WEBVTT 00:00:00.000 --> 00:00:01.976 align:middle line:90% [SQUEAKING] 00:00:01.976 --> 00:00:03.952 align:middle line:90% [RUSTLING] 00:00:03.952 --> 00:00:06.916 align:middle line:90% [CLICKING] 00:00:06.916 --> 00:00:19.270 align:middle line:90% 00:00:19.270 --> 00:00:25.900 align:middle line:84% So let's continue with the big-name theorems, 00:00:25.900 --> 00:00:29.330 align:middle line:84% or the theorems with big names, or the big theorems 00:00:29.330 --> 00:00:38.210 align:middle line:84% with names, which the last one was a uniform boundedness 00:00:38.210 --> 00:00:44.480 align:middle line:84% theorem, which followed from Baire's category theorem, which 00:00:44.480 --> 00:00:46.400 align:middle line:90% let me again recall for you. 00:00:46.400 --> 00:00:52.350 align:middle line:90% 00:00:52.350 --> 00:01:11.150 align:middle line:84% If M is a complete metric space, C k 00:01:11.150 --> 00:01:26.780 align:middle line:84% is a collection of closed sets and M 00:01:26.780 --> 00:01:41.650 align:middle line:84% is equal to the union of these closed sets, then at least one 00:01:41.650 --> 00:01:51.760 align:middle line:84% of these closed sets has an interior point. 00:01:51.760 --> 00:01:54.400 align:middle line:84% Or another way of saying it, at least one of these C k's 00:01:54.400 --> 00:01:59.510 align:middle line:90% contains an open ball. 00:01:59.510 --> 00:02:03.220 align:middle line:90% 00:02:03.220 --> 00:02:06.820 align:middle line:84% So now, what we're going to prove 00:02:06.820 --> 00:02:09.699 align:middle line:84% as a consequence of the Baire category theorem 00:02:09.699 --> 00:02:20.570 align:middle line:84% is so-called "open mapping theorem," which 00:02:20.570 --> 00:02:24.890 align:middle line:84% says that surjective linear, bounded linear operators 00:02:24.890 --> 00:02:26.780 align:middle line:90% send open sets to open sets. 00:02:26.780 --> 00:02:31.410 align:middle line:84% It's kind of the backwards version of continuity. 00:02:31.410 --> 00:02:48.520 align:middle line:84% So if B1, B2 are Banach spaces and T 00:02:48.520 --> 00:02:52.120 align:middle line:84% is a bounded linear operator from B1 00:02:52.120 --> 00:03:04.860 align:middle line:84% to B2, which is surjective, meaning onto, 00:03:04.860 --> 00:03:11.560 align:middle line:84% then T is what's called an "open map." 00:03:11.560 --> 00:03:13.640 align:middle line:84% Or you would just say, "T is open," 00:03:13.640 --> 00:03:26.520 align:middle line:84% which means for all open u, for every open subset of B1, T of u 00:03:26.520 --> 00:03:31.030 align:middle line:90% is open in B2. 00:03:31.030 --> 00:03:33.535 align:middle line:84% So T takes open sets to open sets. 00:03:33.535 --> 00:03:36.830 align:middle line:90% 00:03:36.830 --> 00:03:38.470 align:middle line:90% So statement of the theorem-- 00:03:38.470 --> 00:03:40.840 align:middle line:84% if you have a bounded linear operator between two Banach 00:03:40.840 --> 00:03:45.610 align:middle line:84% spaces which is surjective, then T is an open map. 00:03:45.610 --> 00:03:54.710 align:middle line:90% 00:03:54.710 --> 00:03:58.640 align:middle line:84% So we're first going to specialize 00:03:58.640 --> 00:04:02.120 align:middle line:90% to one type of open set. 00:04:02.120 --> 00:04:12.782 align:middle line:84% And we're going to prove a specialized version of what 00:04:12.782 --> 00:04:13.490 align:middle line:90% we want to prove. 00:04:13.490 --> 00:04:16.490 align:middle line:84% And then we're going to show using the linearity of T, 00:04:16.490 --> 00:04:18.589 align:middle line:84% along with scaling and translation, 00:04:18.589 --> 00:04:22.010 align:middle line:84% that this, then, implies that open sets get 00:04:22.010 --> 00:04:22.930 align:middle line:90% mapped to open sets. 00:04:22.930 --> 00:04:25.420 align:middle line:84% So first, what we're going to prove is that-- 00:04:25.420 --> 00:04:38.240 align:middle line:90% 00:04:38.240 --> 00:04:49.370 align:middle line:84% let me recall this is the set of all B in B1 such that-- 00:04:49.370 --> 00:04:53.920 align:middle line:90% 00:04:53.920 --> 00:04:58.315 align:middle line:90% so this is the open ball in B1-- 00:04:58.315 --> 00:05:02.460 align:middle line:90% 00:05:02.460 --> 00:05:19.115 align:middle line:84% then the image of this open ball contains an open ball. 00:05:19.115 --> 00:05:24.800 align:middle line:90% 00:05:24.800 --> 00:05:33.335 align:middle line:84% So now, the image is in B2, an open ball in B2 centered at 0. 00:05:33.335 --> 00:05:37.920 align:middle line:90% 00:05:37.920 --> 00:05:44.580 align:middle line:84% So T of the open ball could be some of weird set, 00:05:44.580 --> 00:05:47.040 align:middle line:84% and we know that linear maps take 0 to 0. 00:05:47.040 --> 00:05:51.360 align:middle line:84% What we're saying is that there exists an open ball contained 00:05:51.360 --> 00:05:51.945 align:middle line:90% in the image. 00:05:51.945 --> 00:05:54.920 align:middle line:90% 00:05:54.920 --> 00:05:57.595 align:middle line:84% Now, like I said, once we prove this, 00:05:57.595 --> 00:05:58.970 align:middle line:84% we're going to use linearity of T 00:05:58.970 --> 00:06:03.710 align:middle line:84% to be able to shift these balls around, shift and scale 00:06:03.710 --> 00:06:05.450 align:middle line:84% these balls around so that we prove 00:06:05.450 --> 00:06:08.880 align:middle line:90% the result for every open u. 00:06:08.880 --> 00:06:11.270 align:middle line:90% But this is the heart of it. 00:06:11.270 --> 00:06:18.780 align:middle line:90% 00:06:18.780 --> 00:06:28.330 align:middle line:84% So since T is surjective, everything in B2 00:06:28.330 --> 00:06:29.350 align:middle line:90% gets mapped onto. 00:06:29.350 --> 00:06:32.890 align:middle line:90% 00:06:32.890 --> 00:06:38.530 align:middle line:84% So B2 is equal to the union over N natural number 00:06:38.530 --> 00:06:44.880 align:middle line:90% of T of B closure. 00:06:44.880 --> 00:06:49.890 align:middle line:84% So everything in B2 gets mapped to by something in B1. 00:06:49.890 --> 00:06:58.320 align:middle line:84% Everything in B1 lies in some ball centered at 0 in B1. 00:06:58.320 --> 00:07:01.890 align:middle line:84% Just take N bigger than the norm of fixed guy, 00:07:01.890 --> 00:07:04.860 align:middle line:84% and then its image will be contained in this. 00:07:04.860 --> 00:07:08.040 align:middle line:84% And we're taking closure just so that we can write B2 00:07:08.040 --> 00:07:11.160 align:middle line:84% as a union of closed sets, so that we can now 00:07:11.160 --> 00:07:12.630 align:middle line:90% apply Baire's theorem. 00:07:12.630 --> 00:07:16.430 align:middle line:90% 00:07:16.430 --> 00:07:26.880 align:middle line:84% So this implies by Baire that there exists a natural number 00:07:26.880 --> 00:07:41.470 align:middle line:84% n0 such that the image of this ball contains an interior point 00:07:41.470 --> 00:07:50.380 align:middle line:90% or it contains an open ball. 00:07:50.380 --> 00:07:51.895 align:middle line:90% Now, here's the thing-- 00:07:51.895 --> 00:07:55.190 align:middle line:90% 00:07:55.190 --> 00:08:02.000 align:middle line:84% T is linear so the image by T of the ball of radius n0 centered 00:08:02.000 --> 00:08:03.650 align:middle line:90% at 0 is the same as-- 00:08:03.650 --> 00:08:13.170 align:middle line:90% 00:08:13.170 --> 00:08:17.370 align:middle line:84% so what I'm saying is, I'm using a little bit of handy notation 00:08:17.370 --> 00:08:19.380 align:middle line:84% which I didn't introduce yet, but if I 00:08:19.380 --> 00:08:26.910 align:middle line:84% have a subset of a vector space and I have some number outside 00:08:26.910 --> 00:08:30.210 align:middle line:84% of it, what I mean by this is the set obtained 00:08:30.210 --> 00:08:32.070 align:middle line:84% by taking this set and multiplying it 00:08:32.070 --> 00:08:37.179 align:middle line:84% all by the scalar, which is meaningful for vector space. 00:08:37.179 --> 00:08:42.429 align:middle line:84% Now, by scaling, you can check that this thing 00:08:42.429 --> 00:08:45.280 align:middle line:84% under the closure is the same as n0 times 00:08:45.280 --> 00:08:49.210 align:middle line:84% T of the image of the open ball centered at 1. 00:08:49.210 --> 00:08:52.750 align:middle line:84% And then the closure's also with respect to scaling. 00:08:52.750 --> 00:09:10.160 align:middle line:84% So N0 times the closure of the image of B 0, 1 00:09:10.160 --> 00:09:14.200 align:middle line:84% contains an open ball so let me just draw, again, 00:09:14.200 --> 00:09:16.850 align:middle line:90% a picture to convince you. 00:09:16.850 --> 00:09:17.680 align:middle line:90% So now I have-- 00:09:17.680 --> 00:09:21.700 align:middle line:90% 00:09:21.700 --> 00:09:23.330 align:middle line:90% let's draw it like this-- 00:09:23.330 --> 00:09:24.300 align:middle line:90% this is n0. 00:09:24.300 --> 00:09:29.400 align:middle line:90% 00:09:29.400 --> 00:09:39.620 align:middle line:84% Now, if I take this set and now multiply it by 1 over n0, 00:09:39.620 --> 00:09:51.420 align:middle line:84% I just get the rescaled guy, which is, 00:09:51.420 --> 00:09:58.320 align:middle line:84% to use a fancy word, homeomorphic to n0 times 00:09:58.320 --> 00:09:59.580 align:middle line:90% the image of the open ball. 00:09:59.580 --> 00:10:03.750 align:middle line:90% 00:10:03.750 --> 00:10:05.490 align:middle line:84% So this is just a picture to back up 00:10:05.490 --> 00:10:08.190 align:middle line:84% what I'm about to say next, which you can also verify 00:10:08.190 --> 00:10:12.510 align:middle line:84% not just by pictures but by going through definitions, 00:10:12.510 --> 00:10:24.775 align:middle line:84% which implies that T B0 contains an open ball. 00:10:24.775 --> 00:10:43.340 align:middle line:90% 00:10:43.340 --> 00:10:45.300 align:middle line:90% So what's this mean? 00:10:45.300 --> 00:10:58.200 align:middle line:84% This means there exists some point B2 and a number r 00:10:58.200 --> 00:11:09.140 align:middle line:84% positive such that the ball of radius 4r-- 00:11:09.140 --> 00:11:11.510 align:middle line:84% so 4 just because it's going to come in 00:11:11.510 --> 00:11:16.850 align:middle line:84% handy later with arithmetic, so just by choosing r small-- 00:11:16.850 --> 00:11:26.180 align:middle line:84% is contained in the closure of the image 00:11:26.180 --> 00:11:29.680 align:middle line:90% of the open ball of radius 1. 00:11:29.680 --> 00:11:38.550 align:middle line:90% 00:11:38.550 --> 00:11:46.750 align:middle line:84% So this ball is contained in the closure 00:11:46.750 --> 00:11:51.010 align:middle line:84% and therefore, this is contained in the closure. 00:11:51.010 --> 00:12:00.630 align:middle line:84% And therefore, there exists a point v1 00:12:00.630 --> 00:12:09.150 align:middle line:84% which is equal to T of u1, which is in T. 00:12:09.150 --> 00:12:13.890 align:middle line:84% So v0 is in the closure, so I can find points 00:12:13.890 --> 00:12:16.548 align:middle line:84% from the image of this ball close to it. 00:12:16.548 --> 00:12:18.090 align:middle line:84% So that's just what I'm writing here. 00:12:18.090 --> 00:12:23.400 align:middle line:84% So there exists a v1 which is the image of some guy 00:12:23.400 --> 00:12:36.590 align:middle line:84% u1 in the open ball such that it's close to v0. 00:12:36.590 --> 00:12:39.110 align:middle line:90% 00:12:39.110 --> 00:12:42.240 align:middle line:84% And how close, so I don't mess this up? 00:12:42.240 --> 00:12:42.740 align:middle line:90% 2r. 00:12:42.740 --> 00:12:51.190 align:middle line:90% 00:12:51.190 --> 00:12:57.470 align:middle line:90% So why did I do that arithmetic? 00:12:57.470 --> 00:13:02.880 align:middle line:84% Simply because if I look at the ball of radius 00:13:02.880 --> 00:13:06.600 align:middle line:90% 2r centered at v1-- 00:13:06.600 --> 00:13:12.870 align:middle line:84% so v1 is within distance 2r to v0, so everything in here 00:13:12.870 --> 00:13:16.860 align:middle line:84% is going to be within distance 4r to v0. 00:13:16.860 --> 00:13:21.090 align:middle line:84% And since the ball of radius 4r is contained in here, 00:13:21.090 --> 00:13:21.900 align:middle line:90% I obtain-- 00:13:21.900 --> 00:13:24.310 align:middle line:90% 00:13:24.310 --> 00:13:26.060 align:middle line:84% so this is what I just said a minute ago-- 00:13:26.060 --> 00:13:34.250 align:middle line:84% this is contained in the ball of the image of the ball of radius 00:13:34.250 --> 00:13:36.290 align:middle line:90% 1, the closure of that. 00:13:36.290 --> 00:13:43.910 align:middle line:90% 00:13:43.910 --> 00:13:46.610 align:middle line:84% So now, what I'm going to show is that-- 00:13:46.610 --> 00:13:48.320 align:middle line:84% I almost have what I want to prove-- 00:13:48.320 --> 00:13:54.200 align:middle line:84% I'm going to show that the closure contains an open ball. 00:13:54.200 --> 00:13:56.450 align:middle line:84% That's not quite what I want to show because remember, 00:13:56.450 --> 00:14:00.170 align:middle line:84% I want to show that the image of the ball of radius 1 00:14:00.170 --> 00:14:01.520 align:middle line:90% contains an open ball. 00:14:01.520 --> 00:14:06.560 align:middle line:84% But what I'm about to show is that the closure of that 00:14:06.560 --> 00:14:09.470 align:middle line:90% contains an open ball. 00:14:09.470 --> 00:14:22.540 align:middle line:84% Now, if v is less than r, then if I look at this element 00:14:22.540 --> 00:14:31.030 align:middle line:84% 1/2 times 2v plus v1, this is in what? 00:14:31.030 --> 00:14:33.100 align:middle line:90% This is in 1/2-- 00:14:33.100 --> 00:14:38.560 align:middle line:84% so 2v plus v1, 2v has a norm less than 2r. 00:14:38.560 --> 00:14:44.140 align:middle line:84% So this element is in the ball the ball centered 00:14:44.140 --> 00:14:45.940 align:middle line:90% at v1 of radius 2r. 00:14:45.940 --> 00:14:48.460 align:middle line:84% So that's contained in the closure. 00:14:48.460 --> 00:14:50.620 align:middle line:84% So multiplying by 1/2, I get this 00:14:50.620 --> 00:15:06.690 align:middle line:84% is closure, which, as I've said, using the linearity of T 00:15:06.690 --> 00:15:10.650 align:middle line:84% and the homogeneity of the norm tells me 00:15:10.650 --> 00:15:14.430 align:middle line:84% that this element is in the closure 00:15:14.430 --> 00:15:19.930 align:middle line:84% of the ball of the image of the ball of radius 1/2. 00:15:19.930 --> 00:15:22.120 align:middle line:90% So this is not crazy. 00:15:22.120 --> 00:15:24.670 align:middle line:84% This is an element of this, so 1/2 00:15:24.670 --> 00:15:28.400 align:middle line:84% this ought to be an element of 1/2 times this set. 00:15:28.400 --> 00:15:31.990 align:middle line:84% And then the 1/2 comes all the way through because T is linear 00:15:31.990 --> 00:15:37.730 align:middle line:84% and the norm is homogeneous, so the 1/2 can come through. 00:15:37.730 --> 00:15:45.095 align:middle line:84% But as you're seeing this, think it through slowly. 00:15:45.095 --> 00:15:50.350 align:middle line:90% 00:15:50.350 --> 00:15:57.190 align:middle line:84% So that implies that v, which I can write as minus T of u1 00:15:57.190 --> 00:15:58.160 align:middle line:90% over 2. 00:15:58.160 --> 00:16:00.730 align:middle line:84% u1 here was just defined as an element 00:16:00.730 --> 00:16:03.130 align:middle line:84% of the ball of radius 1, which gave me 00:16:03.130 --> 00:16:07.180 align:middle line:90% v1, which was close to v0. 00:16:07.180 --> 00:16:11.140 align:middle line:84% So let me keep up this string of inequalities-- 00:16:11.140 --> 00:16:15.865 align:middle line:90% plus 1/2 times 2v plus v1. 00:16:15.865 --> 00:16:18.550 align:middle line:90% 00:16:18.550 --> 00:16:21.235 align:middle line:90% Now, this is an element of-- 00:16:21.235 --> 00:16:25.040 align:middle line:90% 00:16:25.040 --> 00:16:29.960 align:middle line:84% don't confuse what I'm about to write down with the notation 00:16:29.960 --> 00:16:33.860 align:middle line:84% we used when we were talking about quotients. 00:16:33.860 --> 00:16:35.240 align:middle line:90% That's not what I mean here. 00:16:35.240 --> 00:16:38.870 align:middle line:84% I mean take this set and add this fixed element to it. 00:16:38.870 --> 00:16:41.520 align:middle line:90% 00:16:41.520 --> 00:16:43.760 align:middle line:84% So this set here is the set of all elements 00:16:43.760 --> 00:16:47.540 align:middle line:84% of the form something in here plus this fixed element. 00:16:47.540 --> 00:16:50.430 align:middle line:90% 00:16:50.430 --> 00:16:56.730 align:middle line:84% And now again, by the linearity of T this set 00:16:56.730 --> 00:17:04.260 align:middle line:84% here is exactly the same as T of minus u1 over 2 00:17:04.260 --> 00:17:10.890 align:middle line:84% plus ball of radius 0 centered at 1/2, with the closure 00:17:10.890 --> 00:17:11.670 align:middle line:90% over all of that. 00:17:11.670 --> 00:17:15.089 align:middle line:84% And the closure respects everything 00:17:15.089 --> 00:17:16.170 align:middle line:90% that we're doing here. 00:17:16.170 --> 00:17:19.460 align:middle line:90% 00:17:19.460 --> 00:17:26.060 align:middle line:84% Now, u1 has radius or has norm less than 1. 00:17:26.060 --> 00:17:29.480 align:middle line:84% And therefore, u1 over 2 has norm less than 1/2. 00:17:29.480 --> 00:17:32.120 align:middle line:84% Everything in here has norm less than 1/2. 00:17:32.120 --> 00:17:36.140 align:middle line:84% So something with norm less than 1/2 plus something with norm 00:17:36.140 --> 00:17:39.170 align:middle line:84% less than 1/2 is something with norm less than 1. 00:17:39.170 --> 00:17:51.420 align:middle line:84% So this is contained in the ball of radius 1 centered 00:17:51.420 --> 00:17:56.390 align:middle line:84% at 0, and then closure, and take the image of T. 00:17:56.390 --> 00:18:01.180 align:middle line:84% So here, I'm using that this is contained 00:18:01.180 --> 00:18:03.265 align:middle line:90% in the ball of radius 1. 00:18:03.265 --> 00:18:11.290 align:middle line:90% 00:18:11.290 --> 00:18:13.390 align:middle line:84% So that's almost what we wanted to show. 00:18:13.390 --> 00:18:18.720 align:middle line:84% What we have shown is that the ball of radius r 00:18:18.720 --> 00:18:24.330 align:middle line:84% is contained in the closure of the ball of radius 00:18:24.330 --> 00:18:28.275 align:middle line:84% 1, the closure of the image of the ball of radius 1. 00:18:28.275 --> 00:18:36.920 align:middle line:90% 00:18:36.920 --> 00:18:44.500 align:middle line:84% Now, again, by scaling this, let me just-- 00:18:44.500 --> 00:18:56.780 align:middle line:90% 00:18:56.780 --> 00:19:00.580 align:middle line:90% now, the ball of radius r is-- 00:19:00.580 --> 00:19:03.590 align:middle line:90% 00:19:03.590 --> 00:19:06.410 align:middle line:84% so let me just write down what I'm going to write down. 00:19:06.410 --> 00:19:09.030 align:middle line:90% Then I'll explain it. 00:19:09.030 --> 00:19:13.770 align:middle line:84% And therefore, the ball of radius 2 to the minus 00:19:13.770 --> 00:19:21.390 align:middle line:84% n times r, which is equal to the set of elements of the form 2 00:19:21.390 --> 00:19:25.470 align:middle line:84% to the minus n times the ball of radius-- 00:19:25.470 --> 00:19:30.945 align:middle line:84% elements in the ball of radius r is contained in 2 00:19:30.945 --> 00:19:39.290 align:middle line:90% to the minus n, which equals. 00:19:39.290 --> 00:19:47.220 align:middle line:90% 00:19:47.220 --> 00:19:51.620 align:middle line:84% So all I did right here was say, again, 00:19:51.620 --> 00:19:57.620 align:middle line:84% by homogeneity of the norm, the ball of radius 00:19:57.620 --> 00:20:01.040 align:middle line:84% r or the ball of radius 2 to the minus n times r 00:20:01.040 --> 00:20:04.760 align:middle line:84% is the same as 2 to the minus n times all elements 00:20:04.760 --> 00:20:07.130 align:middle line:90% of the ball of radius r. 00:20:07.130 --> 00:20:11.450 align:middle line:84% And that's contained in this by what we've already proven. 00:20:11.450 --> 00:20:15.350 align:middle line:84% So this is contained in the set of all elements of the form 2 00:20:15.350 --> 00:20:19.550 align:middle line:84% to the minus n times things from this set. 00:20:19.550 --> 00:20:22.190 align:middle line:84% And again, by homogeneity, 2 to the minus n 00:20:22.190 --> 00:20:29.950 align:middle line:84% comes through here and then through here 00:20:29.950 --> 00:20:33.190 align:middle line:90% for all n in natural numbers. 00:20:33.190 --> 00:20:36.030 align:middle line:90% 00:20:36.030 --> 00:20:37.625 align:middle line:90% So let me put-- 00:20:37.625 --> 00:20:41.750 align:middle line:90% 00:20:41.750 --> 00:20:49.480 align:middle line:84% what I've proven is this and that. 00:20:49.480 --> 00:20:56.180 align:middle line:84% So that's almost what I wanted to prove right here, just 00:20:56.180 --> 00:20:57.823 align:middle line:84% I want to be able to drop the closure. 00:20:57.823 --> 00:20:59.240 align:middle line:84% And now what I'm going to prove is 00:20:59.240 --> 00:21:03.260 align:middle line:84% that I can drop the closure if I take 00:21:03.260 --> 00:21:06.920 align:middle line:90% the ball of radius r over 2. 00:21:06.920 --> 00:21:18.650 align:middle line:84% So now, we'll prove that the ball of radius r over 2 00:21:18.650 --> 00:21:29.600 align:middle line:84% is, in fact, contained in the image of the unit ball by T. 00:21:29.600 --> 00:21:33.980 align:middle line:84% Now, we don't have that immediately because we 00:21:33.980 --> 00:21:36.680 align:middle line:84% have the ball of radius 1/2 r is contained 00:21:36.680 --> 00:21:40.280 align:middle line:84% in the closure of the image of T of this. 00:21:40.280 --> 00:21:42.620 align:middle line:84% There's nothing that doesn't say that the closure 00:21:42.620 --> 00:21:45.755 align:middle line:84% of the image of this thing could end up being-- 00:21:45.755 --> 00:21:48.680 align:middle line:90% 00:21:48.680 --> 00:21:53.240 align:middle line:84% there's nothing that says that that has to be contained 00:21:53.240 --> 00:21:54.530 align:middle line:90% strictly within this ball. 00:21:54.530 --> 00:22:13.610 align:middle line:90% 00:22:13.610 --> 00:22:17.580 align:middle line:90% So let's show this. 00:22:17.580 --> 00:22:22.350 align:middle line:84% Let v have norm less than r over 2, so it's in this ball. 00:22:22.350 --> 00:22:27.960 align:middle line:84% Then, as we've proven up here, v is 00:22:27.960 --> 00:22:35.220 align:middle line:84% in the image of by T of the ball of radius 1/2. 00:22:35.220 --> 00:22:40.430 align:middle line:90% 00:22:40.430 --> 00:22:52.370 align:middle line:84% And therefore, there exists a b1 in the ball of 0, 00:22:52.370 --> 00:23:08.580 align:middle line:84% radius 1/2 such that the norm of v minus T of b1 is less than, 00:23:08.580 --> 00:23:14.700 align:middle line:90% let's say, r over 4. 00:23:14.700 --> 00:23:18.930 align:middle line:90% 00:23:18.930 --> 00:23:21.190 align:middle line:84% And now we're going to iterate this. 00:23:21.190 --> 00:23:25.140 align:middle line:84% So now, think of this element as lying in the ball of radius 00:23:25.140 --> 00:23:25.890 align:middle line:90% r over 4. 00:23:25.890 --> 00:23:31.000 align:middle line:90% 00:23:31.000 --> 00:23:33.970 align:middle line:84% And therefore, now I take n equals 2 in here. 00:23:33.970 --> 00:23:43.970 align:middle line:84% Thus, v minus T b1 is in the closure 00:23:43.970 --> 00:23:48.980 align:middle line:84% of the image of the ball of radius of 1/4, 00:23:48.980 --> 00:24:00.860 align:middle line:84% which implies there exists a v2 in the ball of radius 4 such 00:24:00.860 --> 00:24:10.510 align:middle line:84% that v minus T b1 minus T b2 norm is less than half 00:24:10.510 --> 00:24:14.260 align:middle line:84% of what we had before, so r over 8. 00:24:14.260 --> 00:24:18.430 align:middle line:84% But now you see the game that we're playing. 00:24:18.430 --> 00:24:20.320 align:middle line:84% This is now in the ball of radius 00:24:20.320 --> 00:24:24.460 align:middle line:84% r over 8, which is implies this is contained 00:24:24.460 --> 00:24:30.370 align:middle line:84% in the closure of the image of the ball centered 00:24:30.370 --> 00:24:32.410 align:middle line:90% at 0 of radius 1 over 8. 00:24:32.410 --> 00:24:37.330 align:middle line:84% And therefore, I can find a b3 in the ball of radius 1 00:24:37.330 --> 00:24:42.490 align:middle line:84% over 8 so that v minus T b1 minus T b2 minus T b3 00:24:42.490 --> 00:24:44.380 align:middle line:90% is less than r over 16. 00:24:44.380 --> 00:24:47.380 align:middle line:84% And then, we just continue inductively. 00:24:47.380 --> 00:24:57.160 align:middle line:84% So I said I would do the proper induction argument once 00:24:57.160 --> 00:24:58.213 align:middle line:90% and I did it last time. 00:24:58.213 --> 00:24:59.380 align:middle line:90% So I will never do it again. 00:24:59.380 --> 00:25:21.880 align:middle line:84% Continuing, we obtained a sequence b k of elements in B1 00:25:21.880 --> 00:25:32.975 align:middle line:84% such that the norm of b k is less than 2 to the minus k. 00:25:32.975 --> 00:25:35.540 align:middle line:90% 00:25:35.540 --> 00:25:39.370 align:middle line:90% And so there's that. 00:25:39.370 --> 00:25:48.960 align:middle line:84% And if I look at v minus sum from k equals 1 to n T of b k, 00:25:48.960 --> 00:25:57.620 align:middle line:84% this is less than 2 to the minus n minus 1 times r. 00:25:57.620 --> 00:25:59.570 align:middle line:90% Let me make sure my-- 00:25:59.570 --> 00:26:00.070 align:middle line:90% yeah. 00:26:00.070 --> 00:26:10.740 align:middle line:90% 00:26:10.740 --> 00:26:13.080 align:middle line:84% Now, we've used the fact that B2 is complete. 00:26:13.080 --> 00:26:15.405 align:middle line:84% We haven't used the fact that B1 is complete yet. 00:26:15.405 --> 00:26:19.090 align:middle line:90% 00:26:19.090 --> 00:26:21.750 align:middle line:84% So we ought to use it at some point. 00:26:21.750 --> 00:26:23.280 align:middle line:84% We're going to use it now by showing 00:26:23.280 --> 00:26:29.310 align:middle line:84% that v has to be, in fact, inside the image of the radius 00:26:29.310 --> 00:26:31.050 align:middle line:90% of the ball of radius 1. 00:26:31.050 --> 00:26:38.890 align:middle line:84% So these b k's form a Cauchy sequence-- 00:26:38.890 --> 00:26:42.770 align:middle line:90% 00:26:42.770 --> 00:26:44.810 align:middle line:84% or I shouldn't say they form a Cauchy sequence, 00:26:44.810 --> 00:26:46.775 align:middle line:84% but their sum forms a Cauchy sequence. 00:26:46.775 --> 00:26:49.290 align:middle line:90% 00:26:49.290 --> 00:26:50.570 align:middle line:90% Let me write it this way. 00:26:50.570 --> 00:27:03.880 align:middle line:90% 00:27:03.880 --> 00:27:11.610 align:middle line:84% The series is absolutely summable 00:27:11.610 --> 00:27:13.740 align:middle line:84% because the norm is bounded by 2 to the minus 00:27:13.740 --> 00:27:17.280 align:middle line:90% k, which is summable. 00:27:17.280 --> 00:27:21.600 align:middle line:84% And since B1 is a Banach space, the series 00:27:21.600 --> 00:27:27.790 align:middle line:84% must be summable, which implies there exist a B and B1 such 00:27:27.790 --> 00:27:34.890 align:middle line:84% that B is equal to the sum k equals 1 to infinity of b k. 00:27:34.890 --> 00:27:42.910 align:middle line:90% 00:27:42.910 --> 00:27:51.180 align:middle line:84% Moreover, the norm of b, which is equal to the limit 00:27:51.180 --> 00:28:01.170 align:middle line:84% as n goes to infinity of the norm of n k equals 1 b k, 00:28:01.170 --> 00:28:08.380 align:middle line:84% this is, by the triangle inequality, 00:28:08.380 --> 00:28:14.590 align:middle line:84% less than or equal to the limit as n goes to infinity of-- 00:28:14.590 --> 00:28:24.060 align:middle line:84% equals 1 to n b k, which is less than-- well, is equal to-- 00:28:24.060 --> 00:28:28.620 align:middle line:84% sum from k equals 1 to infinity of the norm of b k. 00:28:28.620 --> 00:28:33.660 align:middle line:84% And each of these is less than 2 to the minus k. 00:28:33.660 --> 00:28:37.590 align:middle line:84% So k equals 1 to infinity of 2 to the minus k. 00:28:37.590 --> 00:28:39.405 align:middle line:90% And that sum is just 1. 00:28:39.405 --> 00:28:43.100 align:middle line:90% 00:28:43.100 --> 00:28:51.080 align:middle line:84% So the norm of b is less than 1 using the first property 00:28:51.080 --> 00:28:52.850 align:middle line:90% of this sequence. 00:28:52.850 --> 00:28:54.710 align:middle line:84% The second property of this sequence 00:28:54.710 --> 00:29:04.240 align:middle line:84% then shows that v is equal to the image of b by T. 00:29:04.240 --> 00:29:17.520 align:middle line:84% Moreover, since T is continuous, T b, 00:29:17.520 --> 00:29:21.410 align:middle line:84% which is equal to the limit as n goes 00:29:21.410 --> 00:29:29.950 align:middle line:84% to infinity of t applied to k equals 1 to n b k, 00:29:29.950 --> 00:29:32.950 align:middle line:84% which equals, by linearity, limit 00:29:32.950 --> 00:29:42.180 align:middle line:84% as n goes to infinity of sum from k equals 1 to n T of b k. 00:29:42.180 --> 00:29:46.210 align:middle line:90% 00:29:46.210 --> 00:29:50.230 align:middle line:84% By the second property, the limit as n 00:29:50.230 --> 00:29:58.210 align:middle line:84% goes to infinity of T of b sub k is equal to v. 00:29:58.210 --> 00:30:11.200 align:middle line:84% And therefore, B is in the image of the ball of radius 1 by T. 00:30:11.200 --> 00:30:19.550 align:middle line:84% And thus, the ball of radius r over 2 00:30:19.550 --> 00:30:24.770 align:middle line:84% is contained in the image of the ball of radius 1. 00:30:24.770 --> 00:30:27.630 align:middle line:90% 00:30:27.630 --> 00:30:34.220 align:middle line:84% So that's the special-ish case that I wanted 00:30:34.220 --> 00:30:35.760 align:middle line:90% to prove in terms of open sets. 00:30:35.760 --> 00:30:41.860 align:middle line:84% So what I've shown is that, if you like, 00:30:41.860 --> 00:30:46.180 align:middle line:84% the interior point 0 remains an interior point. 00:30:46.180 --> 00:30:49.375 align:middle line:84% Now let's show that implies the full claim of what 00:30:49.375 --> 00:30:51.250 align:middle line:84% I want to prove for the open mapping theorem, 00:30:51.250 --> 00:30:56.260 align:middle line:84% that every open set gets mapped to an open set in B2. 00:30:56.260 --> 00:30:58.760 align:middle line:90% We'll just use translation. 00:30:58.760 --> 00:31:15.300 align:middle line:84% So in scaling again, suppose u subset of B1 is open and B2 00:31:15.300 --> 00:31:21.660 align:middle line:84% is the image of something in u, so this is in T of u. 00:31:21.660 --> 00:31:28.670 align:middle line:90% 00:31:28.670 --> 00:31:35.570 align:middle line:84% Then there exists an epsilon positive such that B1-- 00:31:35.570 --> 00:31:36.935 align:middle line:90% so remember, u is open-- 00:31:36.935 --> 00:31:40.200 align:middle line:90% 00:31:40.200 --> 00:31:47.770 align:middle line:84% so B1 plus all elements of the ball centered 00:31:47.770 --> 00:31:54.720 align:middle line:84% at 0 of radius epsilon and this, which 00:31:54.720 --> 00:31:59.340 align:middle line:84% is equal to the ball centered at B1 of radius epsilon 00:31:59.340 --> 00:32:01.905 align:middle line:84% is contained in u since it was open. 00:32:01.905 --> 00:32:07.620 align:middle line:90% 00:32:07.620 --> 00:32:14.540 align:middle line:84% Now, since there exists a delta positive 00:32:14.540 --> 00:32:20.480 align:middle line:84% such that ball of radius delta is contained 00:32:20.480 --> 00:32:28.930 align:middle line:84% in the ball of radius 0, 1, this implies 00:32:28.930 --> 00:32:32.680 align:middle line:90% that the ball of radius-- 00:32:32.680 --> 00:32:37.510 align:middle line:84% again, by homogeneity, the ball of radius epsilon times delta 00:32:37.510 --> 00:32:46.620 align:middle line:84% is contained in the image of the ball of radius epsilon, 00:32:46.620 --> 00:32:51.290 align:middle line:84% again, because T is linear and the norm is homogeneous. 00:32:51.290 --> 00:33:05.110 align:middle line:90% 00:33:05.110 --> 00:33:08.860 align:middle line:84% So let's go back to how we had it written down. 00:33:08.860 --> 00:33:21.390 align:middle line:84% So b2 plus-- so this is the of epsilon times delta. 00:33:21.390 --> 00:33:23.640 align:middle line:84% I got my delta and epsilons backwards. 00:33:23.640 --> 00:33:28.110 align:middle line:84% Anyways, this is equal to b2 plus epsilon 00:33:28.110 --> 00:33:32.010 align:middle line:90% times the ball of radius delta. 00:33:32.010 --> 00:33:37.230 align:middle line:84% So this is actually equal to the b2 plus the ball of radius 0 00:33:37.230 --> 00:33:39.690 align:middle line:84% centered at 0 of radius epsilon times delta. 00:33:39.690 --> 00:33:41.895 align:middle line:90% The epsilon I can pull out. 00:33:41.895 --> 00:33:47.930 align:middle line:90% 00:33:47.930 --> 00:33:55.580 align:middle line:84% Now, this is contained in b2 plus epsilon 00:33:55.580 --> 00:34:02.480 align:middle line:90% times the ball of radius 0, 1. 00:34:02.480 --> 00:34:05.920 align:middle line:84% Yeah, I don't know why I wrote this down. 00:34:05.920 --> 00:34:07.900 align:middle line:90% I don't think I needed it. 00:34:07.900 --> 00:34:11.650 align:middle line:90% Anyways, OK. 00:34:11.650 --> 00:34:20.290 align:middle line:84% And this is equal to T of b1 plus epsilon T of B 0, 1. 00:34:20.290 --> 00:34:23.750 align:middle line:84% And again, by linearity and homogeneity, 00:34:23.750 --> 00:34:31.150 align:middle line:84% this is equal to the image of b1 plus epsilon ball 00:34:31.150 --> 00:34:44.400 align:middle line:84% of radius 1, which I can say is b1 plus ball of radius epsilon. 00:34:44.400 --> 00:34:49.580 align:middle line:84% Now, this epsilon, remember, was chosen so that this ball here 00:34:49.580 --> 00:34:51.170 align:middle line:90% is contained within u. 00:34:51.170 --> 00:34:54.679 align:middle line:84% And therefore, this is contained within u 00:34:54.679 --> 00:34:59.330 align:middle line:84% and the image is, therefore, contained in the image of u. 00:34:59.330 --> 00:35:04.370 align:middle line:84% Yes, so I'm not sure why I decided to write this down. 00:35:04.370 --> 00:35:05.920 align:middle line:84% This is what threw me for a loop. 00:35:05.920 --> 00:35:10.000 align:middle line:84% But anyways, so this is the point-- 00:35:10.000 --> 00:35:12.310 align:middle line:84% that we can just take the special case 00:35:12.310 --> 00:35:17.500 align:middle line:84% and shift things around to get the general statement 00:35:17.500 --> 00:35:19.540 align:middle line:84% for open sets getting mapped to open sets. 00:35:19.540 --> 00:35:40.890 align:middle line:90% 00:35:40.890 --> 00:35:43.185 align:middle line:84% So from the open mapping theorem-- 00:35:43.185 --> 00:35:48.520 align:middle line:90% 00:35:48.520 --> 00:35:50.410 align:middle line:84% I don't know, it seems almost topological-- 00:35:50.410 --> 00:35:55.180 align:middle line:84% but we get what's called the closed graph theorem, which 00:35:55.180 --> 00:35:59.290 align:middle line:84% gives you sufficient conditions to be able to check 00:35:59.290 --> 00:36:00.710 align:middle line:90% if something is continuous. 00:36:00.710 --> 00:36:02.697 align:middle line:84% It's a little bit more convenient. 00:36:02.697 --> 00:36:04.280 align:middle line:84% And I'll explain why in just a second. 00:36:04.280 --> 00:36:11.135 align:middle line:84% So this is the closed graph theorem. 00:36:11.135 --> 00:36:13.660 align:middle line:90% 00:36:13.660 --> 00:36:21.683 align:middle line:84% But first, I need to state just a simple theorem 00:36:21.683 --> 00:36:23.725 align:middle line:84% before I actually state the closed graph theorem. 00:36:23.725 --> 00:36:26.530 align:middle line:90% 00:36:26.530 --> 00:36:37.540 align:middle line:84% If B1 and B2 are Banach spaces, then their Cartesian product, 00:36:37.540 --> 00:36:42.280 align:middle line:84% which I can give a natural vector space structure 00:36:42.280 --> 00:36:47.770 align:middle line:84% on from B1 and B2, just the sum of an ordered pair of elements 00:36:47.770 --> 00:36:49.450 align:middle line:90% is just the entry by entry sum. 00:36:49.450 --> 00:36:53.130 align:middle line:90% 00:36:53.130 --> 00:37:02.560 align:middle line:84% But I can also put a norm on it coming from these two, 00:37:02.560 --> 00:37:03.580 align:middle line:90% with norm-- 00:37:03.580 --> 00:37:09.490 align:middle line:84% so for an ordered pair b1 and b2, the norm of this 00:37:09.490 --> 00:37:16.840 align:middle line:84% is just defined to be the sum of the norms 00:37:16.840 --> 00:37:18.730 align:middle line:90% in the respective spaces. 00:37:18.730 --> 00:37:22.430 align:middle line:90% 00:37:22.430 --> 00:37:24.220 align:middle line:90% So this is a norm space. 00:37:24.220 --> 00:37:29.110 align:middle line:84% But moreover, if they're both Banach spaces, 00:37:29.110 --> 00:37:30.220 align:middle line:90% this is a Banach space. 00:37:30.220 --> 00:37:36.880 align:middle line:84% So it's not difficult to prove just based on the definition. 00:37:36.880 --> 00:37:44.065 align:middle line:90% 00:37:44.065 --> 00:37:45.440 align:middle line:84% I'm not going to write the proof. 00:37:45.440 --> 00:37:51.990 align:middle line:84% I will leave it to you, the proof. 00:37:51.990 --> 00:37:55.740 align:middle line:84% Again, it's not difficult. Simply 00:37:55.740 --> 00:37:59.310 align:middle line:84% from how we've defined the norm a Cauchy sequence in B1 00:37:59.310 --> 00:38:05.790 align:middle line:84% cross B2, the first entry will form a Cauchy sequence because 00:38:05.790 --> 00:38:09.750 align:middle line:84% of this definition of norm, and the second entry 00:38:09.750 --> 00:38:12.480 align:middle line:84% of the sequence will form a Cauchy sequence in B2. 00:38:12.480 --> 00:38:14.110 align:middle line:90% Both of those have limits. 00:38:14.110 --> 00:38:19.740 align:middle line:84% So then you can prove the sequence consisting of 00:38:19.740 --> 00:38:21.780 align:middle line:90% ordered pairs has a limit. 00:38:21.780 --> 00:38:24.960 align:middle line:84% It's the same way you can prove that R2 is complete 00:38:24.960 --> 00:38:26.280 align:middle line:90% assuming R1 is complete. 00:38:26.280 --> 00:38:29.090 align:middle line:90% 00:38:29.090 --> 00:38:39.270 align:middle line:84% All right, so now, I can state the closed graph theorem, 00:38:39.270 --> 00:38:42.740 align:middle line:90% which is the following-- 00:38:42.740 --> 00:38:54.380 align:middle line:84% if B1, B2 are Banach spaces and you have a linear operator 00:38:54.380 --> 00:38:56.390 align:middle line:90% from B1 to B2-- 00:38:56.390 --> 00:39:06.495 align:middle line:84% so all you know is it's linear, don't know that it's bounded-- 00:39:06.495 --> 00:39:10.070 align:middle line:90% 00:39:10.070 --> 00:39:12.110 align:middle line:84% then there's an equivalent condition 00:39:12.110 --> 00:39:14.880 align:middle line:84% you can check to see if it is, in fact, 00:39:14.880 --> 00:39:16.370 align:middle line:90% a bounded linear operator. 00:39:16.370 --> 00:39:21.090 align:middle line:90% 00:39:21.090 --> 00:39:25.950 align:middle line:84% Then T is a bounded linear operator 00:39:25.950 --> 00:39:32.190 align:middle line:84% from B1 to B2 if and only if the graph of T 00:39:32.190 --> 00:39:36.000 align:middle line:90% which is defined to be the set-- 00:39:36.000 --> 00:39:39.300 align:middle line:84% let's see, what notation do I use-- 00:39:39.300 --> 00:39:46.200 align:middle line:90% u, v such that v equals T u-- 00:39:46.200 --> 00:39:47.400 align:middle line:90% let me write it this way-- 00:39:47.400 --> 00:39:56.540 align:middle line:90% 00:39:56.540 --> 00:40:01.510 align:middle line:84% which is a subset of B1 cross B2, is closed. 00:40:01.510 --> 00:40:11.300 align:middle line:90% 00:40:11.300 --> 00:40:14.990 align:middle line:84% So a linear operator is a bounded linear operator 00:40:14.990 --> 00:40:21.140 align:middle line:84% if and only if the graph of this linear operator given by u, 00:40:21.140 --> 00:40:23.310 align:middle line:90% T u is closed. 00:40:23.310 --> 00:40:26.690 align:middle line:84% Now, why is this a little bit easier 00:40:26.690 --> 00:40:30.450 align:middle line:84% or I say convenient than just checking if something's 00:40:30.450 --> 00:40:33.910 align:middle line:90% a bounded linear operator? 00:40:33.910 --> 00:40:40.660 align:middle line:84% Well, maybe it's difficult to prove the boundedness property 00:40:40.660 --> 00:40:42.790 align:middle line:84% that we have that's equivalent to continuity. 00:40:42.790 --> 00:40:46.220 align:middle line:90% 00:40:46.220 --> 00:40:50.640 align:middle line:84% So a bounded linear operator is a linear operator 00:40:50.640 --> 00:40:51.480 align:middle line:90% which is continuous. 00:40:51.480 --> 00:40:54.300 align:middle line:84% So maybe it's difficult to prove the bound. 00:40:54.300 --> 00:40:56.480 align:middle line:84% So you have to go back and try and prove-- 00:40:56.480 --> 00:41:02.980 align:middle line:84% just for a God-given or instructor-given operator-- 00:41:02.980 --> 00:41:05.920 align:middle line:84% then you try and go back and prove continuity. 00:41:05.920 --> 00:41:11.170 align:middle line:84% And continuity says, well, take a sequence u n's converging 00:41:11.170 --> 00:41:12.240 align:middle line:90% to u. 00:41:12.240 --> 00:41:21.590 align:middle line:84% You then have to prove that T of u n converges to T of u. 00:41:21.590 --> 00:41:23.930 align:middle line:84% But there's kind of two statements in there. 00:41:23.930 --> 00:41:29.180 align:middle line:84% You have to prove T of u N converges 00:41:29.180 --> 00:41:32.900 align:middle line:90% and that limit is equal to T u. 00:41:32.900 --> 00:41:34.750 align:middle line:84% What the closed graph theorem does 00:41:34.750 --> 00:41:37.340 align:middle line:84% it eliminates one of those steps. 00:41:37.340 --> 00:41:39.440 align:middle line:90% Because to prove that the-- 00:41:39.440 --> 00:41:43.760 align:middle line:84% let's think about that, that the graph is closed in B1 cross B2. 00:41:43.760 --> 00:41:44.910 align:middle line:90% What does that mean? 00:41:44.910 --> 00:41:48.620 align:middle line:84% That means you have to check that it's closed under taking 00:41:48.620 --> 00:41:50.090 align:middle line:90% limits of sequences. 00:41:50.090 --> 00:41:54.230 align:middle line:84% So you have to show that given a sequence u n tending to u 00:41:54.230 --> 00:41:59.090 align:middle line:84% and T of u n converging to v, that v is equal to T u. 00:41:59.090 --> 00:42:03.440 align:middle line:84% So you get to assume already that T of u n converges. 00:42:03.440 --> 00:42:07.310 align:middle line:84% You just now have to check that the thing it converges to is 00:42:07.310 --> 00:42:12.650 align:middle line:84% actually equal to the image of the limit of the u n's. 00:42:12.650 --> 00:42:18.710 align:middle line:84% I hope that makes sense and explains 00:42:18.710 --> 00:42:20.810 align:middle line:84% why I said this is actually a little more 00:42:20.810 --> 00:42:26.105 align:middle line:84% convenient than continuity, or at least useful. 00:42:26.105 --> 00:42:29.390 align:middle line:90% 00:42:29.390 --> 00:42:33.590 align:middle line:90% So this is a two-way street. 00:42:33.590 --> 00:42:37.010 align:middle line:84% There's always, typically, one side 00:42:37.010 --> 00:42:38.450 align:middle line:90% of the street that's plowed. 00:42:38.450 --> 00:42:43.280 align:middle line:84% So let's do this direction, assuming 00:42:43.280 --> 00:42:45.170 align:middle line:84% that T is a bounded linear operator, 00:42:45.170 --> 00:42:46.820 align:middle line:84% and show that the graph is closed. 00:42:46.820 --> 00:42:49.370 align:middle line:90% 00:42:49.370 --> 00:42:53.890 align:middle line:84% So if T is a bounded linear operator-- 00:42:53.890 --> 00:43:00.740 align:middle line:90% 00:43:00.740 --> 00:43:03.935 align:middle line:84% let me just write "suppose" and then start a new sentence. 00:43:03.935 --> 00:43:08.390 align:middle line:90% 00:43:08.390 --> 00:43:27.990 align:middle line:84% So let u n T of u n be a sequence in the graph of T 00:43:27.990 --> 00:43:35.040 align:middle line:84% such that u n converges to u and T of u n converges 00:43:35.040 --> 00:43:39.260 align:middle line:84% to v. To show the graph is closed, 00:43:39.260 --> 00:43:44.780 align:middle line:84% we have to show that that the pair u, v is in the graph, 00:43:44.780 --> 00:43:47.030 align:middle line:90% that v is equal to T of u. 00:43:47.030 --> 00:43:50.120 align:middle line:84% But this follows from continuity. 00:43:50.120 --> 00:43:58.175 align:middle line:84% Then v is equal to the limit as n goes to infinity of T of u n. 00:43:58.175 --> 00:44:00.230 align:middle line:84% And since T is continuous, this is 00:44:00.230 --> 00:44:08.485 align:middle line:84% equal to [INAUDIBLE] u n, which equals T u. 00:44:08.485 --> 00:44:11.490 align:middle line:90% 00:44:11.490 --> 00:44:24.930 align:middle line:84% Thus, the ordered pair u, v is in the graph. 00:44:24.930 --> 00:44:31.370 align:middle line:84% So let's now prove the opposite direction. 00:44:31.370 --> 00:44:50.240 align:middle line:84% It's still not that difficult. So what I'm first going to do 00:44:50.240 --> 00:44:53.530 align:middle line:90% is I'm going to draw a diagram. 00:44:53.530 --> 00:44:56.570 align:middle line:84% This may be the only diagram I ever draw in this-- 00:44:56.570 --> 00:45:03.625 align:middle line:84% a commuting diagram-- maybe the only one I draw in this class. 00:45:03.625 --> 00:45:04.125 align:middle line:90% We'll see. 00:45:04.125 --> 00:45:07.650 align:middle line:90% 00:45:07.650 --> 00:45:12.730 align:middle line:90% So T takes B1 to B2. 00:45:12.730 --> 00:45:16.920 align:middle line:84% I have the graph of T. And now, I'm 00:45:16.920 --> 00:45:20.400 align:middle line:84% going to define two maps going from the graph 00:45:20.400 --> 00:45:24.570 align:middle line:90% to B1 and from the graph to B2. 00:45:24.570 --> 00:45:31.170 align:middle line:84% The graph sits as a subset of B1 cross B2. 00:45:31.170 --> 00:45:35.310 align:middle line:84% So this first map going from the graph 00:45:35.310 --> 00:45:37.230 align:middle line:84% is just going to be the projection 00:45:37.230 --> 00:45:39.210 align:middle line:90% onto the first entry. 00:45:39.210 --> 00:45:46.665 align:middle line:84% And pi 1 and then pi 2 will be the projection onto pi 2. 00:45:46.665 --> 00:45:49.410 align:middle line:90% 00:45:49.410 --> 00:45:51.270 align:middle line:84% Now, to finish this graph, I need 00:45:51.270 --> 00:45:58.415 align:middle line:84% to have an arrow going from B1 up to gamma of T. 00:45:58.415 --> 00:45:59.665 align:middle line:90% So I just first want to note-- 00:45:59.665 --> 00:46:07.250 align:middle line:90% 00:46:07.250 --> 00:46:08.750 align:middle line:84% actually before I note it, I'm going 00:46:08.750 --> 00:46:09.650 align:middle line:90% to go ahead and draw the arrow. 00:46:09.650 --> 00:46:11.060 align:middle line:84% And then I'll tell you what S is. 00:46:11.060 --> 00:46:14.930 align:middle line:90% 00:46:14.930 --> 00:46:22.310 align:middle line:84% So pi 1 with respect to the graph of T-- 00:46:22.310 --> 00:46:30.840 align:middle line:90% this is a surjective map. 00:46:30.840 --> 00:46:34.605 align:middle line:84% So let me actually define these things. 00:46:34.605 --> 00:46:42.130 align:middle line:90% 00:46:42.130 --> 00:46:49.480 align:middle line:84% A graph of T to B1 via pi 1 of an element of the form u, T u 00:46:49.480 --> 00:46:59.010 align:middle line:84% equals u, and then pi 2 from the graph of T to B2 00:46:59.010 --> 00:47:05.850 align:middle line:84% is just take the second element, u T u equals T u. 00:47:05.850 --> 00:47:12.800 align:middle line:90% 00:47:12.800 --> 00:47:16.460 align:middle line:90% So my point here-- 00:47:16.460 --> 00:47:29.640 align:middle line:84% so first note-- gamma of T is a Banach space. 00:47:29.640 --> 00:47:30.630 align:middle line:90% Why? 00:47:30.630 --> 00:47:38.080 align:middle line:84% Gamma of t is a subspace of B1 cross B2 because T is linear 00:47:38.080 --> 00:47:39.340 align:middle line:90% and it's closed. 00:47:39.340 --> 00:47:42.160 align:middle line:84% So a closed subspace of a Banach space 00:47:42.160 --> 00:47:58.180 align:middle line:84% is, again, a Banach space, since it's 00:47:58.180 --> 00:48:08.830 align:middle line:84% a closed subspace of the Banach space B1 00:48:08.830 --> 00:48:15.330 align:middle line:84% cross B2 with this norm that I defined in the earlier theorem. 00:48:15.330 --> 00:48:20.170 align:middle line:84% Now, pi 1 and pi 2-- these are both continuous, 00:48:20.170 --> 00:48:23.380 align:middle line:84% viewed now as maps from the Banach space 00:48:23.380 --> 00:48:26.710 align:middle line:90% of the graph to B1 and B2. 00:48:26.710 --> 00:48:32.770 align:middle line:90% 00:48:32.770 --> 00:48:42.250 align:middle line:84% Pi 1 is a bounded linear operator from the graph T, B1. 00:48:42.250 --> 00:48:52.230 align:middle line:84% And pi 2 is a bounded linear operator to B2. 00:48:52.230 --> 00:48:53.250 align:middle line:90% Why is this? 00:48:53.250 --> 00:48:57.890 align:middle line:90% I mean, this is pretty clear. 00:48:57.890 --> 00:49:03.860 align:middle line:84% So the graph of the norm of things in here are just-- 00:49:03.860 --> 00:49:10.190 align:middle line:84% so this is because if I take the norm of pi of-- let me write it 00:49:10.190 --> 00:49:14.600 align:middle line:84% u, v now, where v is standing for T u-- 00:49:14.600 --> 00:49:22.710 align:middle line:84% this is equal to v, which is less than or equal to-- 00:49:22.710 --> 00:49:26.840 align:middle line:90% so this is pi 2 of this one-- 00:49:26.840 --> 00:49:37.300 align:middle line:84% which is less than or equal to, and then the same thing 00:49:37.300 --> 00:49:38.140 align:middle line:90% with pi 1. 00:49:38.140 --> 00:49:44.270 align:middle line:90% 00:49:44.270 --> 00:49:50.120 align:middle line:84% So pi 1 and pi 2 going from the graph to B1 and B2-- 00:49:50.120 --> 00:49:53.310 align:middle line:84% these are bounded linear operators. 00:49:53.310 --> 00:49:59.940 align:middle line:84% And pi 1, when restricted to the graph is, in fact, bijective. 00:49:59.940 --> 00:50:03.780 align:middle line:90% It's one to one and onto. 00:50:03.780 --> 00:50:10.043 align:middle line:90% So I've used "moreover" again. 00:50:10.043 --> 00:50:11.710 align:middle line:84% So let me just write moreover more over. 00:50:11.710 --> 00:50:15.310 align:middle line:90% 00:50:15.310 --> 00:50:33.690 align:middle line:84% Pi one going from the graph to B1 is one to one and onto, 00:50:33.690 --> 00:50:34.620 align:middle line:90% bijective. 00:50:34.620 --> 00:50:37.510 align:middle line:90% 00:50:37.510 --> 00:50:42.820 align:middle line:84% Everything in B1 gets mapped to by pi 1 from the graph. 00:50:42.820 --> 00:50:47.680 align:middle line:84% If you have something u1 here, then its image is u1 T of u1. 00:50:47.680 --> 00:50:52.750 align:middle line:84% That's a unique element in the graph, since T is a function. 00:50:52.750 --> 00:50:56.350 align:middle line:84% There's only one element in the graph corresponding 00:50:56.350 --> 00:50:58.000 align:middle line:90% to a given u. 00:50:58.000 --> 00:51:03.090 align:middle line:90% 00:51:03.090 --> 00:51:08.520 align:middle line:84% And so let's pause the proof here 00:51:08.520 --> 00:51:10.500 align:middle line:84% because I forgot to write a corollary 00:51:10.500 --> 00:51:13.170 align:middle line:90% after the open mapping theorem. 00:51:13.170 --> 00:51:14.985 align:middle line:84% In fact, let's state it over here. 00:51:14.985 --> 00:51:18.450 align:middle line:90% 00:51:18.450 --> 00:51:23.570 align:middle line:84% So this was the end of the proof of the open mapping theorem. 00:51:23.570 --> 00:51:26.300 align:middle line:84% So there's space for the corollary, 00:51:26.300 --> 00:51:30.040 align:middle line:90% which I wanted to write here. 00:51:30.040 --> 00:51:44.110 align:middle line:84% It's the following-- if B1 B2 are Banach spaces, 00:51:44.110 --> 00:51:47.890 align:middle line:84% T is a bounded linear operator from B1 00:51:47.890 --> 00:51:55.740 align:middle line:84% to B2, which is bijective, meaning one to one and onto, 00:51:55.740 --> 00:52:03.780 align:middle line:84% so it has an inverse, then T inverse 00:52:03.780 --> 00:52:11.591 align:middle line:84% is a bounded linear operator from B2 to B1. 00:52:11.591 --> 00:52:16.360 align:middle line:90% 00:52:16.360 --> 00:52:19.540 align:middle line:84% So if I have a bounded linear operator from one Banach 00:52:19.540 --> 00:52:21.880 align:middle line:84% space to another and it's bijective, 00:52:21.880 --> 00:52:24.690 align:middle line:84% its inverse is automatically continuous. 00:52:24.690 --> 00:52:29.230 align:middle line:90% 00:52:29.230 --> 00:52:31.870 align:middle line:84% And the proof of that just follows from the open mapping 00:52:31.870 --> 00:52:32.920 align:middle line:90% theorem. 00:52:32.920 --> 00:52:36.700 align:middle line:84% So I'm going to write it in one line 00:52:36.700 --> 00:52:39.580 align:middle line:84% because that's all the space I'm going to give myself. 00:52:39.580 --> 00:52:46.330 align:middle line:84% T inverse is continuous if and only if, 00:52:46.330 --> 00:52:56.010 align:middle line:84% for all u that's open, for all open sets 00:52:56.010 --> 00:53:02.610 align:middle line:84% u and v1 which is in the image, the inverse image of T 00:53:02.610 --> 00:53:04.110 align:middle line:90% inverse-- 00:53:04.110 --> 00:53:06.240 align:middle line:84% or I should say the inverse image of u 00:53:06.240 --> 00:53:12.300 align:middle line:84% by T inverse, which you can just check is equal to T of u-- 00:53:12.300 --> 00:53:14.430 align:middle line:90% is open. 00:53:14.430 --> 00:53:17.880 align:middle line:84% And that's true by the open mapping theorem 00:53:17.880 --> 00:53:20.880 align:middle line:84% since a bijective map is surjective. 00:53:20.880 --> 00:53:27.990 align:middle line:84% So every bijective bounded linear operator automatically 00:53:27.990 --> 00:53:32.430 align:middle line:90% has a bounded inverse. 00:53:32.430 --> 00:53:33.420 align:middle line:90% It's also linear. 00:53:33.420 --> 00:53:37.320 align:middle line:84% I mean, I didn't say that, but if I 00:53:37.320 --> 00:53:39.870 align:middle line:84% have a linear operator which is bijective, 00:53:39.870 --> 00:53:43.150 align:middle line:90% then its inverse is also linear. 00:53:43.150 --> 00:53:48.840 align:middle line:84% So now coming back to this proof of the closed graph theorem. 00:53:48.840 --> 00:53:54.960 align:middle line:84% So we have the graph, which is a Banach space in its own right 00:53:54.960 --> 00:54:00.224 align:middle line:84% as a subset of B1 cross B2, as a closed subset of B1 cross B2. 00:54:00.224 --> 00:54:06.600 align:middle line:84% I have pi 1 and pi 2, which are bounded linear operators 00:54:06.600 --> 00:54:07.770 align:middle line:90% between B1 and B2. 00:54:07.770 --> 00:54:09.930 align:middle line:84% I didn't say that they're linear, 00:54:09.930 --> 00:54:12.690 align:middle line:90% but that should be clear. 00:54:12.690 --> 00:54:19.480 align:middle line:90% 00:54:19.480 --> 00:54:27.870 align:middle line:84% And pi 1, when restricted to the graph, is one to one and onto. 00:54:27.870 --> 00:54:32.100 align:middle line:84% Pi 1 here just takes the first element of what's in the graph 00:54:32.100 --> 00:54:33.060 align:middle line:90% and spits out-- 00:54:33.060 --> 00:54:38.130 align:middle line:84% let me, instead of having T u there, let me have v like that. 00:54:38.130 --> 00:54:41.510 align:middle line:90% 00:54:41.510 --> 00:54:43.490 align:middle line:84% We just know that v is equal to T u. 00:54:43.490 --> 00:54:46.850 align:middle line:90% 00:54:46.850 --> 00:54:50.000 align:middle line:84% And so this is one to one and onto. 00:54:50.000 --> 00:54:51.110 align:middle line:90% It's bijective. 00:54:51.110 --> 00:54:54.860 align:middle line:90% 00:54:54.860 --> 00:54:59.270 align:middle line:84% Thus, it has an inverse which is a bounded linear operator, 00:54:59.270 --> 00:55:01.310 align:middle line:84% by the corollary that I stated over there. 00:55:01.310 --> 00:55:13.610 align:middle line:90% 00:55:13.610 --> 00:55:25.450 align:middle line:84% It is a bounded linear operator, which implies 00:55:25.450 --> 00:55:32.330 align:middle line:90% that T, which I can write as-- 00:55:32.330 --> 00:55:35.780 align:middle line:84% I shouldn't have S inverse, defined to be the inverse. 00:55:35.780 --> 00:55:40.370 align:middle line:84% And therefore, T, which is equal to S of pi 1 00:55:40.370 --> 00:55:47.420 align:middle line:84% going from B1 to B2 is now the product 00:55:47.420 --> 00:55:52.640 align:middle line:84% of two bounded linear operators, pi 1 restricted from B1 00:55:52.640 --> 00:55:54.540 align:middle line:90% from B1 up to the graph. 00:55:54.540 --> 00:55:59.570 align:middle line:90% And then S-- no, no, no. 00:55:59.570 --> 00:56:00.860 align:middle line:90% I'm messing this all up. 00:56:00.860 --> 00:56:10.300 align:middle line:90% Hold on-- S pi 2. 00:56:10.300 --> 00:56:18.130 align:middle line:90% 00:56:18.130 --> 00:56:19.760 align:middle line:90% OK, now this makes sense. 00:56:19.760 --> 00:56:23.830 align:middle line:84% So S, which is the inverse of pi 1, 00:56:23.830 --> 00:56:26.260 align:middle line:90% is a bounded linear operator. 00:56:26.260 --> 00:56:28.180 align:middle line:84% Pi 2 is a bounded linear operator. 00:56:28.180 --> 00:56:29.890 align:middle line:84% And therefore, their composition is also 00:56:29.890 --> 00:56:33.790 align:middle line:84% a bounded linear operator, which implies that T 00:56:33.790 --> 00:56:35.440 align:middle line:90% is a bounded linear operator. 00:56:35.440 --> 00:56:44.848 align:middle line:90% 00:56:44.848 --> 00:56:47.390 align:middle line:84% So I made kind of a mess of that having to go back and forth, 00:56:47.390 --> 00:56:50.075 align:middle line:90% but the proof is simple enough. 00:56:50.075 --> 00:56:57.400 align:middle line:90% 00:56:57.400 --> 00:57:04.860 align:middle line:84% I'm sure Professor Melrose would have just drawn the picture, 00:57:04.860 --> 00:57:06.750 align:middle line:84% but I decided to make a mess of it. 00:57:06.750 --> 00:57:10.970 align:middle line:90% 00:57:10.970 --> 00:57:17.480 align:middle line:84% So those are some pretty important theorems 00:57:17.480 --> 00:57:19.580 align:middle line:84% that follow from the Baire category theorem. 00:57:19.580 --> 00:57:22.610 align:middle line:84% So we got uniform boundedness from Baire category. 00:57:22.610 --> 00:57:24.530 align:middle line:84% We got open mapping from Baire category. 00:57:24.530 --> 00:57:29.150 align:middle line:84% We got closed graph from open mapping. 00:57:29.150 --> 00:57:36.790 align:middle line:84% If you're a lover of logic, think about it a little bit-- 00:57:36.790 --> 00:57:39.730 align:middle line:84% open mapping implies closed graph, 00:57:39.730 --> 00:57:45.130 align:middle line:84% but you can also show that closed graph implies 00:57:45.130 --> 00:57:45.790 align:middle line:90% open mapping. 00:57:45.790 --> 00:57:48.435 align:middle line:84% So as logical statements, they're equivalent. 00:57:48.435 --> 00:57:51.160 align:middle line:90% 00:57:51.160 --> 00:58:05.490 align:middle line:84% Now, we're going to move on to the Hahn Banach theorem. 00:58:05.490 --> 00:58:07.170 align:middle line:84% So I haven't done many examples here 00:58:07.170 --> 00:58:11.410 align:middle line:84% going into this kind of general theory, but don't worry. 00:58:11.410 --> 00:58:16.110 align:middle line:84% There will be plenty of examples in the assignments of using 00:58:16.110 --> 00:58:18.110 align:middle line:90% these theorems, and so on. 00:58:18.110 --> 00:58:19.585 align:middle line:90% So the Hahn Banach theorem-- 00:58:19.585 --> 00:58:25.620 align:middle line:90% 00:58:25.620 --> 00:58:29.950 align:middle line:84% these theorems before were all kind of answering a question. 00:58:29.950 --> 00:58:34.515 align:middle line:84% Maybe I didn't state the questions as clearly. 00:58:34.515 --> 00:58:37.230 align:middle line:90% 00:58:37.230 --> 00:58:39.750 align:middle line:90% Closed graph is kind of-- 00:58:39.750 --> 00:58:41.790 align:middle line:84% well, it doesn't so much answer question 00:58:41.790 --> 00:58:45.330 align:middle line:84% as give us an alternative to proving continuity. 00:58:45.330 --> 00:58:48.240 align:middle line:84% Open mapping you can think of as trying 00:58:48.240 --> 00:58:49.830 align:middle line:84% to answer this question-- if I have 00:58:49.830 --> 00:58:52.140 align:middle line:84% a bijective bounded linear operator, 00:58:52.140 --> 00:58:56.790 align:middle line:84% is its inverse a bounded linear operator? 00:58:56.790 --> 00:59:00.270 align:middle line:84% And uniform boundedness is the answer 00:59:00.270 --> 00:59:02.640 align:middle line:84% to the question for at least a sequence 00:59:02.640 --> 00:59:05.280 align:middle line:84% of bounded linear operators, does pointwise convergence 00:59:05.280 --> 00:59:09.150 align:middle line:84% imply, or pointwise boundedness imply uniform boundedness? 00:59:09.150 --> 00:59:15.000 align:middle line:90% 00:59:15.000 --> 00:59:18.930 align:middle line:84% Now, the question that the Hahn Banach theorem tries to answer 00:59:18.930 --> 00:59:20.310 align:middle line:90% is the following-- 00:59:20.310 --> 00:59:23.900 align:middle line:90% 00:59:23.900 --> 00:59:49.410 align:middle line:84% given the general non-trivial normed norm space 00:59:49.410 --> 01:00:05.300 align:middle line:84% V, is the dual space given by simply the zero vector? 01:00:05.300 --> 01:00:09.320 align:middle line:84% So at the end of last class, I defined the dual space. 01:00:09.320 --> 01:00:14.930 align:middle line:84% Recall, this is equal to the bounded linear operators 01:00:14.930 --> 01:00:20.810 align:middle line:84% from the norm space V to the field of scalars which 01:00:20.810 --> 01:00:24.740 align:middle line:84% is a Banach space because it's a space 01:00:24.740 --> 01:00:30.230 align:middle line:84% of bounded linear operators from a norm space to a Banach space, 01:00:30.230 --> 01:00:32.890 align:middle line:90% so it's a Banach space. 01:00:32.890 --> 01:00:37.645 align:middle line:84% And we usually refer to elements of the dual as-- 01:00:37.645 --> 01:00:40.610 align:middle line:90% 01:00:40.610 --> 01:00:42.290 align:middle line:84% I'm not sure if I said this last time, 01:00:42.290 --> 01:00:45.590 align:middle line:84% but we don't refer to them as bounded linear operators 01:00:45.590 --> 01:00:48.860 align:middle line:84% from the vector space to the field of scalars, 01:00:48.860 --> 01:00:51.380 align:middle line:90% but as functionals. 01:00:51.380 --> 01:00:54.710 align:middle line:84% Because the classical space is where function spaces-- 01:00:54.710 --> 01:00:59.700 align:middle line:84% the classical Banach spaces were spaces of function. 01:00:59.700 --> 01:01:03.260 align:middle line:84% So the things that ate them and spat out a number 01:01:03.260 --> 01:01:08.750 align:middle line:84% were called functionals, so evolving 01:01:08.750 --> 01:01:12.500 align:middle line:84% from function of functions and functions of lines. 01:01:12.500 --> 01:01:16.470 align:middle line:90% 01:01:16.470 --> 01:01:19.710 align:middle line:84% So the question is, if I have just the norm space, 01:01:19.710 --> 01:01:24.510 align:middle line:84% is the dual space kind of nontrivial in general? 01:01:24.510 --> 01:01:28.590 align:middle line:84% So last time, I hinted that for certain spaces, 01:01:28.590 --> 01:01:33.690 align:middle line:84% you can actually write down the dual space explicitly. 01:01:33.690 --> 01:01:37.935 align:middle line:84% You can at least identify the dual space in an explicit way. 01:01:37.935 --> 01:01:45.100 align:middle line:90% 01:01:45.100 --> 01:01:49.060 align:middle line:84% I hinted last time that the dual of little l 01:01:49.060 --> 01:01:55.150 align:middle line:84% p prime is, in fact, equal to l p prime, 01:01:55.150 --> 01:02:03.740 align:middle line:84% where p prime is defined as the dual exponent. 01:02:03.740 --> 01:02:06.620 align:middle line:84% So 1 over p plus 1 over p prime equals 1. 01:02:06.620 --> 01:02:11.210 align:middle line:84% And this is for p bigger than or equal to 1 and less 01:02:11.210 --> 01:02:16.580 align:middle line:84% than infinity, but not for p equals infinity. 01:02:16.580 --> 01:02:18.820 align:middle line:84% And if you remember last time, C0, 01:02:18.820 --> 01:02:24.295 align:middle line:84% which is a set of sequences which converts to 0-- 01:02:24.295 --> 01:02:26.920 align:middle line:84% I can't remember if I wrote this down at the end of last class, 01:02:26.920 --> 01:02:32.860 align:middle line:84% but you can also identify its dual space with little l1. 01:02:32.860 --> 01:02:39.140 align:middle line:90% 01:02:39.140 --> 01:02:42.020 align:middle line:84% So just to give you some examples of spaces that do 01:02:42.020 --> 01:02:47.330 align:middle line:90% have nontrivial dual space-- 01:02:47.330 --> 01:02:50.870 align:middle line:84% examples of norm spaces which have nontrivial dual space 01:02:50.870 --> 01:02:52.595 align:middle line:84% are given by the little l p spaces. 01:02:52.595 --> 01:02:56.500 align:middle line:90% 01:02:56.500 --> 01:02:58.420 align:middle line:84% But now the question is, in general 01:02:58.420 --> 01:03:01.740 align:middle line:84% for a norm space, is the space of functionals, 01:03:01.740 --> 01:03:04.270 align:middle line:90% is the dual space nontrivial? 01:03:04.270 --> 01:03:07.570 align:middle line:84% And this is a statement of the Hahn Banach theorem 01:03:07.570 --> 01:03:12.850 align:middle line:84% that there's, in fact, a lot of elements in the dual space. 01:03:12.850 --> 01:03:17.260 align:middle line:84% Now, we're not going to get to the statement or proof of that 01:03:17.260 --> 01:03:18.010 align:middle line:90% in this lecture. 01:03:18.010 --> 01:03:26.270 align:middle line:90% 01:03:26.270 --> 01:03:33.420 align:middle line:84% Because we first need to at least go over, or state, 01:03:33.420 --> 01:03:38.188 align:middle line:84% or you can say result, axiom from set theory 01:03:38.188 --> 01:03:38.855 align:middle line:90% that we'll need. 01:03:38.855 --> 01:03:48.710 align:middle line:90% 01:03:48.710 --> 01:03:54.710 align:middle line:84% So first, like I said, we need an axiom 01:03:54.710 --> 01:03:58.110 align:middle line:84% or recall an axiom, a certain lemma, from set theory. 01:03:58.110 --> 01:04:06.100 align:middle line:84% So first, let me set down the appropriate terminology. 01:04:06.100 --> 01:04:22.900 align:middle line:84% So partial order on a set E is a relation, 01:04:22.900 --> 01:04:27.910 align:middle line:84% meaning just a subset of E cross E, denoted by this-- 01:04:27.910 --> 01:04:31.300 align:middle line:84% but we usually don't identify it as a subset of E cross E-- 01:04:31.300 --> 01:04:34.820 align:middle line:90% 01:04:34.820 --> 01:04:40.005 align:middle line:90% such that three things occur. 01:04:40.005 --> 01:04:42.255 align:middle line:84% So you think of it kind of as a less than or equal to. 01:04:42.255 --> 01:04:47.790 align:middle line:90% 01:04:47.790 --> 01:04:54.770 align:middle line:84% For all E in E, E is related to E. I will say less than 01:04:54.770 --> 01:04:56.780 align:middle line:84% or equal to, even though this may have nothing 01:04:56.780 --> 01:05:00.690 align:middle line:84% to do with less than or equal to. 01:05:00.690 --> 01:05:11.360 align:middle line:84% For all e and f in capital E, E less than or equal to f and f 01:05:11.360 --> 01:05:14.750 align:middle line:84% less than or equal to e, these two assumptions 01:05:14.750 --> 01:05:17.810 align:middle line:84% imply that e is the same element as f. 01:05:17.810 --> 01:05:20.350 align:middle line:90% 01:05:20.350 --> 01:05:26.060 align:middle line:84% And transitivity-- so this is reflexivity. 01:05:26.060 --> 01:05:28.920 align:middle line:84% I'm not sure what would you call this one. 01:05:28.920 --> 01:05:31.230 align:middle line:90% I can't exactly remember. 01:05:31.230 --> 01:05:37.800 align:middle line:84% For all e, f, g in E, the two assumptions E less than 01:05:37.800 --> 01:05:45.630 align:middle line:84% or equal to f and f less than or equal to g implies e less than 01:05:45.630 --> 01:05:47.530 align:middle line:90% or equal to g. 01:05:47.530 --> 01:05:51.585 align:middle line:84% So this is the definition of a partial order. 01:05:51.585 --> 01:05:57.070 align:middle line:90% 01:05:57.070 --> 01:06:13.690 align:middle line:84% So to go with this, we say an upper bound of a set 01:06:13.690 --> 01:06:27.680 align:middle line:84% D contained in E is an element e in E such 01:06:27.680 --> 01:06:35.040 align:middle line:84% that for all d in capital D, d is less than or equal to e. 01:06:35.040 --> 01:06:48.720 align:middle line:84% And a maximal element of the set E 01:06:48.720 --> 01:06:59.500 align:middle line:84% is an element e in E, which nothing lies bigger 01:06:59.500 --> 01:07:00.790 align:middle line:90% than it, essentially. 01:07:00.790 --> 01:07:05.240 align:middle line:84% Nothing maximizes it or majorizes it, 01:07:05.240 --> 01:07:19.860 align:middle line:84% such that if f is in E and e is less than or equal to f, then e 01:07:19.860 --> 01:07:21.940 align:middle line:90% is equal to f. 01:07:21.940 --> 01:07:33.240 align:middle line:84% And a similar definition for a minimal element. 01:07:33.240 --> 01:07:35.270 align:middle line:84% So this was the definition of a maximal element. 01:07:35.270 --> 01:07:39.140 align:middle line:90% 01:07:39.140 --> 01:07:44.450 align:middle line:84% And this is the definition for a minimal element. 01:07:44.450 --> 01:07:47.270 align:middle line:84% Now, note the maximum element may not 01:07:47.270 --> 01:07:50.190 align:middle line:84% sit above everything in E necessarily. 01:07:50.190 --> 01:07:54.770 align:middle line:84% It may just be kind of off to the side of everything in E. 01:07:54.770 --> 01:07:59.650 align:middle line:84% Because this doesn't assert that you can always 01:07:59.650 --> 01:08:03.850 align:middle line:84% check to see if between two elements, 01:08:03.850 --> 01:08:05.260 align:middle line:90% if one is bigger than the other. 01:08:05.260 --> 01:08:09.010 align:middle line:84% That's something a little more restrictive, 01:08:09.010 --> 01:08:10.450 align:middle line:90% which is the following-- 01:08:10.450 --> 01:08:14.040 align:middle line:90% 01:08:14.040 --> 01:08:24.109 align:middle line:84% if E and less than or equal to is a partially ordered set, 01:08:24.109 --> 01:08:41.840 align:middle line:84% meaning a set with a partial order, a chain in E-- 01:08:41.840 --> 01:08:46.189 align:middle line:90% 01:08:46.189 --> 01:08:50.600 align:middle line:84% so maybe a better way to say this is a set C is a chain if, 01:08:50.600 --> 01:08:55.510 align:middle line:84% but a chain in E is a set C such that for all e, 01:08:55.510 --> 01:09:06.910 align:middle line:84% f in C, either e is less than or equal to f or f 01:09:06.910 --> 01:09:10.660 align:middle line:90% is less than or equal to e. 01:09:10.660 --> 01:09:13.210 align:middle line:84% So a chain is something so that you 01:09:13.210 --> 01:09:15.580 align:middle line:84% can, for any two elements in the set, 01:09:15.580 --> 01:09:18.520 align:middle line:84% compare whether one is bigger than the other. 01:09:18.520 --> 01:09:21.040 align:middle line:90% That's what a chain is. 01:09:21.040 --> 01:09:25.420 align:middle line:84% But for a general partial order, it doesn't necessarily 01:09:25.420 --> 01:09:27.154 align:middle line:84% need to be the case that you can always 01:09:27.154 --> 01:09:29.029 align:middle line:84% check to see if one is bigger than the other. 01:09:29.029 --> 01:09:31.210 align:middle line:84% For example, your partial order could be 01:09:31.210 --> 01:09:33.609 align:middle line:90% on the power set of some set. 01:09:33.609 --> 01:09:36.319 align:middle line:84% And the partial order is inclusion, 01:09:36.319 --> 01:09:40.899 align:middle line:84% whether one subset is a contained in another. 01:09:40.899 --> 01:09:42.760 align:middle line:84% Then it satisfies these three properties, 01:09:42.760 --> 01:09:46.180 align:middle line:84% but there's sets that cannot be compared to each other. 01:09:46.180 --> 01:09:52.790 align:middle line:90% 01:09:52.790 --> 01:10:00.265 align:middle line:84% So let me write this as lemma due to Zorn. 01:10:00.265 --> 01:10:01.515 align:middle line:90% We're not going to prove this. 01:10:01.515 --> 01:10:05.080 align:middle line:90% 01:10:05.080 --> 01:10:10.110 align:middle line:84% Even though I'm writing "lemma," just take it as an axiom, 01:10:10.110 --> 01:10:17.490 align:middle line:84% an axiom of Fraenkel set theory that goes with it, 01:10:17.490 --> 01:10:29.310 align:middle line:84% which is the following-- that if every chain in a nonempty-- 01:10:29.310 --> 01:10:32.190 align:middle line:84% of course, we're considering nontrivial stuff, so 01:10:32.190 --> 01:10:33.360 align:middle line:90% in a nonempty-- 01:10:33.360 --> 01:10:39.070 align:middle line:90% 01:10:39.070 --> 01:10:51.250 align:middle line:84% partially ordered set, E has an upper bound, 01:10:51.250 --> 01:11:02.515 align:middle line:90% then E has a maximal element. 01:11:02.515 --> 01:11:05.550 align:middle line:90% 01:11:05.550 --> 01:11:09.720 align:middle line:84% So if you can check that every chain has an upper bound, 01:11:09.720 --> 01:11:13.950 align:middle line:84% then you get to conclude that the partially ordered 01:11:13.950 --> 01:11:17.380 align:middle line:90% set has a maximal element. 01:11:17.380 --> 01:11:23.590 align:middle line:84% Now, we'll give a simple application 01:11:23.590 --> 01:11:28.850 align:middle line:84% of this at the start of the next lecture. 01:11:28.850 --> 01:11:36.550 align:middle line:84% But first, let me put this into your brains to marinate on. 01:11:36.550 --> 01:11:40.270 align:middle line:84% So we're going to use Zorn's Lemma to prove the Hahn Banach 01:11:40.270 --> 01:11:40.770 align:middle line:90% theorem. 01:11:40.770 --> 01:11:46.810 align:middle line:84% And I'll go into why that is next time. 01:11:46.810 --> 01:11:49.240 align:middle line:84% But you can use Zorn's Lemma to prove other things, 01:11:49.240 --> 01:11:51.470 align:middle line:84% and it is used to prove other things. 01:11:51.470 --> 01:11:55.390 align:middle line:84% And first off, from Zorn's Lemma you 01:11:55.390 --> 01:12:00.743 align:middle line:84% can, in fact, prove the axiom of choice, 01:12:00.743 --> 01:12:02.410 align:middle line:84% which says given any collection of sets, 01:12:02.410 --> 01:12:11.730 align:middle line:84% you can essentially choose an element from each set stated 01:12:11.730 --> 01:12:13.470 align:middle line:90% in a very precise way. 01:12:13.470 --> 01:12:15.240 align:middle line:84% Another way, which we're going to use 01:12:15.240 --> 01:12:18.300 align:middle line:84% at the beginning of next time, is to prove the following-- 01:12:18.300 --> 01:12:21.540 align:middle line:84% but first let me make a definition. 01:12:21.540 --> 01:12:26.770 align:middle line:90% 01:12:26.770 --> 01:12:32.780 align:middle line:84% A Hamel basis, H which is a subset-- 01:12:32.780 --> 01:12:34.670 align:middle line:90% not a subspace, but a subset-- 01:12:34.670 --> 01:12:55.610 align:middle line:84% of V, a vector space, is a linearly independent set 01:12:55.610 --> 01:13:20.560 align:middle line:84% such that every element of v is a finite linear combination 01:13:20.560 --> 01:13:24.070 align:middle line:90% of elements of H. 01:13:24.070 --> 01:13:30.540 align:middle line:84% So from linear algebra, for finite dimensions, 01:13:30.540 --> 01:13:35.850 align:middle line:84% this is in some sense how one can define the dimension is you 01:13:35.850 --> 01:13:40.130 align:middle line:84% find a basis and then the cardinality of that basis 01:13:40.130 --> 01:13:42.620 align:middle line:90% is always the same. 01:13:42.620 --> 01:13:48.410 align:middle line:84% So a Hamel basis for R n is just the vectors 01:13:48.410 --> 01:13:53.190 align:middle line:84% with 1 in one of the entries and 0 otherwise. 01:13:53.190 --> 01:13:57.710 align:middle line:84% So for R 2, it's just 1, 0, and 0, 1. 01:13:57.710 --> 01:14:02.910 align:middle line:84% A Hamel basis for, let's say, little l1 would 01:14:02.910 --> 01:14:07.560 align:middle line:84% be one in the first, followed by 0's, 1 01:14:07.560 --> 01:14:11.640 align:middle line:84% in the second, 0 elsewhere, 1 in the third, 0 elsewhere-- 01:14:11.640 --> 01:14:12.750 align:middle line:90% the set of these elements. 01:14:12.750 --> 01:14:16.060 align:middle line:90% 01:14:16.060 --> 01:14:18.970 align:middle line:84% Now, the question is, does every vector space 01:14:18.970 --> 01:14:21.370 align:middle line:90% have a Hamel basis? 01:14:21.370 --> 01:14:24.220 align:middle line:84% And using, next time, Zorn's Lemma, 01:14:24.220 --> 01:14:29.110 align:middle line:84% we'll show that indeed, every vector space has a Hamel basis. 01:14:29.110 --> 01:14:37.050 align:middle line:84% And in fact, that Hamel basis can be quite big. 01:14:37.050 --> 01:14:42.450 align:middle line:84% But that'll be a simple application we do next time, 01:14:42.450 --> 01:14:46.530 align:middle line:84% is that via Zorn, you can show that every vector 01:14:46.530 --> 01:14:48.870 align:middle line:90% space has a Hamel basis. 01:14:48.870 --> 01:14:50.870 align:middle line:90% We'll stop there. 01:14:50.870 --> 01:14:57.000 align:middle line:90%