% How could a seminorm arise? 00:12:14.080 --> 00:12:17.980 align:middle line:84% Well, think of taking the maximum 00:12:17.980 --> 00:12:22.600 align:middle line:84% of the derivative of a function as being a norm 00:12:22.600 --> 00:12:24.880 align:middle line:90% or as being a potential norm. 00:12:24.880 --> 00:12:32.330 align:middle line:84% Then it satisfies homogeneity and the triangle inequality. 00:12:32.330 --> 00:12:36.670 align:middle line:84% But it is not a norm because the derivative 00:12:36.670 --> 00:12:40.810 align:middle line:90% of constant functions is 0. 00:12:40.810 --> 00:12:43.540 align:middle line:84% But constant functions aren't identically 0, 00:12:43.540 --> 00:12:46.210 align:middle line:84% if you think of them as elements of, say, 00:12:46.210 --> 00:12:49.180 align:middle line:84% the space of continuously differentiable functions. 00:12:49.180 --> 00:12:52.330 align:middle line:90% 00:12:52.330 --> 00:12:55.180 align:middle line:84% But this next theorem says that if you mod out 00:12:55.180 --> 00:13:00.850 align:middle line:84% by the elements in a vector space on which the seminorm is 00:13:00.850 --> 00:13:06.100 align:middle line:84% 0, then you get essentially a normed space where the norm is 00:13:06.100 --> 00:13:07.850 align:middle line:90% given by the seminorm. 00:13:07.850 --> 00:13:12.370 align:middle line:84% So if you mod out by the elements with 0 seminorm, 00:13:12.370 --> 00:13:15.160 align:middle line:90% you get an actual norm space. 00:13:15.160 --> 00:13:16.530 align:middle line:90% The statement is let-- 00:13:16.530 --> 00:13:21.702 align:middle line:90% 00:13:21.702 --> 00:13:36.600 align:middle line:84% let's take a seminorm on vector space V. 00:13:36.600 --> 00:13:44.580 align:middle line:84% Then we define E to be the set of all v in capital V such 00:13:44.580 --> 00:13:52.565 align:middle line:84% that this seminorm of these vectors is 0. 00:13:52.565 --> 00:13:57.840 align:middle line:90% 00:13:57.840 --> 00:14:11.070 align:middle line:84% This is a subspace of V. And if I define the following function 00:14:11.070 --> 00:14:15.170 align:middle line:90% on v-mod E-- 00:14:15.170 --> 00:14:18.130 align:middle line:90% 00:14:18.130 --> 00:14:22.010 align:middle line:90% so what is it? 00:14:22.010 --> 00:14:26.570 align:middle line:84% This is simply the norm of the representative v 00:14:26.570 --> 00:14:29.750 align:middle line:90% of this equivalence class. 00:14:29.750 --> 00:14:34.100 align:middle line:90% 00:14:34.100 --> 00:14:36.920 align:middle line:84% So if I define a function like this, 00:14:36.920 --> 00:14:55.060 align:middle line:84% so then this function defines a norm 00:14:55.060 --> 00:15:00.930 align:middle line:84% on the space v-mod E. So in short, if I have a seminorm 00:15:00.930 --> 00:15:05.820 align:middle line:84% and I mod out by all the elements with seminorm 00:15:05.820 --> 00:15:10.410 align:middle line:84% equal to 0, then I get an actual normed space, where this norm 00:15:10.410 --> 00:15:12.900 align:middle line:90% is essentially the seminorm. 00:15:12.900 --> 00:15:23.070 align:middle line:90% 00:15:23.070 --> 00:15:25.090 align:middle line:90% OK, so let's prove this. 00:15:25.090 --> 00:15:28.060 align:middle line:90% 00:15:28.060 --> 00:15:30.040 align:middle line:90% So why is this a subspace? 00:15:30.040 --> 00:15:49.300 align:middle line:84% Well, if I take two elements in this space of elements that 00:15:49.300 --> 00:16:03.720 align:middle line:84% have 0 seminorm and two scalars, and if I 00:16:03.720 --> 00:16:12.600 align:middle line:84% take the seminorm of the linear combination form, 00:16:12.600 --> 00:16:14.280 align:middle line:84% I want to show this is equal to 0. 00:16:14.280 --> 00:16:17.670 align:middle line:84% And this follows essentially by homogeneity and the triangle 00:16:17.670 --> 00:16:18.400 align:middle line:90% inequality. 00:16:18.400 --> 00:16:21.630 align:middle line:90% This is less than or equal to-- 00:16:21.630 --> 00:16:23.142 align:middle line:84% if I apply the triangle inequality 00:16:23.142 --> 00:16:24.600 align:middle line:84% and then pull the scalars out, this 00:16:24.600 --> 00:16:26.775 align:middle line:90% is less than or equal to plus-- 00:16:26.775 --> 00:16:30.380 align:middle line:90% 00:16:30.380 --> 00:16:34.820 align:middle line:84% and if these both have seminorm equal to 0, this is equal to 0. 00:16:34.820 --> 00:16:38.160 align:middle line:84% And now, remember, a seminorm is always non-negative. 00:16:38.160 --> 00:16:42.890 align:middle line:84% So if I show something is less than or equal to 0, it's also-- 00:16:42.890 --> 00:16:44.270 align:middle line:90% then it must be equal to 0. 00:16:44.270 --> 00:16:58.240 align:middle line:90% 00:16:58.240 --> 00:17:02.690 align:middle line:84% So first off, we define this function. 00:17:02.690 --> 00:17:07.839 align:middle line:84% So now, let's show that this defines a norm on v-mod E. 00:17:07.839 --> 00:17:10.720 align:middle line:84% We need to first show it's well-defined because I'm 00:17:10.720 --> 00:17:15.490 align:middle line:84% defining it in terms of a representative 00:17:15.490 --> 00:17:17.424 align:middle line:84% of this equivalence class, all right? 00:17:17.424 --> 00:17:35.180 align:middle line:90% 00:17:35.180 --> 00:17:37.070 align:middle line:90% And what does this mean? i.e. 00:17:37.070 --> 00:17:43.550 align:middle line:84% If, v plus-- if I have two equivalence classes with-- 00:17:43.550 --> 00:17:46.280 align:middle line:84% or if I have the same equivalence class with two 00:17:46.280 --> 00:17:50.210 align:middle line:84% different representatives, v and v-prime, then-- 00:17:50.210 --> 00:18:03.360 align:middle line:84% and I should write E, not W. Then the norm of v 00:18:03.360 --> 00:18:07.830 align:middle line:90% equals the norm of v-prime. 00:18:07.830 --> 00:18:10.100 align:middle line:84% And therefore, this function is well-defined. 00:18:10.100 --> 00:18:14.110 align:middle line:90% Now, how do we do that? 00:18:14.110 --> 00:18:16.110 align:middle line:84% Again, it's going to be essentially the triangle 00:18:16.110 --> 00:18:16.920 align:middle line:90% inequality. 00:18:16.920 --> 00:18:22.040 align:middle line:90% 00:18:22.040 --> 00:18:31.440 align:middle line:84% So suppose v plus e equals v-prime plus em 00:18:31.440 --> 00:18:33.510 align:middle line:84% meaning I have two representatives 00:18:33.510 --> 00:18:34.950 align:middle line:90% for the same equivalence class. 00:18:34.950 --> 00:18:38.090 align:middle line:90% 00:18:38.090 --> 00:18:40.840 align:middle line:84% And that means there exists a little e capital 00:18:40.840 --> 00:18:47.890 align:middle line:84% E such that v is equal to v-prime plus e. 00:18:47.890 --> 00:18:52.360 align:middle line:84% Then if I take the norm of v, this 00:18:52.360 --> 00:18:55.270 align:middle line:84% is equal to norm of v-prime plus e. 00:18:55.270 --> 00:18:57.580 align:middle line:84% And by the triangle inequality, this 00:18:57.580 --> 00:19:06.180 align:middle line:84% is less than or equal to norm of v-prime plus seminorm of E. 00:19:06.180 --> 00:19:10.560 align:middle line:84% And now, since E is coming from this set of all vectors 00:19:10.560 --> 00:19:14.430 align:middle line:84% with seminorm equal to 0, this is equal to 0. 00:19:14.430 --> 00:19:18.310 align:middle line:84% And therefore, I get the seminorm of v-prime. 00:19:18.310 --> 00:19:22.350 align:middle line:84% So I've shown that the seminorm of v is less than 00:19:22.350 --> 00:19:24.510 align:middle line:84% or equal to the seminorm of v-prime. 00:19:24.510 --> 00:19:25.980 align:middle line:90% Now, there was nothing-- 00:19:25.980 --> 00:19:28.920 align:middle line:84% this argument is symmetric in v and v-prime, all right? 00:19:28.920 --> 00:19:32.580 align:middle line:84% If v is equal to v-prime plus e for some element in capital E, 00:19:32.580 --> 00:19:33.640 align:middle line:90% so is v-prime. 00:19:33.640 --> 00:19:41.460 align:middle line:90% So I can also switch the-- 00:19:41.460 --> 00:19:43.740 align:middle line:84% I can also switch v and v-prime, meaning 00:19:43.740 --> 00:19:48.870 align:middle line:84% I've shown that v is less than or equal to v-prime 00:19:48.870 --> 00:19:53.490 align:middle line:84% and the norm of v-prime is less than or equal to v. 00:19:53.490 --> 00:20:00.980 align:middle line:84% And therefore, their two norms agree. 00:20:00.980 --> 00:20:09.190 align:middle line:84% So this function here is well-defined. 00:20:09.190 --> 00:20:12.220 align:middle line:84% And so I'm going to leave it to you 00:20:12.220 --> 00:20:18.250 align:middle line:84% now to check that this function now, which is well-defined 00:20:18.250 --> 00:20:22.720 align:middle line:84% on v-mod E, is, in fact-- does, in fact, satisfy all 00:20:22.720 --> 00:20:25.000 align:middle line:90% the properties of a seminorm. 00:20:25.000 --> 00:20:27.230 align:middle line:90% I mean, it's non-negative. 00:20:27.230 --> 00:20:37.880 align:middle line:84% So from essentially the homogeneity and triangle 00:20:37.880 --> 00:20:40.850 align:middle line:84% inequality property of the seminorm, 00:20:40.850 --> 00:20:44.570 align:middle line:84% the triangle inequality and homogeneity of this 00:20:44.570 --> 00:20:46.250 align:middle line:90% function then follow. 00:20:46.250 --> 00:20:51.470 align:middle line:84% And we've essentially equated all elements with seminorm 00:20:51.470 --> 00:20:53.040 align:middle line:90% equal to 0 to 0. 00:20:53.040 --> 00:20:55.290 align:middle line:84% So this is why it's also positive definite. 00:20:55.290 --> 00:20:57.472 align:middle line:84% So I'm going to leave that to you, OK? 00:20:57.472 --> 00:21:07.110 align:middle line:90% 00:21:07.110 --> 00:21:10.560 align:middle line:90% OK, now, there's one other way-- 00:21:10.560 --> 00:21:14.520 align:middle line:84% if we're given-- so in this process, 00:21:14.520 --> 00:21:17.910 align:middle line:84% we started with a seminorm on the space, identified 00:21:17.910 --> 00:21:22.740 align:middle line:84% the subspace of all, if you like, zero norm elements, 00:21:22.740 --> 00:21:28.440 align:middle line:90% and obtained a new normed space. 00:21:28.440 --> 00:21:32.580 align:middle line:84% But you could start off with a normed space, a closed subspace 00:21:32.580 --> 00:21:40.290 align:middle line:84% of that normed space, and obtain a new normed space on v-mod W, 00:21:40.290 --> 00:21:48.300 align:middle line:84% if w is a closed space, in similar fashion, where 00:21:48.300 --> 00:21:50.040 align:middle line:90% I define this norm this way. 00:21:50.040 --> 00:21:56.850 align:middle line:84% And well, the norm on that space won't be the same as this one. 00:21:56.850 --> 00:21:59.700 align:middle line:84% But that'll be in the exercises, OK? 00:21:59.700 --> 00:22:02.560 align:middle line:90% 00:22:02.560 --> 00:22:03.060 align:middle line:90% OK. 00:22:03.060 --> 00:22:09.600 align:middle line:90% 00:22:09.600 --> 00:22:19.650 align:middle line:84% So that's about-- this is kind of concluding 00:22:19.650 --> 00:22:21.840 align:middle line:84% the elementary section of functional analysis, 00:22:21.840 --> 00:22:27.060 align:middle line:84% meaning the bare bones of Banach spaces and normed spaces. 00:22:27.060 --> 00:22:31.180 align:middle line:84% And now, with the rest of this lecture and then next lecture, 00:22:31.180 --> 00:22:35.640 align:middle line:84% we're going to get into the fundamental results 00:22:35.640 --> 00:22:41.140 align:middle line:84% of functional analysis related to Banach spaces. 00:22:41.140 --> 00:22:44.750 align:middle line:84% So these theorems that I'll now be stating and proving 00:22:44.750 --> 00:22:48.410 align:middle line:84% in the coming couple of lectures have names attached to them. 00:22:48.410 --> 00:22:51.400 align:middle line:84% So you should definitely know what they are, what they state, 00:22:51.400 --> 00:22:54.470 align:middle line:90% what they don't state. 00:22:54.470 --> 00:22:57.050 align:middle line:84% But to prove them, I first need a result 00:22:57.050 --> 00:23:04.005 align:middle line:84% from metric spaces, which I did teach last semester. 00:23:04.005 --> 00:23:05.380 align:middle line:84% But we didn't cover this theorem. 00:23:05.380 --> 00:23:09.340 align:middle line:84% I can't remember if I covered it when I taught 18.100B. 00:23:09.340 --> 00:23:10.390 align:middle line:90% I don't think I did. 00:23:10.390 --> 00:23:16.960 align:middle line:90% 00:23:16.960 --> 00:23:23.500 align:middle line:84% So this is about when can you write-- 00:23:23.500 --> 00:23:27.290 align:middle line:90% 00:23:27.290 --> 00:23:28.340 align:middle line:90% or in what way-- 00:23:28.340 --> 00:23:31.550 align:middle line:90% 00:23:31.550 --> 00:23:35.680 align:middle line:84% OK, so let me just state that them. 00:23:35.680 --> 00:23:37.700 align:middle line:84% And then I'll interpret it for you. 00:23:37.700 --> 00:23:39.565 align:middle line:90% So this is Baire's theorem. 00:23:39.565 --> 00:23:44.480 align:middle line:90% 00:23:44.480 --> 00:23:48.200 align:middle line:84% It also goes by the name of the Baire category theorem, 00:23:48.200 --> 00:23:51.560 align:middle line:84% although it has nothing to do with category theory 00:23:51.560 --> 00:23:57.130 align:middle line:90% as is, I guess, used today. 00:23:57.130 --> 00:23:59.860 align:middle line:84% I don't know if category theory gets used 00:23:59.860 --> 00:24:02.170 align:middle line:84% or if it just is created for itself. 00:24:02.170 --> 00:24:05.010 align:middle line:90% 00:24:05.010 --> 00:24:13.920 align:middle line:84% But Baire's category theorem is the only category theorem 00:24:13.920 --> 00:24:16.808 align:middle line:90% that I know. 00:24:16.808 --> 00:24:17.725 align:middle line:90% So what does it state? 00:24:17.725 --> 00:24:21.290 align:middle line:90% 00:24:21.290 --> 00:24:26.900 align:middle line:84% If M-- so again, this is a theorem about metric spaces, so 00:24:26.900 --> 00:24:29.450 align:middle line:90% not necessarily normed spaces. 00:24:29.450 --> 00:24:45.410 align:middle line:84% So if M is a complete metric space 00:24:45.410 --> 00:25:06.460 align:middle line:84% and Cn is a collection of closed subsets of M 00:25:06.460 --> 00:25:08.950 align:middle line:90% such that M is their union-- 00:25:08.950 --> 00:25:12.770 align:middle line:90% 00:25:12.770 --> 00:25:13.820 align:middle line:90% so this is a-- 00:25:13.820 --> 00:25:16.950 align:middle line:90% 00:25:16.950 --> 00:25:17.940 align:middle line:90% so I have an N here. 00:25:17.940 --> 00:25:24.290 align:middle line:84% So I should specify this is N as a natural number-- 00:25:24.290 --> 00:25:27.250 align:middle line:90% 00:25:27.250 --> 00:25:51.280 align:middle line:84% then at least one Cn contains an open ball B x, r. 00:25:51.280 --> 00:25:56.650 align:middle line:84% So recall this is a set of all y in M such 00:25:56.650 --> 00:26:01.630 align:middle line:84% that the distance from x to y is less than r. 00:26:01.630 --> 00:26:06.620 align:middle line:84% So at least one Cn contains an open ball 00:26:06.620 --> 00:26:20.480 align:middle line:84% could be stated more succinctly as at least one Cn has 00:26:20.480 --> 00:26:21.455 align:middle line:90% an interior point. 00:26:21.455 --> 00:26:28.450 align:middle line:90% 00:26:28.450 --> 00:26:30.970 align:middle line:84% So sometimes in applications, these Cn's are not 00:26:30.970 --> 00:26:31.930 align:middle line:90% necessarily closed. 00:26:31.930 --> 00:26:35.170 align:middle line:84% But if M is equal to the union of these sets, 00:26:35.170 --> 00:26:38.320 align:middle line:84% then it's also equal to the union of their closures. 00:26:38.320 --> 00:26:41.380 align:middle line:84% And so what this theorem says is that if you can write 00:26:41.380 --> 00:26:44.470 align:middle line:84% a metric space as a union of sets, 00:26:44.470 --> 00:26:47.770 align:middle line:84% then the closure of one of those sets 00:26:47.770 --> 00:26:49.390 align:middle line:90% has to contain an open ball. 00:26:49.390 --> 00:26:52.880 align:middle line:90% 00:26:52.880 --> 00:26:58.100 align:middle line:84% Now, let's think a little bit of what it means for-- 00:26:58.100 --> 00:27:02.090 align:middle line:84% or there's a specific terminology for when a closed-- 00:27:02.090 --> 00:27:05.000 align:middle line:84% when the closure of a set does not contain an open ball, 00:27:05.000 --> 00:27:07.760 align:middle line:90% that means it's nowhere dense. 00:27:07.760 --> 00:27:10.400 align:middle line:84% Or that's the phrase we usually use for that, 00:27:10.400 --> 00:27:12.590 align:middle line:90% is that it's nowhere dense. 00:27:12.590 --> 00:27:15.290 align:middle line:84% So Baire's theorem says that if you 00:27:15.290 --> 00:27:20.570 align:middle line:84% have M equal to the union of a collection of sets, 00:27:20.570 --> 00:27:26.170 align:middle line:84% then the closure of one of those sets has to be dense somewhere. 00:27:26.170 --> 00:27:29.380 align:middle line:84% Or you can't write M, a complete metric space, 00:27:29.380 --> 00:27:32.815 align:middle line:84% as the union of nowhere dense subsets. 00:27:32.815 --> 00:27:43.630 align:middle line:90% 00:27:43.630 --> 00:27:46.750 align:middle line:84% And so this theorem is quite simple to state. 00:27:46.750 --> 00:27:53.560 align:middle line:84% It's quite very useful and, I mean, 00:27:53.560 --> 00:27:55.250 align:middle line:90% quite powerful in applications. 00:27:55.250 --> 00:27:56.800 align:middle line:84% In fact, you can use this theorem 00:27:56.800 --> 00:28:01.750 align:middle line:84% to give an alternative proof to something that hopefully you 00:28:01.750 --> 00:28:03.250 align:middle line:84% saw in your analysis class, which 00:28:03.250 --> 00:28:06.040 align:middle line:84% is that there exists a continuous function which 00:28:06.040 --> 00:28:09.247 align:middle line:90% is nowhere differentiable. 00:28:09.247 --> 00:28:12.110 align:middle line:90% 00:28:12.110 --> 00:28:13.700 align:middle line:84% Maybe I'll put that in the exercises. 00:28:13.700 --> 00:28:17.690 align:middle line:84% Maybe I'll just point you to it somewhere. 00:28:17.690 --> 00:28:19.810 align:middle line:90% We'll see. 00:28:19.810 --> 00:28:21.880 align:middle line:84% All right, so the proof is by contradiction. 00:28:21.880 --> 00:28:28.802 align:middle line:90% 00:28:28.802 --> 00:28:30.760 align:middle line:84% I believe this is the first contradiction proof 00:28:30.760 --> 00:28:33.140 align:middle line:90% that we've done in the class. 00:28:33.140 --> 00:28:36.820 align:middle line:84% So in 18.100A and B, I mean, you start off-- 00:28:36.820 --> 00:28:38.440 align:middle line:90% everything is by contradiction. 00:28:38.440 --> 00:28:41.240 align:middle line:90% 00:28:41.240 --> 00:28:46.570 align:middle line:90% So let's suppose not, i.e. 00:28:46.570 --> 00:28:50.680 align:middle line:84% you can find there is a collection 00:28:50.680 --> 00:28:55.732 align:middle line:84% of closed subsets of M such that M is equal to this union. 00:28:55.732 --> 00:28:57.190 align:middle line:84% And what we're going to do is we're 00:28:57.190 --> 00:28:59.830 align:middle line:84% going to find a point that's not in this union 00:28:59.830 --> 00:29:03.130 align:middle line:84% and therefore not in M. And that will be our contradiction. 00:29:03.130 --> 00:29:07.570 align:middle line:84% We'll use the completeness of M to obtain this point 00:29:07.570 --> 00:29:10.030 align:middle line:84% because how we're going to obtain this point 00:29:10.030 --> 00:29:13.090 align:middle line:84% is as a limit of a certain sequence. 00:29:13.090 --> 00:29:16.690 align:middle line:84% And to be able to say that this limit exists, 00:29:16.690 --> 00:29:20.620 align:middle line:84% we're going to show the sequence as Cauchy. 00:29:20.620 --> 00:29:22.570 align:middle line:84% All right, so we're going to build up 00:29:22.570 --> 00:29:25.920 align:middle line:90% this sequence inductively. 00:29:25.920 --> 00:29:30.300 align:middle line:84% And I will write this inductive proof kind 00:29:30.300 --> 00:29:31.920 align:middle line:90% of carefully this first time. 00:29:31.920 --> 00:29:33.960 align:middle line:84% And then in the future, I'm just going 00:29:33.960 --> 00:29:37.590 align:middle line:84% to say, OK, choose p1 this way, choose p2 this way. 00:29:37.590 --> 00:29:40.920 align:middle line:84% Continuing in this manner, we obtain a sequence of points, 00:29:40.920 --> 00:29:41.920 align:middle line:90% blah, blah, blah. 00:29:41.920 --> 00:29:46.200 align:middle line:84% But at this stage, I'll write this inductive proof 00:29:46.200 --> 00:29:47.530 align:middle line:90% kind of carefully. 00:29:47.530 --> 00:29:49.560 align:middle line:90% OK, so suppose not. 00:29:49.560 --> 00:29:55.430 align:middle line:90% 00:29:55.430 --> 00:30:10.640 align:middle line:84% Since M certainly contains an open ball and C1 cannot contain 00:30:10.640 --> 00:30:15.930 align:middle line:84% an open ball, this implies M does not equal to the first 00:30:15.930 --> 00:30:16.500 align:middle line:90% closed set. 00:30:16.500 --> 00:30:19.380 align:middle line:84% Otherwise, the first closed set would contain an open ball. 00:30:19.380 --> 00:30:23.860 align:middle line:90% 00:30:23.860 --> 00:30:29.660 align:middle line:84% Thus, there exists an element p1 in M take away C1. 00:30:29.660 --> 00:30:31.340 align:middle line:90% Now, C1 is closed. 00:30:31.340 --> 00:30:36.420 align:middle line:90% 00:30:36.420 --> 00:30:38.670 align:middle line:84% This implies that its complement is open. 00:30:38.670 --> 00:30:43.910 align:middle line:90% 00:30:43.910 --> 00:30:51.940 align:middle line:84% And therefore, I can find a small epsilon 1 such 00:30:51.940 --> 00:30:59.710 align:middle line:84% that the ball centered at p1 of radius epsilon 00:30:59.710 --> 00:31:06.025 align:middle line:84% 1 intersect C1 equals the empty set. 00:31:06.025 --> 00:31:14.910 align:middle line:90% 00:31:14.910 --> 00:31:16.260 align:middle line:90% Now, I'm going to pick C2. 00:31:16.260 --> 00:31:23.610 align:middle line:90% 00:31:23.610 --> 00:31:33.420 align:middle line:84% Now the ball centered at p1 of radius epsilon 1 over 3, 00:31:33.420 --> 00:31:37.740 align:middle line:84% this is not contained in C2 because, again, this 00:31:37.740 --> 00:31:38.800 align:middle line:90% is by contradiction. 00:31:38.800 --> 00:31:40.320 align:middle line:84% So we're assuming that M is written 00:31:40.320 --> 00:31:42.930 align:middle line:84% as a union of closed sets and none of these closed sets 00:31:42.930 --> 00:31:44.130 align:middle line:90% contain an open ball. 00:31:44.130 --> 00:31:46.440 align:middle line:84% So this open ball cannot be contained in C2. 00:31:46.440 --> 00:31:51.640 align:middle line:90% 00:31:51.640 --> 00:31:57.830 align:middle line:84% And therefore, there exists some point p2 00:31:57.830 --> 00:32:05.150 align:middle line:84% in this open ball centered at p1 of radius epsilon 1 over 3 00:32:05.150 --> 00:32:12.890 align:middle line:90% such that p2 is not in C2. 00:32:12.890 --> 00:32:23.960 align:middle line:84% Now, again, C2-closed implies that there exists an epsilon 2 00:32:23.960 --> 00:32:27.950 align:middle line:84% less than epsilon over 3-- so we can make it very small if we 00:32:27.950 --> 00:32:29.720 align:middle line:90% wish-- 00:32:29.720 --> 00:32:37.800 align:middle line:84% such that the ball centered at p2 of radius epsilon 2 00:32:37.800 --> 00:32:41.880 align:middle line:84% intersect C2 equals the empty set. 00:32:41.880 --> 00:32:44.420 align:middle line:90% 00:32:44.420 --> 00:32:47.680 align:middle line:90% So now, I've picked p2-- 00:32:47.680 --> 00:32:48.640 align:middle line:90% p1 and p2. 00:32:48.640 --> 00:32:50.950 align:middle line:84% Now, at this point, I would usually 00:32:50.950 --> 00:32:54.670 align:middle line:84% say, continuing in this manner, we obtain a sequence of points. 00:32:54.670 --> 00:32:57.280 align:middle line:84% But let me write out the argument carefully. 00:32:57.280 --> 00:33:02.820 align:middle line:90% 00:33:02.820 --> 00:33:09.450 align:middle line:90% Suppose there exists k points-- 00:33:09.450 --> 00:33:31.210 align:middle line:84% p1, pk and positive epsilon 1, epsilon k such that two things 00:33:31.210 --> 00:33:32.120 align:middle line:90% occur. 00:33:32.120 --> 00:33:41.150 align:middle line:84% So epsilon k is less than epsilon k minus 1 over 3 and-- 00:33:41.150 --> 00:33:43.810 align:middle line:90% 00:33:43.810 --> 00:33:48.610 align:middle line:84% which is less than epsilon k minus 2 over 3 and so 00:33:48.610 --> 00:33:55.255 align:middle line:84% on all the way down to epsilon 1 over 3 k minus 1. 00:33:55.255 --> 00:33:58.650 align:middle line:90% 00:33:58.650 --> 00:34:01.800 align:middle line:84% So if you like, k here is bigger than or equal to 2. 00:34:01.800 --> 00:34:05.860 align:middle line:90% 00:34:05.860 --> 00:34:17.139 align:middle line:84% And pj satisfies-- pj is in the ball centered at pj minus 1 00:34:17.139 --> 00:34:23.010 align:middle line:84% of radius epsilon j minus 1 over 3. 00:34:23.010 --> 00:34:30.179 align:middle line:84% And the ball centered at pj epsilon j 00:34:30.179 --> 00:34:34.350 align:middle line:84% intersects Cj equals the empty set. 00:34:34.350 --> 00:34:37.389 align:middle line:90% 00:34:37.389 --> 00:34:45.120 align:middle line:84% So let me star those two properties right there. 00:34:45.120 --> 00:34:50.219 align:middle line:84% We're going to obtain another point satisfying those two 00:34:50.219 --> 00:34:52.020 align:middle line:90% properties. 00:34:52.020 --> 00:34:59.720 align:middle line:84% And I don't know why I'm being a stickler. 00:34:59.720 --> 00:35:01.020 align:middle line:90% It doesn't really matter. 00:35:01.020 --> 00:35:05.390 align:middle line:84% But let me put a 2 because that's not 00:35:05.390 --> 00:35:07.160 align:middle line:90% necessarily satisfied for p1. 00:35:07.160 --> 00:35:11.880 align:middle line:84% But OK, now, I want to show there exists-- 00:35:11.880 --> 00:35:17.060 align:middle line:84% I can obtain a k plus first point satisfying those two 00:35:17.060 --> 00:35:19.010 align:middle line:90% properties. 00:35:19.010 --> 00:35:20.780 align:middle line:84% Again, this is just formally saying 00:35:20.780 --> 00:35:24.320 align:middle line:84% that once I've chosen p2, then I can choose 00:35:24.320 --> 00:35:31.480 align:middle line:90% a p3 doing this argument again. 00:35:31.480 --> 00:35:35.680 align:middle line:84% And it'll satisfy everything p2 satisfied with epsilon 1, 00:35:35.680 --> 00:35:37.750 align:middle line:90% except now with epsilon 2. 00:35:37.750 --> 00:35:39.860 align:middle line:84% But now, I'm just making it more formal. 00:35:39.860 --> 00:35:46.460 align:middle line:84% So now, again, M is the union of all these closed sets, none 00:35:46.460 --> 00:35:49.960 align:middle line:90% of which contain an open ball. 00:35:49.960 --> 00:35:59.830 align:middle line:84% So since the ball of radius pk epsilon k over 3 00:35:59.830 --> 00:36:06.890 align:middle line:84% is not contained in Ck plus 1, there 00:36:06.890 --> 00:36:15.900 align:middle line:84% exists an element pk plus 1 not in Ck plus 1. 00:36:15.900 --> 00:36:16.800 align:middle line:90% Or I should say-- 00:36:16.800 --> 00:36:27.756 align:middle line:90% 00:36:27.756 --> 00:36:42.280 align:middle line:84% over 3-- such that pk plus 1 is not in Ck plus 1. 00:36:42.280 --> 00:36:59.950 align:middle line:84% So let me-- there's pk epsilon k over 3, Ck plus 1. 00:36:59.950 --> 00:37:01.955 align:middle line:90% So maybe they're not disjoint. 00:37:01.955 --> 00:37:03.080 align:middle line:90% Maybe there's some overlap. 00:37:03.080 --> 00:37:16.890 align:middle line:84% But then I can find a point pk plus 1 00:37:16.890 --> 00:37:19.050 align:middle line:84% in there that's not in Ck plus 1. 00:37:19.050 --> 00:37:24.780 align:middle line:90% 00:37:24.780 --> 00:37:31.250 align:middle line:84% So then there exists an epsilon k plus 1, 00:37:31.250 --> 00:37:33.410 align:middle line:84% which I can choose very small, say smaller 00:37:33.410 --> 00:37:38.400 align:middle line:84% than epsilon k over 3, such that-- 00:37:38.400 --> 00:37:42.800 align:middle line:84% again, because Ck plus 1 is closed, pk plus 1 00:37:42.800 --> 00:37:44.810 align:middle line:90% is not in Ck plus 1. 00:37:44.810 --> 00:37:47.900 align:middle line:84% So I can find a small ball around pk 00:37:47.900 --> 00:37:52.028 align:middle line:90% plus 1 disjoint from Ck plus 1-- 00:37:52.028 --> 00:38:08.660 align:middle line:84% a ball-- we said Ck equals the empty set. 00:38:08.660 --> 00:38:15.170 align:middle line:90% 00:38:15.170 --> 00:38:18.740 align:middle line:84% So then given k points, I can then 00:38:18.740 --> 00:38:23.600 align:middle line:84% choose a k plus first point satisfying these two bullet 00:38:23.600 --> 00:38:25.610 align:middle line:90% points here, all right? 00:38:25.610 --> 00:38:28.240 align:middle line:90% 00:38:28.240 --> 00:38:54.860 align:middle line:84% So by induction, we have found a sequence of points, pk, 00:38:54.860 --> 00:38:58.000 align:middle line:90% in M such that-- 00:38:58.000 --> 00:39:03.210 align:middle line:90% and epsilon k. 00:39:03.210 --> 00:39:05.160 align:middle line:90% So if you like, k starts at 2. 00:39:05.160 --> 00:39:08.060 align:middle line:90% 00:39:08.060 --> 00:39:19.230 align:middle line:84% Epsilon k and 0 epsilon 1 such that for all k, 00:39:19.230 --> 00:39:21.780 align:middle line:84% those two bullet points hold, such that-- 00:39:21.780 --> 00:39:26.380 align:middle line:90% 00:39:26.380 --> 00:39:28.470 align:middle line:84% which I denoted-- which I put a little star by, 00:39:28.470 --> 00:39:30.762 align:middle line:84% star [? hold. ?] So I'm talking about these two points. 00:39:30.762 --> 00:39:34.800 align:middle line:90% 00:39:34.800 --> 00:39:40.615 align:middle line:84% So when I say star, I mean these two statements here, OK? 00:39:40.615 --> 00:39:57.140 align:middle line:90% 00:39:57.140 --> 00:40:04.700 align:middle line:84% OK, now, let's show that the sequence is Cauchy. 00:40:04.700 --> 00:40:05.630 align:middle line:90% So now, I claim-- 00:40:05.630 --> 00:40:16.010 align:middle line:90% 00:40:16.010 --> 00:40:16.765 align:middle line:90% how do we do that? 00:40:16.765 --> 00:40:21.540 align:middle line:90% 00:40:21.540 --> 00:40:24.500 align:middle line:90% This follows. 00:40:24.500 --> 00:40:27.173 align:middle line:84% So I'm not going to write out the full epsilon M argument. 00:40:27.173 --> 00:40:29.590 align:middle line:84% But I'm just going to write down the crucial estimate that 00:40:29.590 --> 00:40:30.340 align:middle line:90% proves this. 00:40:30.340 --> 00:40:40.380 align:middle line:84% This follows from the fact that for all k 00:40:40.380 --> 00:40:47.470 align:middle line:84% for all l, If I look at the distance between pk and pk 00:40:47.470 --> 00:40:50.290 align:middle line:84% plus l because in the end, when I look at-- 00:40:50.290 --> 00:40:52.060 align:middle line:84% if I want to show something's Cauchy, 00:40:52.060 --> 00:40:53.950 align:middle line:84% I need to take the difference of two-- 00:40:53.950 --> 00:40:57.470 align:middle line:84% or the distance between two points. 00:40:57.470 --> 00:41:00.530 align:middle line:84% One I can choose as the point occurring 00:41:00.530 --> 00:41:03.680 align:middle line:84% earlier in the sequence, and then one occurring later. 00:41:03.680 --> 00:41:07.700 align:middle line:84% Anyways, then by the triangle inequality repeated, 00:41:07.700 --> 00:41:10.580 align:middle line:84% I can write that this is less than or equal to the distance 00:41:10.580 --> 00:41:16.610 align:middle line:84% from pk pk plus 1 plus the distance from pk 00:41:16.610 --> 00:41:19.100 align:middle line:90% plus 1 pk plus l. 00:41:19.100 --> 00:41:22.010 align:middle line:84% And now, I could do the triangle inequality again. 00:41:22.010 --> 00:41:27.710 align:middle line:84% And so I get that this is less than or equal to the distance 00:41:27.710 --> 00:41:33.350 align:middle line:84% from pk to pk plus 1 plus the distance from pk plus 1 00:41:33.350 --> 00:41:41.570 align:middle line:84% to pk plus 2 and so on until I get to the distance between pk 00:41:41.570 --> 00:41:45.612 align:middle line:90% plus l minus 1 and pk plus l. 00:41:45.612 --> 00:41:50.960 align:middle line:90% 00:41:50.960 --> 00:41:56.930 align:middle line:84% Now, by star, what's in yellow, pk plus 1 00:41:56.930 --> 00:42:01.670 align:middle line:84% is in the epsilon k over 3 ball centered at pk. 00:42:01.670 --> 00:42:06.410 align:middle line:84% So this is less than epsilon k over 3. 00:42:06.410 --> 00:42:12.110 align:middle line:84% And the same thing here, less than epsilon k plus 1 over 3 00:42:12.110 --> 00:42:20.350 align:middle line:84% plus epsilon k plus l minus 1 over 3. 00:42:20.350 --> 00:42:24.730 align:middle line:84% And now, all of those are less than-- 00:42:24.730 --> 00:42:27.220 align:middle line:84% if you go to the first one, it's less than epsilon 1 00:42:27.220 --> 00:42:30.520 align:middle line:90% over 3 k minus 1. 00:42:30.520 --> 00:42:37.910 align:middle line:84% So then this is less than epsilon 1 over 3 k 00:42:37.910 --> 00:42:46.790 align:middle line:84% plus epsilon 1 over 3 k plus l, yeah. 00:42:46.790 --> 00:42:52.190 align:middle line:90% 00:42:52.190 --> 00:42:55.610 align:middle line:84% And now, I can actually sum this up. 00:42:55.610 --> 00:42:59.990 align:middle line:84% This is strictly less than epsilon 1. 00:42:59.990 --> 00:43:06.260 align:middle line:84% And if I sum now from l equals k to infinity of 1 over 3 k, 00:43:06.260 --> 00:43:16.950 align:middle line:84% this is equal to 1 over epsilon 1 3 k 1 over 1 minus 1/3, 00:43:16.950 --> 00:43:27.680 align:middle line:84% which equals epsilon 1 over 2 3 to the minus k plus 1. 00:43:27.680 --> 00:43:31.850 align:middle line:84% So the distance between a point pk and pk plus l 00:43:31.850 --> 00:43:38.210 align:middle line:84% is less than some constant times 3 to the minus k plus 1. 00:43:38.210 --> 00:43:44.000 align:middle line:84% So if k is very large, this is very small independent of l. 00:43:44.000 --> 00:43:45.870 align:middle line:90% So I shouldn't have used l here. 00:43:45.870 --> 00:43:55.405 align:middle line:84% And that's M, M. So this shows that the sequence is Cauchy. 00:43:55.405 --> 00:44:05.690 align:middle line:90% 00:44:05.690 --> 00:44:14.430 align:middle line:84% So since M is complete, there exists a p in M such 00:44:14.430 --> 00:44:20.100 align:middle line:84% that these elements pk converge to p. 00:44:20.100 --> 00:44:24.120 align:middle line:84% And now, we're going to show this point p does not lie 00:44:24.120 --> 00:44:26.195 align:middle line:90% in any of the CJ's. 00:44:26.195 --> 00:44:29.810 align:middle line:90% 00:44:29.810 --> 00:44:32.990 align:middle line:84% And we'll do it by showing it's essentially in all 00:44:32.990 --> 00:44:37.510 align:middle line:90% of these balls, pj epsilon j. 00:44:37.510 --> 00:44:42.030 align:middle line:90% 00:44:42.030 --> 00:44:44.220 align:middle line:84% All right, and it's kind a similar computation 00:44:44.220 --> 00:44:45.610 align:middle line:90% to what we just did. 00:44:45.610 --> 00:45:02.170 align:middle line:84% So now, for all k natural number, 00:45:02.170 --> 00:45:07.210 align:middle line:84% if you look at the distance between pk plus 1, 00:45:07.210 --> 00:45:09.130 align:middle line:90% pk plus 1 plus l-- 00:45:09.130 --> 00:45:12.390 align:middle line:90% 00:45:12.390 --> 00:45:20.210 align:middle line:84% again, so if I just go back to this computation right here, 00:45:20.210 --> 00:45:26.000 align:middle line:84% this, we proved, was less than epsilon k plus 1 times 1 00:45:26.000 --> 00:45:31.310 align:middle line:90% over 3 k plus 1 plus-- 00:45:31.310 --> 00:45:39.070 align:middle line:84% or actually, 1 over 3 plus 1 over 3 00:45:39.070 --> 00:45:45.250 align:middle line:90% squared plus 1 over 3 k plus 1-- 00:45:45.250 --> 00:45:48.660 align:middle line:90% 00:45:48.660 --> 00:45:49.545 align:middle line:90% k plus l. 00:45:49.545 --> 00:45:56.850 align:middle line:90% 00:45:56.850 --> 00:45:58.755 align:middle line:84% Or I'm sorry, I think that's just l. 00:45:58.755 --> 00:46:02.440 align:middle line:90% 00:46:02.440 --> 00:46:06.710 align:middle line:84% And now, this thing is less than if I replace it 00:46:06.710 --> 00:46:08.220 align:middle line:90% by the infinite sum. 00:46:08.220 --> 00:46:13.970 align:middle line:84% So this is less than epsilon k plus 1 times the infinite sum 00:46:13.970 --> 00:46:18.380 align:middle line:84% again, m equals 0 to infinity 3 to the minus 00:46:18.380 --> 00:46:21.785 align:middle line:90% m, which equals what? 00:46:21.785 --> 00:46:26.490 align:middle line:90% 00:46:26.490 --> 00:46:32.310 align:middle line:84% So this is, again, equal to epsilon k plus 1 times 3/2. 00:46:32.310 --> 00:46:57.220 align:middle line:90% 00:46:57.220 --> 00:47:02.120 align:middle line:84% OK, so we've proven that the distance between pk plus 1 00:47:02.120 --> 00:47:06.290 align:middle line:84% pk plus 1 plus l is less than epsilon 00:47:06.290 --> 00:47:11.600 align:middle line:90% k plus 1 times 3 over 2. 00:47:11.600 --> 00:47:15.120 align:middle line:90% 00:47:15.120 --> 00:47:18.480 align:middle line:84% Let me take the limit as l goes to infinity. 00:47:18.480 --> 00:47:23.720 align:middle line:90% 00:47:23.720 --> 00:47:27.050 align:middle line:84% pk plus 1 plus l converges to the point p. 00:47:27.050 --> 00:47:33.710 align:middle line:84% So I get that the distance between pk plus 1 and the point 00:47:33.710 --> 00:47:42.540 align:middle line:84% p is less than or equal to 3/2 epsilon k plus 1. 00:47:42.540 --> 00:47:49.330 align:middle line:84% And remember, epsilon k plus 1 is less than a third epsilon k. 00:47:49.330 --> 00:47:57.210 align:middle line:84% So this is less than 1/2 epsilon k, all right? 00:47:57.210 --> 00:47:59.460 align:middle line:84% And since pk plus 1-- remember, this 00:47:59.460 --> 00:48:04.800 align:middle line:84% is in the ball of radius 13 epsilon k. 00:48:04.800 --> 00:48:10.630 align:middle line:84% So I get that the distance between pk and p 00:48:10.630 --> 00:48:16.350 align:middle line:84% is less than or equal to the distance between pk 00:48:16.350 --> 00:48:25.240 align:middle line:90% and pk plus 1 plus 1 and p. 00:48:25.240 --> 00:48:28.510 align:middle line:84% We get this is less than or equal to 1/3 epsilon 00:48:28.510 --> 00:48:31.120 align:middle line:90% k plus 1/2 epsilon k. 00:48:31.120 --> 00:48:34.510 align:middle line:84% And this is less than epsilon k, which 00:48:34.510 --> 00:48:46.070 align:middle line:84% means that p is in the ball of radius pk of radius epsilon k, 00:48:46.070 --> 00:48:52.020 align:middle line:84% which by the second property up in star-- 00:48:52.020 --> 00:48:54.590 align:middle line:84% the fact that all these balls are disjoint from-- 00:48:54.590 --> 00:48:59.750 align:middle line:84% each of the balls is disjoint from Ck means p is not in Ck. 00:48:59.750 --> 00:49:03.420 align:middle line:90% Now, k was arbitrary here. 00:49:03.420 --> 00:49:12.320 align:middle line:84% So we conclude that p is not in this union, which 00:49:12.320 --> 00:49:15.890 align:middle line:84% is equal to M. And that's a contradiction. 00:49:15.890 --> 00:49:26.000 align:middle line:90% 00:49:26.000 --> 00:49:30.620 align:middle line:84% So I mean, I said the strategy at the top 00:49:30.620 --> 00:49:34.850 align:middle line:84% of the proof, the technical argument maybe muddled 00:49:34.850 --> 00:49:35.760 align:middle line:90% what was going on. 00:49:35.760 --> 00:49:42.540 align:middle line:84% But again, the point is you can build-- 00:49:42.540 --> 00:49:44.190 align:middle line:84% if this conclusion does not hold, 00:49:44.190 --> 00:49:49.890 align:middle line:84% you can build a sequence which has this disjointness property 00:49:49.890 --> 00:49:53.930 align:middle line:84% from this collection of open sets 00:49:53.930 --> 00:49:56.150 align:middle line:84% such that the sequence is Cauchy. 00:49:56.150 --> 00:49:58.520 align:middle line:84% And because M is a complete metric space, 00:49:58.520 --> 00:50:00.530 align:middle line:90% you can then extract a limit p. 00:50:00.530 --> 00:50:04.160 align:middle line:84% This limit p will have all the properties that these pk's had, 00:50:04.160 --> 00:50:07.310 align:middle line:84% namely that they're not in any of the Ck's. 00:50:07.310 --> 00:50:10.490 align:middle line:84% And therefore, this point p does not 00:50:10.490 --> 00:50:13.670 align:middle line:84% lie in the metric space, which is nonsense. 00:50:13.670 --> 00:50:15.260 align:middle line:84% So we arrived at our contradiction. 00:50:15.260 --> 00:50:19.190 align:middle line:84% And therefore, Baire's category theorem is proven. 00:50:19.190 --> 00:50:28.320 align:middle line:90% 00:50:28.320 --> 00:50:35.300 align:middle line:84% OK, so that's Baire categories theorem. 00:50:35.300 --> 00:50:38.840 align:middle line:90% 00:50:38.840 --> 00:50:41.950 align:middle line:84% Let's now use this to prove some fundamental results 00:50:41.950 --> 00:50:43.220 align:middle line:90% of functional analysis. 00:50:43.220 --> 00:50:46.570 align:middle line:84% So the first result we'll prove is 00:50:46.570 --> 00:50:56.430 align:middle line:84% what's called the uniform boundedness 00:50:56.430 --> 00:51:03.220 align:middle line:84% theorem, which says that if you have a sequence of linear-- 00:51:03.220 --> 00:51:07.460 align:middle line:84% bounded linear operators on a Banach space, 00:51:07.460 --> 00:51:09.560 align:middle line:84% then pointwise boundedness implies 00:51:09.560 --> 00:51:15.910 align:middle line:84% uniform mindedness or uniform boundedness 00:51:15.910 --> 00:51:17.350 align:middle line:90% in the operator norm. 00:51:17.350 --> 00:51:23.750 align:middle line:90% So let B be a Banach space. 00:51:23.750 --> 00:51:30.080 align:middle line:90% 00:51:30.080 --> 00:51:40.500 align:middle line:84% And let Tn be a sequence of bounded linear operators, which 00:51:40.500 --> 00:51:47.980 align:middle line:84% we denoted last time by this script B, 00:51:47.980 --> 00:51:51.580 align:middle line:84% to some other normed space, V. So I didn't write that, 00:51:51.580 --> 00:51:54.805 align:middle line:90% but V is a normed space. 00:51:54.805 --> 00:51:58.030 align:middle line:90% 00:51:58.030 --> 00:52:10.920 align:middle line:84% If for all b in capital B, sup n of Tn v-norm is less than 00:52:10.920 --> 00:52:16.500 align:middle line:84% infinity-- so if I assume that this sequence is pointwise 00:52:16.500 --> 00:52:18.350 align:middle line:90% bounded-- 00:52:18.350 --> 00:52:24.980 align:middle line:84% pointwise bounded, then I can conclude 00:52:24.980 --> 00:52:27.170 align:middle line:84% that they're uniformly bounded, namely 00:52:27.170 --> 00:52:33.675 align:middle line:84% the sup of the operator norms are bounded. 00:52:33.675 --> 00:52:43.750 align:middle line:90% 00:52:43.750 --> 00:52:45.800 align:middle line:84% So we're going to use the Baire category theorem. 00:52:45.800 --> 00:52:49.330 align:middle line:84% So we're going to write the Banach space 00:52:49.330 --> 00:52:52.690 align:middle line:90% B as a union of closed subsets. 00:52:52.690 --> 00:52:54.700 align:middle line:84% What will be these closed subsets? 00:52:54.700 --> 00:53:00.370 align:middle line:84% Remember, we're trying to find a uniform bound on the Tn. 00:53:00.370 --> 00:53:09.460 align:middle line:84% So just playing around, so let's define the subset of B to Ck. 00:53:09.460 --> 00:53:17.810 align:middle line:84% This is a set of all elements b in B such that the norm of B 00:53:17.810 --> 00:53:27.320 align:middle line:84% is less than or equal to 1 and sup n Tn of B 00:53:27.320 --> 00:53:28.595 align:middle line:90% is less than or equal to k. 00:53:28.595 --> 00:53:37.830 align:middle line:90% 00:53:37.830 --> 00:53:42.495 align:middle line:84% All right, so first off, these are closed. 00:53:42.495 --> 00:53:47.790 align:middle line:90% 00:53:47.790 --> 00:53:51.780 align:middle line:84% So I should say, what is k? k here is a natural number. 00:53:51.780 --> 00:53:52.860 align:middle line:90% Ck is closed. 00:53:52.860 --> 00:53:53.620 align:middle line:90% Why is this? 00:53:53.620 --> 00:53:56.640 align:middle line:84% Well, we need to show that a sequence converging-- 00:53:56.640 --> 00:53:58.950 align:middle line:84% a sequence of elements from Ck that converges 00:53:58.950 --> 00:54:00.990 align:middle line:90% converges to an element of Ck. 00:54:00.990 --> 00:54:09.330 align:middle line:84% So if Bn-- this is a sequence of elements in Ck-- 00:54:09.330 --> 00:54:18.710 align:middle line:84% and Bn converges to B, then the norm of b is equal to the limit 00:54:18.710 --> 00:54:23.840 align:middle line:84% as n goes to infinity of the norm of bn. 00:54:23.840 --> 00:54:26.930 align:middle line:84% And each of these is less than or equal to 1. 00:54:26.930 --> 00:54:29.900 align:middle line:84% So the limit has to be less than or equal to 1. 00:54:29.900 --> 00:54:36.140 align:middle line:90% 00:54:36.140 --> 00:54:41.380 align:middle line:90% And so I shouldn't-- 00:54:41.380 --> 00:54:43.810 align:middle line:90% maybe I'll use AI a-- 00:54:43.810 --> 00:54:45.000 align:middle line:90% I have the Tn's. 00:54:45.000 --> 00:54:45.900 align:middle line:90% I have the Bn's. 00:54:45.900 --> 00:54:52.980 align:middle line:84% So let me use M. And for all M in natural number, 00:54:52.980 --> 00:54:58.890 align:middle line:84% Tm applied to be and normed, this is equal to the limit 00:54:58.890 --> 00:55:05.310 align:middle line:84% as m goes to infinity of Tm Bn in norm 00:55:05.310 --> 00:55:07.920 align:middle line:84% because these are bounded linear operators 00:55:07.920 --> 00:55:10.910 align:middle line:90% and therefore continuous. 00:55:10.910 --> 00:55:24.510 align:middle line:90% 00:55:24.510 --> 00:55:29.770 align:middle line:84% OK, so Tm of B is equal to the limit 00:55:29.770 --> 00:55:32.140 align:middle line:84% as n goes to infinity of Tm of Bn. 00:55:32.140 --> 00:55:34.870 align:middle line:90% 00:55:34.870 --> 00:55:41.130 align:middle line:84% And now, the norm of Tm of Bn is always-- 00:55:41.130 --> 00:55:43.950 align:middle line:84% well, this is always less than or equal to k 00:55:43.950 --> 00:55:48.975 align:middle line:84% because the Bn's are in Ck and sup of all these guys 00:55:48.975 --> 00:55:50.100 align:middle line:90% is less than or equal to k. 00:55:50.100 --> 00:55:54.495 align:middle line:84% So that's also less than or equal to k. 00:55:54.495 --> 00:55:59.290 align:middle line:90% 00:55:59.290 --> 00:56:07.840 align:middle line:84% Thus, B is in Ck, which implies Ck is closed. 00:56:07.840 --> 00:56:12.520 align:middle line:90% 00:56:12.520 --> 00:56:15.550 align:middle line:90% Now, the closed ball-- 00:56:15.550 --> 00:56:18.340 align:middle line:84% or I shouldn't say-- it's not the closure of the open ball, 00:56:18.340 --> 00:56:21.790 align:middle line:84% but the closed ball, b in capital B, such 00:56:21.790 --> 00:56:25.420 align:middle line:84% that b is less than or equal to 1. 00:56:25.420 --> 00:56:31.690 align:middle line:84% This is equal to the union over all k of Ck. 00:56:31.690 --> 00:56:33.370 align:middle line:90% Why? 00:56:33.370 --> 00:56:36.670 align:middle line:84% So first off, each of these is contained in this closed ball. 00:56:36.670 --> 00:56:41.470 align:middle line:84% But I'm assuming by star that no matter what 00:56:41.470 --> 00:56:46.090 align:middle line:84% b is in the Banach space, sup over n of these 00:56:46.090 --> 00:56:47.450 align:middle line:90% is less than infinity. 00:56:47.450 --> 00:56:51.760 align:middle line:84% So there's always some integer k given a b-- 00:56:51.760 --> 00:56:54.640 align:middle line:84% so given a certain b in this set, 00:56:54.640 --> 00:56:58.780 align:middle line:84% I can always find a large enough integer such that this 00:56:58.780 --> 00:57:01.820 align:middle line:90% is less than that integer. 00:57:01.820 --> 00:57:03.490 align:middle line:90% So it has to-- 00:57:03.490 --> 00:57:06.760 align:middle line:84% every element in here has to lie in one of these sets C sub k 00:57:06.760 --> 00:57:08.695 align:middle line:90% So this is equal to the union. 00:57:08.695 --> 00:57:11.640 align:middle line:90% 00:57:11.640 --> 00:57:17.390 align:middle line:84% So this is a complete metric space 00:57:17.390 --> 00:57:20.090 align:middle line:84% because it's a closed subset of M. 00:57:20.090 --> 00:57:31.360 align:middle line:84% So this is a complete metric space written 00:57:31.360 --> 00:57:35.086 align:middle line:90% as the union of a-- 00:57:35.086 --> 00:57:37.150 align:middle line:90% a union of closed subsets. 00:57:37.150 --> 00:57:50.780 align:middle line:84% So by Baire's theorem, there exists 00:57:50.780 --> 00:58:10.500 align:middle line:84% one of these sets containing an open ball of the form B, B0, 00:58:10.500 --> 00:58:11.190 align:middle line:90% delta 0. 00:58:11.190 --> 00:58:14.870 align:middle line:90% 00:58:14.870 --> 00:58:18.577 align:middle line:84% So one of these Ck's contains an open ball. 00:58:18.577 --> 00:58:19.660 align:middle line:90% Now, we're going to show-- 00:58:19.660 --> 00:58:25.450 align:middle line:84% we're going to use this k to derive a uniform bound. 00:58:25.450 --> 00:58:32.155 align:middle line:90% So if b is in B and-- 00:58:32.155 --> 00:58:32.655 align:middle line:90% let's see. 00:58:32.655 --> 00:58:39.680 align:middle line:90% 00:58:39.680 --> 00:58:41.060 align:middle line:90% So let's write it this way. 00:58:41.060 --> 00:58:45.450 align:middle line:90% 00:58:45.450 --> 00:58:53.720 align:middle line:84% If b is in the open ball of radius delta 0 centered at 0, 00:58:53.720 --> 00:58:55.820 align:middle line:90% so this means i.e. 00:58:55.820 --> 00:58:59.270 align:middle line:84% the norm of b is less than delta 0. 00:58:59.270 --> 00:59:02.370 align:middle line:90% 00:59:02.370 --> 00:59:11.870 align:middle line:84% Then b0 plus b is in this open ball centered at B0. 00:59:11.870 --> 00:59:14.500 align:middle line:90% 00:59:14.500 --> 00:59:29.650 align:middle line:84% And therefore, sup n of Tn of b0 plus b norm is less than 00:59:29.650 --> 00:59:31.540 align:middle line:90% or equal to k. 00:59:31.540 --> 00:59:34.660 align:middle line:90% So this ball is contained in Ck. 00:59:34.660 --> 00:59:37.060 align:middle line:84% And therefore, if I take the sup over n 00:59:37.060 --> 00:59:38.840 align:middle line:84% of T sub n applied to this thing, 00:59:38.840 --> 00:59:40.300 align:middle line:90% this is less than or equal to k. 00:59:40.300 --> 00:59:49.560 align:middle line:90% 00:59:49.560 --> 00:59:54.720 align:middle line:84% But then I conclude that sup over n of Tn 00:59:54.720 --> 00:59:59.890 align:middle line:84% applied to B, which is less than or equal to sup n-- 00:59:59.890 --> 01:00:02.200 align:middle line:84% so I'm going to add and subtract B0. 01:00:02.200 --> 01:00:06.850 align:middle line:90% 01:00:06.850 --> 01:00:19.040 align:middle line:84% So let me actually write this as sup of Tn B0 plus Tn B0 01:00:19.040 --> 01:00:24.080 align:middle line:84% plus B. This is less than or equal to 01:00:24.080 --> 01:00:26.420 align:middle line:90% by the triangle inequality. 01:00:26.420 --> 01:00:28.310 align:middle line:84% And then carrying the sup through, 01:00:28.310 --> 01:00:40.880 align:middle line:84% this is less than or equal to sup n Tn B0 plus sup n Tn B0 01:00:40.880 --> 01:00:45.160 align:middle line:90% plus B. 01:00:45.160 --> 01:00:51.610 align:middle line:84% And so B0 is certainly in this ball, which is contained in Ck. 01:00:51.610 --> 01:00:59.210 align:middle line:84% So this is less than or equal to k. 01:00:59.210 --> 01:01:04.070 align:middle line:84% And this is still in that ball of radius 01:01:04.070 --> 01:01:05.990 align:middle line:90% delta 0 centered at B0. 01:01:05.990 --> 01:01:08.180 align:middle line:90% So it lies in CK. 01:01:08.180 --> 01:01:10.100 align:middle line:84% And therefore, this sup is also less than 01:01:10.100 --> 01:01:11.960 align:middle line:90% or equal to k equals 2k. 01:01:11.960 --> 01:01:17.450 align:middle line:84% So I've shown that if I take any element in the open ball 01:01:17.450 --> 01:01:28.150 align:middle line:84% of radius delta 0, then the sup over n of Tn of B is less than 01:01:28.150 --> 01:01:33.070 align:middle line:90% or equal to 2k, all right? 01:01:33.070 --> 01:01:35.170 align:middle line:84% And now, it's just a simple rescaling argument 01:01:35.170 --> 01:01:38.600 align:middle line:84% to show that the sup of the norms are now bounded. 01:01:38.600 --> 01:01:45.970 align:middle line:90% So suppose norm of B equals 1. 01:01:45.970 --> 01:01:59.350 align:middle line:84% Then the norm of Tn of delta 0 over 2 applied to B 01:01:59.350 --> 01:02:02.610 align:middle line:90% is less than or equal to-- 01:02:02.610 --> 01:02:07.320 align:middle line:84% so for all n certainly less than or equal to the sup over n 01:02:07.320 --> 01:02:10.110 align:middle line:84% of all of these, which is less than or equal to 2k 01:02:10.110 --> 01:02:13.110 align:middle line:84% because this is an element with norm delta 01:02:13.110 --> 01:02:15.270 align:middle line:84% 0 over 2, which is less than delta 0. 01:02:15.270 --> 01:02:19.440 align:middle line:84% So this is less than or equal to 2k, which implies-- 01:02:19.440 --> 01:02:21.120 align:middle line:90% so, again Tn is linear. 01:02:21.120 --> 01:02:31.820 align:middle line:90% So I should have reversed this. 01:02:31.820 --> 01:02:35.512 align:middle line:90% 01:02:35.512 --> 01:02:37.130 align:middle line:90% Let M be a natural number. 01:02:37.130 --> 01:02:40.480 align:middle line:90% 01:02:40.480 --> 01:02:44.230 align:middle line:90% Let B equal to 1. 01:02:44.230 --> 01:02:46.380 align:middle line:90% Yeah, OK. 01:02:46.380 --> 01:02:48.810 align:middle line:84% So then Tn applied to B is less than 01:02:48.810 --> 01:02:52.920 align:middle line:90% or equal to 4k over delta 0. 01:02:52.920 --> 01:02:58.950 align:middle line:84% Now, this holds for all B with length equal to 1. 01:02:58.950 --> 01:03:00.880 align:middle line:84% And therefore, the operator norm, 01:03:00.880 --> 01:03:04.230 align:middle line:84% which is the sup over all B, would normally equal to 1, 01:03:04.230 --> 01:03:08.430 align:middle line:84% is less than or equal to 4k over delta 0. 01:03:08.430 --> 01:03:10.080 align:middle line:90% This holds for all n. 01:03:10.080 --> 01:03:15.120 align:middle line:84% And therefore, the sup over n of the operator norm 01:03:15.120 --> 01:03:19.740 align:middle line:84% is less than or equal to 4k over delta 0, 01:03:19.740 --> 01:03:23.115 align:middle line:84% and therefore giving us the uniform bound. 01:03:23.115 --> 01:03:37.360 align:middle line:90% 01:03:37.360 --> 01:03:38.320 align:middle line:90% I have time for this? 01:03:38.320 --> 01:03:46.030 align:middle line:90% 01:03:46.030 --> 01:03:50.650 align:middle line:84% OK, I don't think I have time for the entire proof of what's 01:03:50.650 --> 01:03:52.530 align:middle line:90% to come. 01:03:52.530 --> 01:03:55.720 align:middle line:84% So I think we'll just stop there for now. 01:03:55.720 --> 01:04:02.000 align:middle line:90%