WEBVTT 00:00:00.000 --> 00:00:02.475 align:middle line:90% [SQUEAKING] 00:00:02.475 --> 00:00:03.960 align:middle line:90% [RUSTLING] 00:00:03.960 --> 00:00:06.930 align:middle line:90% [CLICKING] 00:00:06.930 --> 00:00:18.810 align:middle line:90% 00:00:18.810 --> 00:00:22.770 align:middle line:84% CASEY RODRIGUEZ: All right, so today, we 00:00:22.770 --> 00:00:26.890 align:middle line:84% are going to prove the theorem that I mentioned last time, 00:00:26.890 --> 00:00:30.310 align:middle line:84% the Hahn-Banach theorem, which is 00:00:30.310 --> 00:00:34.480 align:middle line:84% a theorem about being able to extend 00:00:34.480 --> 00:00:39.100 align:middle line:84% a bounded linear functional on some subspace of a norm space 00:00:39.100 --> 00:00:40.850 align:middle line:90% to all of the norm space. 00:00:40.850 --> 00:00:43.960 align:middle line:84% And this will, therefore, answer the question 00:00:43.960 --> 00:00:47.680 align:middle line:84% that we posed at the beginning of this topic, 00:00:47.680 --> 00:00:51.640 align:middle line:84% whether or not the dual, which is 00:00:51.640 --> 00:00:54.820 align:middle line:84% the space of all bounded linear functions on a norm space, 00:00:54.820 --> 00:01:00.012 align:middle line:84% is nontrivial for every norm space. 00:01:00.012 --> 00:01:01.720 align:middle line:84% Now, one of the tools we're going to need 00:01:01.720 --> 00:01:06.560 align:middle line:84% is this axiom or lemma from set theory, 00:01:06.560 --> 00:01:10.510 align:middle line:84% which is due to Zorn, which is as following-- 00:01:10.510 --> 00:01:12.407 align:middle line:90% 00:01:12.407 --> 00:01:14.740 align:middle line:84% and you're going to have to recall from the last lecture 00:01:14.740 --> 00:01:16.780 align:middle line:90% what some of these words mean. 00:01:16.780 --> 00:01:19.330 align:middle line:84% So Zorn's Lemma states that if every chain 00:01:19.330 --> 00:01:23.980 align:middle line:84% in a nonempty, partially ordered set E has an upper bound, 00:01:23.980 --> 00:01:25.660 align:middle line:90% then E has a maximal element. 00:01:25.660 --> 00:01:28.210 align:middle line:90% So a partially ordered set E-- 00:01:28.210 --> 00:01:31.600 align:middle line:84% that means the set E with a relation 00:01:31.600 --> 00:01:33.130 align:middle line:84% that's basically like a less than 00:01:33.130 --> 00:01:36.110 align:middle line:84% or equal to that satisfies three properties, 00:01:36.110 --> 00:01:39.820 align:middle line:84% so that it's like an extended less than or equal to. 00:01:39.820 --> 00:01:43.930 align:middle line:84% And a chain is a subset of E so that any two elements 00:01:43.930 --> 00:01:49.860 align:middle line:90% in that subset can be compared. 00:01:49.860 --> 00:01:53.455 align:middle line:84% One is either bigger than or equal to the other. 00:01:53.455 --> 00:01:55.455 align:middle line:84% One is always bigger than or equal to the other. 00:01:55.455 --> 00:01:58.060 align:middle line:90% 00:01:58.060 --> 00:02:01.650 align:middle line:84% And so this theorem says that if every chain of a partially 00:02:01.650 --> 00:02:07.850 align:middle line:84% ordered set has an upper bound-- that has a pretty clear 00:02:07.850 --> 00:02:08.449 align:middle line:90% meaning-- 00:02:08.449 --> 00:02:10.340 align:middle line:90% then E has a maximal element. 00:02:10.340 --> 00:02:12.470 align:middle line:90% So a maximal element of E-- 00:02:12.470 --> 00:02:15.260 align:middle line:84% that means an element that is not 00:02:15.260 --> 00:02:20.520 align:middle line:84% less than or equal to something other than itself. 00:02:20.520 --> 00:02:24.050 align:middle line:84% So anything bigger than or equal to this maximal element 00:02:24.050 --> 00:02:26.690 align:middle line:90% has to be that element. 00:02:26.690 --> 00:02:30.590 align:middle line:84% A maximal element is not necessarily an upper bound. 00:02:30.590 --> 00:02:33.670 align:middle line:84% A maximal element just means it cannot be-- 00:02:33.670 --> 00:02:36.650 align:middle line:90% nothing can get over its head. 00:02:36.650 --> 00:02:39.330 align:middle line:90% 00:02:39.330 --> 00:02:42.840 align:middle line:84% And so as a warmup, we're going to use Zorn's Lemma 00:02:42.840 --> 00:02:48.550 align:middle line:84% to prove a fact about vector spaces. 00:02:48.550 --> 00:02:51.980 align:middle line:90% So a Hamel basis-- 00:02:51.980 --> 00:02:54.480 align:middle line:84% and I think I went over this at the end of class last time-- 00:02:54.480 --> 00:03:00.390 align:middle line:84% of a vector space V, this is a linear independent set, H, 00:03:00.390 --> 00:03:03.870 align:middle line:84% such that every element of V is a finite linear combination 00:03:03.870 --> 00:03:06.900 align:middle line:84% of the elements from H. So for example, 00:03:06.900 --> 00:03:10.260 align:middle line:84% this set consisting of the vector 1, 00:03:10.260 --> 00:03:14.580 align:middle line:84% 0, and the vector 0, 1, this is a Hamel basis for R2. 00:03:14.580 --> 00:03:16.440 align:middle line:84% Every element in R2 can be written 00:03:16.440 --> 00:03:21.720 align:middle line:84% as a finite linear combination of these guys. 00:03:21.720 --> 00:03:26.730 align:middle line:84% And the fact that finite-dimensional spaces have 00:03:26.730 --> 00:03:30.720 align:middle line:84% bases is something you discuss in linear algebra, 00:03:30.720 --> 00:03:32.530 align:middle line:84% but now we're in functional analysis, 00:03:32.530 --> 00:03:36.150 align:middle line:84% which is linear algebra in infinite dimensions. 00:03:36.150 --> 00:03:42.340 align:middle line:84% It's not so clear that every vector space has a Hamel basis. 00:03:42.340 --> 00:03:44.580 align:middle line:84% And so what we're going to do is we're 00:03:44.580 --> 00:03:47.910 align:middle line:84% going to apply Zorn's Lemma to prove that every vector 00:03:47.910 --> 00:03:49.510 align:middle line:90% space does have a Hamel basis. 00:03:49.510 --> 00:03:52.050 align:middle line:84% So this is the following theorem, 00:03:52.050 --> 00:03:55.610 align:middle line:84% and this argument will kind of be 00:03:55.610 --> 00:03:59.240 align:middle line:84% a warmup for how we'll apply Zorn to prove 00:03:59.240 --> 00:04:00.680 align:middle line:90% the Hahn-Banach theorem. 00:04:00.680 --> 00:04:14.460 align:middle line:84% So if V is a vector space, then V has a Hamel basis. 00:04:14.460 --> 00:04:19.279 align:middle line:90% 00:04:19.279 --> 00:04:28.570 align:middle line:84% So for the proof, I am going to apply Zorn. 00:04:28.570 --> 00:04:30.780 align:middle line:84% So I need to have some ordered set. 00:04:30.780 --> 00:04:36.600 align:middle line:84% And my ordered set is going to be 00:04:36.600 --> 00:04:53.180 align:middle line:84% let E be the set of all linearly independent subsets of V. 00:04:53.180 --> 00:04:55.760 align:middle line:84% And we're going to define an order on E. 00:04:55.760 --> 00:04:59.090 align:middle line:84% So E is the set of all subsets of V 00:04:59.090 --> 00:05:01.140 align:middle line:90% that are linearly independent. 00:05:01.140 --> 00:05:22.090 align:middle line:84% And we define partial order on E by inclusion. 00:05:22.090 --> 00:05:25.300 align:middle line:84% So the elements of E are subsets of V. 00:05:25.300 --> 00:05:28.810 align:middle line:84% So we'll say subset is less than or equal to another subset 00:05:28.810 --> 00:05:30.770 align:middle line:90% if one is included in the other. 00:05:30.770 --> 00:05:43.110 align:middle line:84% So e and e prime in V, then we'll say e is less than 00:05:43.110 --> 00:05:47.760 align:middle line:84% or equal to e prime if and only if this subset of V e 00:05:47.760 --> 00:05:51.450 align:middle line:90% is a subset of e prime. 00:05:51.450 --> 00:05:54.570 align:middle line:84% And again-- I think I said this in a previous lecture-- 00:05:54.570 --> 00:05:58.320 align:middle line:84% I'm kind of used to using this notation for a subset, 00:05:58.320 --> 00:06:02.850 align:middle line:84% not necessarily a strict subset, just from teaching 18.100A last 00:06:02.850 --> 00:06:04.780 align:middle line:90% semester. 00:06:04.780 --> 00:06:06.990 align:middle line:84% So this does not mean a strict subset, 00:06:06.990 --> 00:06:09.090 align:middle line:90% so maybe put that in there. 00:06:09.090 --> 00:06:12.740 align:middle line:90% 00:06:12.740 --> 00:06:14.630 align:middle line:84% So this is my partially ordered set 00:06:14.630 --> 00:06:17.880 align:middle line:84% which I hope to apply the Zorn's Lemma on. 00:06:17.880 --> 00:06:21.020 align:middle line:84% And then what I will show is that the maximal element 00:06:21.020 --> 00:06:24.230 align:middle line:84% of this set, once I show it exists, in fact 00:06:24.230 --> 00:06:29.960 align:middle line:84% has to be a Hamel basis for V. So now we'll 00:06:29.960 --> 00:06:43.520 align:middle line:84% apply Zorn towards applying Zorn. 00:06:43.520 --> 00:06:51.960 align:middle line:84% Let C be a chain in E. That means any two elements of C 00:06:51.960 --> 00:06:52.860 align:middle line:90% can be compared. 00:06:52.860 --> 00:06:58.690 align:middle line:90% 00:06:58.690 --> 00:07:06.130 align:middle line:84% Define little c to be equal to union over all e in capital 00:07:06.130 --> 00:07:08.020 align:middle line:90% C e. 00:07:08.020 --> 00:07:10.450 align:middle line:84% So each of these little e's is a subset 00:07:10.450 --> 00:07:13.810 align:middle line:84% of V consisting of linearly independent elements. 00:07:13.810 --> 00:07:16.000 align:middle line:84% And now what I'm taking little c to be 00:07:16.000 --> 00:07:20.020 align:middle line:84% is the union of all these subsets of linearly independent 00:07:20.020 --> 00:07:21.820 align:middle line:90% elements. 00:07:21.820 --> 00:07:33.530 align:middle line:84% And I claim that this is a linearly independent subset 00:07:33.530 --> 00:07:42.180 align:middle line:84% which, since this subset of V contains every element of c, 00:07:42.180 --> 00:07:47.840 align:middle line:84% means c is bigger than or equal to all of E. 00:07:47.840 --> 00:08:11.030 align:middle line:84% Thus, c is an upper bound for C. So we just 00:08:11.030 --> 00:08:13.190 align:middle line:84% need to show that little c is now 00:08:13.190 --> 00:08:17.630 align:middle line:84% linearly independent subset, and therefore, 00:08:17.630 --> 00:08:22.490 align:middle line:84% all of these e's in this capital C are bounded above by c. 00:08:22.490 --> 00:08:27.130 align:middle line:84% And therefore, c is an upper bound for capital C. 00:08:27.130 --> 00:08:31.090 align:middle line:84% Now, to show it's a linearly independent subset, 00:08:31.090 --> 00:08:35.260 align:middle line:84% we're going to use the fact that capital C is a chain-- that you 00:08:35.260 --> 00:08:37.480 align:middle line:90% can always compare two things. 00:08:37.480 --> 00:08:41.230 align:middle line:84% So let's show that little c is a linearly independent subset. 00:08:41.230 --> 00:08:43.750 align:middle line:90% That's something very specific. 00:08:43.750 --> 00:08:52.585 align:middle line:84% So let v 1 up to v N be in little c. 00:08:52.585 --> 00:09:00.990 align:middle line:90% 00:09:00.990 --> 00:09:08.700 align:middle line:84% Then there exists e1, e2, e n in capital C. 00:09:08.700 --> 00:09:10.920 align:middle line:84% So little c is the union of all the e's. 00:09:10.920 --> 00:09:18.614 align:middle line:84% So these have to come from somewhere such that for all j, 00:09:18.614 --> 00:09:21.190 align:middle line:90% v j is in e j. 00:09:21.190 --> 00:09:25.330 align:middle line:90% 00:09:25.330 --> 00:09:29.920 align:middle line:84% Now, it's not difficult to show by induction 00:09:29.920 --> 00:09:34.480 align:middle line:84% that since I can compare any two elements in C, 00:09:34.480 --> 00:09:41.220 align:middle line:84% I can compare any n elements of C, 00:09:41.220 --> 00:09:47.670 align:middle line:84% meaning I can actually order any finitely many elements in C. So 00:09:47.670 --> 00:09:53.580 align:middle line:84% to say this is a chain means I can always order two elements, 00:09:53.580 --> 00:09:56.940 align:middle line:84% but by induction, it's not difficult to show I can always 00:09:56.940 --> 00:10:02.230 align:middle line:84% order n elements of C. So I'm just going to skip to that. 00:10:02.230 --> 00:10:07.910 align:middle line:84% And I'll leave it to you, meaning that I can always 00:10:07.910 --> 00:10:11.330 align:middle line:84% find a biggest element out of any finite collection 00:10:11.330 --> 00:10:11.960 align:middle line:90% of these guys. 00:10:11.960 --> 00:10:14.970 align:middle line:90% 00:10:14.970 --> 00:10:26.110 align:middle line:84% So since C is a chain, there exists capital J such 00:10:26.110 --> 00:10:33.520 align:middle line:84% that for all j equals 1 up to N, e j 00:10:33.520 --> 00:10:38.110 align:middle line:84% is less than or equal to e capital J, which again, 00:10:38.110 --> 00:10:42.010 align:middle line:84% remember, this means that e j is a subset of e capital J, 00:10:42.010 --> 00:10:44.575 align:middle line:84% just by how we've defined this partial order. 00:10:44.575 --> 00:10:47.960 align:middle line:90% 00:10:47.960 --> 00:10:53.060 align:middle line:84% And therefore, since all of these e j's for j from 1 to N 00:10:53.060 --> 00:10:55.760 align:middle line:84% are contained in this e sub capital J, 00:10:55.760 --> 00:11:07.700 align:middle line:84% that means that v1 up to v n are in E capital J. 00:11:07.700 --> 00:11:11.900 align:middle line:84% So I had this finitely many from C. 00:11:11.900 --> 00:11:14.010 align:middle line:84% I can always compare any two of them. 00:11:14.010 --> 00:11:15.890 align:middle line:84% And therefore, I pick the biggest linearly 00:11:15.890 --> 00:11:18.320 align:middle line:84% independent subset out of this finitely many. 00:11:18.320 --> 00:11:20.570 align:middle line:84% And all of these vectors have to then come 00:11:20.570 --> 00:11:25.250 align:middle line:84% from that linearly independent subset. 00:11:25.250 --> 00:11:29.360 align:middle line:84% And therefore, since this is a linearly independent subset, 00:11:29.360 --> 00:11:44.720 align:middle line:84% that means these are linearly independent, since e J is 00:11:44.720 --> 00:11:46.295 align:middle line:90% a linearly independent subset. 00:11:46.295 --> 00:11:49.790 align:middle line:90% 00:11:49.790 --> 00:11:54.260 align:middle line:84% And so we've shown every finite collection of vectors in C 00:11:54.260 --> 00:11:55.770 align:middle line:90% is linearly independent. 00:11:55.770 --> 00:12:02.340 align:middle line:84% So we've concluded that little c is a collection of linearly 00:12:02.340 --> 00:12:03.585 align:middle line:90% independent vectors. 00:12:03.585 --> 00:12:07.580 align:middle line:90% 00:12:07.580 --> 00:12:11.240 align:middle line:84% So we've now shown that the hypotheses of Zorn 00:12:11.240 --> 00:12:15.380 align:middle line:84% are verified-- that every chain has an upper bound, 00:12:15.380 --> 00:12:36.730 align:middle line:84% and therefore, this set E has a maximal element, which I'll 00:12:36.730 --> 00:12:48.330 align:middle line:84% call H. So I claim that H now spans the vector space V, 00:12:48.330 --> 00:12:51.630 align:middle line:84% meaning every element of V can be written 00:12:51.630 --> 00:12:58.240 align:middle line:84% as a finite linear combination of elements of H. 00:12:58.240 --> 00:13:10.920 align:middle line:84% So I claim that H spans V. So when I say H spans V, that's 00:13:10.920 --> 00:13:14.100 align:middle line:84% just a short way of saying that every element of V 00:13:14.100 --> 00:13:16.560 align:middle line:84% can be written as a finite linear combination of elements 00:13:16.560 --> 00:13:17.790 align:middle line:90% of H. 00:13:17.790 --> 00:13:18.705 align:middle line:90% So suppose not. 00:13:18.705 --> 00:13:22.300 align:middle line:90% 00:13:22.300 --> 00:13:43.410 align:middle line:84% Then there exists an element v in capital v such that v cannot 00:13:43.410 --> 00:14:03.150 align:middle line:84% be written as a finite linear combination of elements of H. 00:14:03.150 --> 00:14:06.480 align:middle line:84% Now, it's something from linear algebra. 00:14:06.480 --> 00:14:08.310 align:middle line:84% I'm sure they went over this before, 00:14:08.310 --> 00:14:11.820 align:middle line:84% but if I have a linearly independent subset 00:14:11.820 --> 00:14:13.500 align:middle line:84% and an element that can't be written 00:14:13.500 --> 00:14:17.130 align:middle line:84% as a finite linear combination of elements from that subset, 00:14:17.130 --> 00:14:20.100 align:middle line:84% then just by adding that element, 00:14:20.100 --> 00:14:23.055 align:middle line:84% I now obtain a new linearly independent subset. 00:14:23.055 --> 00:14:25.900 align:middle line:90% 00:14:25.900 --> 00:14:32.930 align:middle line:84% And therefore, I conclude that the set H union 00:14:32.930 --> 00:14:45.700 align:middle line:84% v is linearly independent, a linearly independent subset 00:14:45.700 --> 00:14:56.800 align:middle line:84% of v. But then, H will be less than, 00:14:56.800 --> 00:15:02.290 align:middle line:84% meaning it is less than or equal to, but not equal to, 00:15:02.290 --> 00:15:14.990 align:middle line:84% which implies H is not maximum, which is a contradiction. 00:15:14.990 --> 00:15:17.120 align:middle line:84% Remember, H was supposed to be the maximal element 00:15:17.120 --> 00:15:19.670 align:middle line:90% of E. Nothing sits above H. 00:15:19.670 --> 00:15:22.970 align:middle line:84% And if we assume that H does not span v, 00:15:22.970 --> 00:15:25.640 align:middle line:84% then we can tack onto H something 00:15:25.640 --> 00:15:28.880 align:middle line:84% making a bigger linearly independent subset of v. 00:15:28.880 --> 00:15:32.920 align:middle line:84% And that results in a contradiction. 00:15:32.920 --> 00:15:39.540 align:middle line:84% So thus, that must contradict our initial assumption 00:15:39.540 --> 00:15:50.830 align:middle line:84% that H did not span v. And therefore, by definition, 00:15:50.830 --> 00:15:51.685 align:middle line:90% it's a Hamel basis. 00:15:51.685 --> 00:16:03.750 align:middle line:90% 00:16:03.750 --> 00:16:07.890 align:middle line:84% So we've seen this kind of exercise, first exercise, 00:16:07.890 --> 00:16:14.160 align:middle line:84% of using this very powerful weapon, Zorn's Lemma 00:16:14.160 --> 00:16:19.290 align:middle line:84% to prove this fact that every vector space has a Hamel basis. 00:16:19.290 --> 00:16:23.730 align:middle line:84% And now we're going to use it to prove the Hahn-Banach theorem. 00:16:23.730 --> 00:16:28.650 align:middle line:84% So let me state the Hahn-Banach theorem for you, 00:16:28.650 --> 00:16:30.660 align:middle line:84% and then we're going to discuss the strategy. 00:16:30.660 --> 00:16:31.620 align:middle line:84% Actually, what I'm going to do is 00:16:31.620 --> 00:16:33.420 align:middle line:84% I'm going to state the Hahn-Banach theorem, 00:16:33.420 --> 00:16:34.920 align:middle line:84% I'm going to state a lemma, and then 00:16:34.920 --> 00:16:41.280 align:middle line:84% I'm going to give you what the plan is for proving 00:16:41.280 --> 00:16:43.200 align:middle line:90% the Hahn-Banach theorem. 00:16:43.200 --> 00:16:54.060 align:middle line:84% So the Hahn-Banach theorem is if V is a normed space, 00:16:54.060 --> 00:17:18.579 align:middle line:84% M is a subspace of V, and u going from M to C is linear 00:17:18.579 --> 00:17:19.640 align:middle line:90% such that-- 00:17:19.640 --> 00:17:21.490 align:middle line:84% so it's a bounded linear functional-- 00:17:21.490 --> 00:17:24.810 align:middle line:90% 00:17:24.810 --> 00:17:29.130 align:middle line:90% for all t in M u of t-- 00:17:29.130 --> 00:17:31.080 align:middle line:90% so this is now a complex number. 00:17:31.080 --> 00:17:35.970 align:middle line:84% Its absolute value is less than or equal to constant times 00:17:35.970 --> 00:17:47.890 align:middle line:84% norm of t, then there exists a continuous extension of u 00:17:47.890 --> 00:17:51.430 align:middle line:84% to the entire space and a continuous extension that 00:17:51.430 --> 00:17:53.420 align:middle line:90% has the same constant here. 00:17:53.420 --> 00:17:57.550 align:middle line:84% So remember, think of this as being the norm of little u. 00:17:57.550 --> 00:18:00.430 align:middle line:84% And so we can extend it to a bounded linear functional 00:18:00.430 --> 00:18:02.470 align:middle line:90% with the same norm, essentially. 00:18:02.470 --> 00:18:07.880 align:middle line:84% Then there exists a capital U which 00:18:07.880 --> 00:18:15.080 align:middle line:84% is a bounded linear functional from V to C, 00:18:15.080 --> 00:18:20.750 align:middle line:84% so it's an element of the dual space such that now, 00:18:20.750 --> 00:18:27.010 align:middle line:84% for all t in V, U of t is less than 00:18:27.010 --> 00:18:29.650 align:middle line:90% or equal to the same constant t. 00:18:29.650 --> 00:18:33.850 align:middle line:84% And I should have said before, such that capital U, when 00:18:33.850 --> 00:18:39.700 align:middle line:84% I restrict to M, gives me little u, and for all T in V, 00:18:39.700 --> 00:18:42.370 align:middle line:84% capital U of t is less than or equal to a constant times 00:18:42.370 --> 00:18:44.380 align:middle line:90% the norm of t. 00:18:44.380 --> 00:18:48.840 align:middle line:84% So I should have put this here at the start of the theorem, 00:18:48.840 --> 00:18:54.070 align:middle line:84% but this is the Hahn-Banach theorem. 00:18:54.070 --> 00:18:56.940 align:middle line:90% 00:18:56.940 --> 00:19:00.180 align:middle line:84% And this is a very, very useful theorem to have. 00:19:00.180 --> 00:19:03.990 align:middle line:84% In fact, in the exercises, you can use this theorem 00:19:03.990 --> 00:19:11.700 align:middle line:84% to prove that the dual of little l infinity is not little l1. 00:19:11.700 --> 00:19:14.650 align:middle line:84% So remember from the second assignment 00:19:14.650 --> 00:19:16.740 align:middle line:84% and from what I've said in lectures, 00:19:16.740 --> 00:19:19.890 align:middle line:84% the duel of little l p is little l q, 00:19:19.890 --> 00:19:22.680 align:middle line:84% where 1 over p plus 1 over q equals 1, 00:19:22.680 --> 00:19:26.040 align:middle line:84% as long as p is bigger than or equal to 1 00:19:26.040 --> 00:19:27.060 align:middle line:90% and less than infinity. 00:19:27.060 --> 00:19:29.350 align:middle line:84% And it doesn't work for p equals infinity. 00:19:29.350 --> 00:19:31.140 align:middle line:84% So using the Hahn-Banach theorem, 00:19:31.140 --> 00:19:35.135 align:middle line:84% you can show why it doesn't work for little l infinity. 00:19:35.135 --> 00:19:36.510 align:middle line:84% And that'll be in the assignment. 00:19:36.510 --> 00:19:45.020 align:middle line:90% 00:19:45.020 --> 00:19:49.220 align:middle line:84% Now, so I don't have to keep writing all of this 00:19:49.220 --> 00:19:51.920 align:middle line:84% over and over again, I'm going to refer to this 00:19:51.920 --> 00:20:05.850 align:middle line:84% as U is a continuous extension of little u, 00:20:05.850 --> 00:20:07.920 align:middle line:84% although that's not quite precise because we 00:20:07.920 --> 00:20:13.870 align:middle line:84% are extending little u to a continuous-- 00:20:13.870 --> 00:20:16.680 align:middle line:84% so a bounded linear operator on V, 00:20:16.680 --> 00:20:23.580 align:middle line:84% but we're also extending it so that the capital U satisfies 00:20:23.580 --> 00:20:25.530 align:middle line:90% the same bound is little u. 00:20:25.530 --> 00:20:28.940 align:middle line:90% 00:20:28.940 --> 00:20:33.320 align:middle line:84% So this is a little imprecise, but I think you get my meaning. 00:20:33.320 --> 00:20:49.040 align:middle line:90% 00:20:49.040 --> 00:20:55.900 align:middle line:84% So this assumption let me denote by star, namely, 00:20:55.900 --> 00:20:58.900 align:middle line:84% that we have a subspace and we have 00:20:58.900 --> 00:21:01.960 align:middle line:84% a bounded linear functional on the subspace that 00:21:01.960 --> 00:21:03.380 align:middle line:90% satisfies this bound. 00:21:03.380 --> 00:21:09.617 align:middle line:84% So I'm not going to prove the Hahn-Banach theorem just yet. 00:21:09.617 --> 00:21:11.950 align:middle line:84% I'm going to prove a lemma or I'm going to state a lemma 00:21:11.950 --> 00:21:13.900 align:middle line:84% and then tell you how we're going to use it to prove 00:21:13.900 --> 00:21:14.920 align:middle line:90% the Hahn-Banach theorem. 00:21:14.920 --> 00:21:23.160 align:middle line:90% 00:21:23.160 --> 00:21:27.307 align:middle line:84% Actually, we'll state it slightly different. 00:21:27.307 --> 00:21:28.390 align:middle line:90% I'm not going to use that. 00:21:28.390 --> 00:21:33.220 align:middle line:90% 00:21:33.220 --> 00:21:54.210 align:middle line:84% So if V is a normed space, M, a subset of V is a subspace, 00:21:54.210 --> 00:22:05.850 align:middle line:84% and u from M to C is linear such that little u of t 00:22:05.850 --> 00:22:09.600 align:middle line:84% is less than or equal to a constant times 00:22:09.600 --> 00:22:12.615 align:middle line:84% the norm of t, so for all t in M-- 00:22:12.615 --> 00:22:21.760 align:middle line:90% 00:22:21.760 --> 00:22:23.980 align:middle line:84% I think the difference between capital U and little u 00:22:23.980 --> 00:22:26.560 align:middle line:90% is clear enough-- 00:22:26.560 --> 00:22:31.310 align:middle line:84% then I can extend it at least in one direction. 00:22:31.310 --> 00:22:51.060 align:middle line:84% So in other words, then so if M is-- 00:22:51.060 --> 00:22:54.360 align:middle line:84% and one last assumption-- and I take something that's 00:22:54.360 --> 00:23:01.000 align:middle line:84% not in the linear subspace, then I can extend it 00:23:01.000 --> 00:23:07.090 align:middle line:84% to the subspace of V consisting of M and also the direction x. 00:23:07.090 --> 00:23:22.150 align:middle line:84% So then, there exists u prime in the dual-- 00:23:22.150 --> 00:23:24.100 align:middle line:90% so I shouldn't use u prime. 00:23:24.100 --> 00:23:24.910 align:middle line:90% Let's say v-- 00:23:24.910 --> 00:23:28.600 align:middle line:90% 00:23:28.600 --> 00:23:30.700 align:middle line:90% OK, well, let's use u prime-- 00:23:30.700 --> 00:23:45.160 align:middle line:84% u prime from M prime to C, which is linear. 00:23:45.160 --> 00:23:53.280 align:middle line:84% And here, M prime is defined to be the subset or the subspace M 00:23:53.280 --> 00:23:53.940 align:middle line:90% plus-- 00:23:53.940 --> 00:23:59.880 align:middle line:84% so this does not mean quotient space. 00:23:59.880 --> 00:24:02.280 align:middle line:84% We were using pluses for quotient spaces. 00:24:02.280 --> 00:24:06.150 align:middle line:84% But this is the subspace of V consisting of M 00:24:06.150 --> 00:24:12.700 align:middle line:84% plus elements of the form a constant times x. 00:24:12.700 --> 00:24:26.070 align:middle line:84% So this is elements of the form t plus ax, where t is in M, 00:24:26.070 --> 00:24:43.280 align:middle line:84% a is in C, such that u prime when restricted to M 00:24:43.280 --> 00:24:51.970 align:middle line:84% gives me u for all t prime in M prime, u prime of t 00:24:51.970 --> 00:24:54.430 align:middle line:84% prime, absolute value, is less than 00:24:54.430 --> 00:25:01.690 align:middle line:84% or equal to the same constant times the norm of t prime. 00:25:01.690 --> 00:25:05.893 align:middle line:84% So maybe I made a mess of that, speaking 00:25:05.893 --> 00:25:07.060 align:middle line:90% while I was writing a lemma. 00:25:07.060 --> 00:25:10.990 align:middle line:84% But anyways, let's say you have a bounded linear functional 00:25:10.990 --> 00:25:16.990 align:middle line:84% on a subspace of V, and you take an element that's not in M. 00:25:16.990 --> 00:25:21.910 align:middle line:84% So the way I'll draw the picture is there's M, 00:25:21.910 --> 00:25:27.920 align:middle line:90% and here's something x. 00:25:27.920 --> 00:25:33.410 align:middle line:84% Then, I can extend this bounded linear functional 00:25:33.410 --> 00:25:39.190 align:middle line:84% which lives on M to now the subspace consisting 00:25:39.190 --> 00:25:44.110 align:middle line:84% of all elements of M plus elements plus scalar 00:25:44.110 --> 00:25:47.410 align:middle line:90% multiples of x. 00:25:47.410 --> 00:25:49.450 align:middle line:84% And I can do this in a continuous way, 00:25:49.450 --> 00:25:52.030 align:middle line:84% meaning I get a bounded linear functional 00:25:52.030 --> 00:25:55.945 align:middle line:84% on M prime, which satisfies the same bound as u did. 00:25:55.945 --> 00:26:06.710 align:middle line:90% 00:26:06.710 --> 00:26:10.520 align:middle line:84% So what's the strategy for using this to prove 00:26:10.520 --> 00:26:13.700 align:middle line:90% the Hahn-Banach theorem? 00:26:13.700 --> 00:26:19.640 align:middle line:84% Just so that we're clear on why we would be interested in such 00:26:19.640 --> 00:26:38.050 align:middle line:84% a thing, so we apply Zorn's Lemma 00:26:38.050 --> 00:26:51.800 align:middle line:84% to all continuous extensions of little u. 00:26:51.800 --> 00:26:54.930 align:middle line:84% So now, I'm talking about how we prove Hahn-Banach theorem. 00:26:54.930 --> 00:27:01.140 align:middle line:84% So we define as our partially ordered 00:27:01.140 --> 00:27:04.500 align:middle line:84% set would be the set of all continuous extensions of u. 00:27:04.500 --> 00:27:06.240 align:middle line:84% And then we would put a partial order 00:27:06.240 --> 00:27:09.180 align:middle line:84% on that, where one extension is bigger 00:27:09.180 --> 00:27:11.070 align:middle line:84% than or equal to another extension 00:27:11.070 --> 00:27:16.320 align:middle line:84% if that extension extends the smaller extension. 00:27:16.320 --> 00:27:21.450 align:middle line:84% And using the argument we did for proving the vector 00:27:21.450 --> 00:27:23.520 align:middle line:84% space has a Hamel basis, we can then 00:27:23.520 --> 00:27:25.980 align:middle line:90% show that Zorn's Lemma applies. 00:27:25.980 --> 00:27:29.640 align:middle line:84% And then, we'll have a maximal element of this set 00:27:29.640 --> 00:27:34.970 align:middle line:90% of continuous extensions of u. 00:27:34.970 --> 00:27:37.250 align:middle line:84% Now, what we would like to conclude 00:27:37.250 --> 00:27:41.330 align:middle line:84% is that this maximal element, this maximal continuous 00:27:41.330 --> 00:27:48.180 align:middle line:84% extension, is defined on all of V. 00:27:48.180 --> 00:27:54.240 align:middle line:90% And so how would we prove that? 00:27:54.240 --> 00:27:56.030 align:middle line:84% Well, we would do it just kind of like we 00:27:56.030 --> 00:27:57.900 align:middle line:90% did for the Hamel basis case. 00:27:57.900 --> 00:28:00.020 align:middle line:84% We would suppose not, and then we 00:28:00.020 --> 00:28:01.580 align:middle line:84% would show that if it was not defined 00:28:01.580 --> 00:28:05.210 align:middle line:84% on the entire normed space, then we 00:28:05.210 --> 00:28:13.170 align:middle line:84% can extend that maximal element, again using this lemma, 00:28:13.170 --> 00:28:17.250 align:middle line:84% and therefore, contradicting the fact that that extension was 00:28:17.250 --> 00:28:20.430 align:middle line:90% a maximal element. 00:28:20.430 --> 00:28:23.580 align:middle line:84% So in short, we apply Zorn's Lemma 00:28:23.580 --> 00:28:32.930 align:middle line:84% to all continuous extensions of u to get a maximal element, 00:28:32.930 --> 00:28:50.720 align:middle line:84% capital U. Two, we use the lemma to show U is this extension, 00:28:50.720 --> 00:28:55.610 align:middle line:90% is defined on all of capital V. 00:28:55.610 --> 00:28:59.760 align:middle line:84% So these extensions will come with two pieces of information. 00:28:59.760 --> 00:29:04.190 align:middle line:84% One is the subspace, which is bigger than M, 00:29:04.190 --> 00:29:07.140 align:middle line:84% and then also, the functional itself. 00:29:07.140 --> 00:29:08.840 align:middle line:84% And so we'll use the lemma to show 00:29:08.840 --> 00:29:11.720 align:middle line:84% that the subspace that this capital U is defined on 00:29:11.720 --> 00:29:14.970 align:middle line:84% is all of V by showing that if it's not, 00:29:14.970 --> 00:29:18.230 align:middle line:84% then we can extend it to a slightly bigger subspace using 00:29:18.230 --> 00:29:20.870 align:middle line:84% the lemma, which would contradict the fact that this 00:29:20.870 --> 00:29:24.140 align:middle line:90% is a maximal element. 00:29:24.140 --> 00:29:25.205 align:middle line:90% So that's the plan. 00:29:25.205 --> 00:29:31.910 align:middle line:90% 00:29:31.910 --> 00:29:39.960 align:middle line:84% So in fact, since we have this lemma already here, 00:29:39.960 --> 00:29:42.480 align:middle line:84% and since I've said it so many times, let's go ahead 00:29:42.480 --> 00:29:45.660 align:middle line:84% and just prove the Hahn-Banach theorem assuming this lemma 00:29:45.660 --> 00:29:48.000 align:middle line:90% holds. 00:29:48.000 --> 00:29:51.680 align:middle line:84% We don't need the proof of this lemma to actually prove 00:29:51.680 --> 00:29:52.680 align:middle line:90% the Hahn-Banach theorem. 00:29:52.680 --> 00:29:53.875 align:middle line:90% We just need this statement. 00:29:53.875 --> 00:29:55.500 align:middle line:84% So that's what we're going to do first, 00:29:55.500 --> 00:29:57.975 align:middle line:84% and then we'll go back and prove this lemma. 00:29:57.975 --> 00:30:05.110 align:middle line:90% 00:30:05.110 --> 00:30:07.810 align:middle line:84% So this is the proof of the Hahn-Banach theorem. 00:30:07.810 --> 00:30:10.770 align:middle line:84% We'll go back and prove the lemma in a second. 00:30:10.770 --> 00:30:16.740 align:middle line:84% So let E-- like I said, this will 00:30:16.740 --> 00:30:20.520 align:middle line:84% be the set of all continuous extensions of little u. 00:30:20.520 --> 00:30:36.124 align:middle line:84% So v, comma, let's say N such that N is a subspace of V, 00:30:36.124 --> 00:30:38.750 align:middle line:84% M is contained in N-- not strictly, 00:30:38.750 --> 00:30:40.580 align:middle line:90% but it's a subset of N-- 00:30:40.580 --> 00:30:44.210 align:middle line:90% 00:30:44.210 --> 00:30:59.630 align:middle line:84% and v is a continuous extension of u to capital N, 00:30:59.630 --> 00:31:01.880 align:middle line:90% meaning it satisfies-- 00:31:01.880 --> 00:31:07.310 align:middle line:84% its a bounded linear functional on capital N. When you restrict 00:31:07.310 --> 00:31:12.050 align:middle line:84% it to capital M, it equals U. And is satisfies the same bound 00:31:12.050 --> 00:31:21.750 align:middle line:84% that little u does on the bigger subspace N. And note, 00:31:21.750 --> 00:31:28.750 align:middle line:84% this is nonempty because u and M are in this. 00:31:28.750 --> 00:31:35.770 align:middle line:84% I'm not saying M has to be a strict subspace of capital N. 00:31:35.770 --> 00:31:51.880 align:middle line:84% And so we'll define a partial order on E 00:31:51.880 --> 00:31:55.820 align:middle line:90% by the following definition-- 00:31:55.820 --> 00:32:06.880 align:middle line:84% we will say v1, N1 is less than or equal to v2, N2 00:32:06.880 --> 00:32:16.510 align:middle line:84% if N1 is contained in N2 and v2, when restricted to N1, 00:32:16.510 --> 00:32:18.160 align:middle line:90% gives me v1. 00:32:18.160 --> 00:32:25.690 align:middle line:84% So v2 is, if you like, a continuous extension of v1. 00:32:25.690 --> 00:32:27.370 align:middle line:84% Remember, all of these functionals 00:32:27.370 --> 00:32:32.050 align:middle line:84% are assumed to be satisfying the same bound as u did, 00:32:32.050 --> 00:32:33.370 align:middle line:90% so with the same constant. 00:32:33.370 --> 00:32:37.090 align:middle line:90% 00:32:37.090 --> 00:32:40.860 align:middle line:84% So it's not difficult to check, just like I 00:32:40.860 --> 00:32:43.090 align:middle line:90% didn't check it for inclusion. 00:32:43.090 --> 00:32:49.290 align:middle line:84% But it's not hard to check that this 00:32:49.290 --> 00:32:51.960 align:middle line:90% is, in fact, a partial order. 00:32:51.960 --> 00:33:02.010 align:middle line:90% 00:33:02.010 --> 00:33:05.630 align:middle line:84% So now, we want to apply Zorn's Lemma 00:33:05.630 --> 00:33:08.600 align:middle line:84% to get a maximal element of E, which we want to show 00:33:08.600 --> 00:33:15.880 align:middle line:84% is, in fact, this u that we say exists. 00:33:15.880 --> 00:33:19.430 align:middle line:84% So we have to show every chain in E has an upper bound. 00:33:19.430 --> 00:33:35.770 align:middle line:84% So let C, which I'll denote as v i N i for i in some index, 00:33:35.770 --> 00:33:44.580 align:middle line:84% be a chain in E. So this is a set of extensions, 00:33:44.580 --> 00:33:49.180 align:middle line:84% and we can always compare any two extensions in there. 00:33:49.180 --> 00:33:53.610 align:middle line:84% So let me just repeat or write down again 00:33:53.610 --> 00:33:55.890 align:middle line:90% what this means to be a chain. 00:33:55.890 --> 00:34:02.130 align:middle line:84% This means, then, for all i1, i2 in this index 00:34:02.130 --> 00:34:09.540 align:middle line:84% that I'm using just to index these elements of the chain, 00:34:09.540 --> 00:34:23.020 align:middle line:84% either v i1 in i1 is less than or equal to v i2 in i2 00:34:23.020 --> 00:34:27.288 align:middle line:90% or vice versa. 00:34:27.288 --> 00:34:29.080 align:middle line:84% I can't remember if a C goes there or an S, 00:34:29.080 --> 00:34:31.989 align:middle line:84% so I'm going to put an S. One is either bigger 00:34:31.989 --> 00:34:33.820 align:middle line:84% than-- whenever I have two elements, 00:34:33.820 --> 00:34:35.260 align:middle line:90% I can always compare them. 00:34:35.260 --> 00:34:42.110 align:middle line:90% 00:34:42.110 --> 00:34:49.250 align:middle line:84% Let N be the union of all these subspaces coming 00:34:49.250 --> 00:34:51.139 align:middle line:90% from this collection. 00:34:51.139 --> 00:34:57.530 align:middle line:90% 00:34:57.530 --> 00:35:03.310 align:middle line:84% Now, again, it's not difficult to show 00:35:03.310 --> 00:35:05.430 align:middle line:90% that N is, in fact, a subspace. 00:35:05.430 --> 00:35:12.840 align:middle line:84% So I claim N is a subspace, and again, we're 00:35:12.840 --> 00:35:16.020 align:middle line:84% going to use the fact that C is a chain 00:35:16.020 --> 00:35:19.480 align:middle line:90% to be able to verify this. 00:35:19.480 --> 00:35:33.480 align:middle line:84% Let v1, v2 be in N. And let's take two scalars, a1, a2 in C. 00:35:33.480 --> 00:35:41.430 align:middle line:84% I want to show that a1 times v1 plus a2 times v2 remains in N. 00:35:41.430 --> 00:35:54.570 align:middle line:84% Then there exists i1 i2 indices such that v1 is in N i1 and v2 00:35:54.570 --> 00:35:59.730 align:middle line:90% to is in N i2. 00:35:59.730 --> 00:36:05.550 align:middle line:84% Now, I can always compare any two subspaces 00:36:05.550 --> 00:36:09.230 align:middle line:84% that are appearing in this set of ordered pairs 00:36:09.230 --> 00:36:10.460 align:middle line:90% forming this chain. 00:36:10.460 --> 00:36:13.040 align:middle line:84% So one of these subspaces is bigger than the other. 00:36:13.040 --> 00:36:16.770 align:middle line:90% 00:36:16.770 --> 00:36:26.980 align:middle line:84% Then just by flipping 1 and 2, if I need to, 00:36:26.980 --> 00:36:38.090 align:middle line:84% and since C is a chain, N i1 is contained in N i2 00:36:38.090 --> 00:36:39.440 align:middle line:90% without loss of generality. 00:36:39.440 --> 00:36:43.390 align:middle line:84% So it will either be N i1 will be contained in N 00:36:43.390 --> 00:36:45.390 align:middle line:90% i2 or the other way around. 00:36:45.390 --> 00:36:47.780 align:middle line:84% And if it's the other way around, 00:36:47.780 --> 00:36:50.180 align:middle line:90% just flip the numbers 1 and 2. 00:36:50.180 --> 00:36:52.670 align:middle line:84% So I'm going to assume N i1 is contained in N i2. 00:36:52.670 --> 00:36:56.705 align:middle line:84% Then that means both of these elements are contained in N i2. 00:36:56.705 --> 00:37:00.349 align:middle line:90% 00:37:00.349 --> 00:37:03.730 align:middle line:84% v1, v2 are both in the bigger one, 00:37:03.730 --> 00:37:06.700 align:middle line:84% and since this is a subspace, this 00:37:06.700 --> 00:37:15.850 align:middle line:84% means that a1v1 plus a2v2 is in N i2, which remember, 00:37:15.850 --> 00:37:25.465 align:middle line:84% is a subset of N. And therefore, N is a subspace. 00:37:25.465 --> 00:37:36.200 align:middle line:90% 00:37:36.200 --> 00:37:39.190 align:middle line:84% So I now have a subspace which contains 00:37:39.190 --> 00:37:41.830 align:middle line:84% all of these subspaces coming from the chain. 00:37:41.830 --> 00:37:45.160 align:middle line:84% Now I need to define a linear functional 00:37:45.160 --> 00:37:48.400 align:middle line:84% on this subspace N that extends all 00:37:48.400 --> 00:37:51.100 align:middle line:90% the v i's in a continuous way. 00:37:51.100 --> 00:37:55.870 align:middle line:84% And this would give me an upper bound for this chain. 00:37:55.870 --> 00:37:59.230 align:middle line:84% But it's not difficult to guess what 00:37:59.230 --> 00:38:01.330 align:middle line:90% that linear functional will be. 00:38:01.330 --> 00:38:06.130 align:middle line:90% 00:38:06.130 --> 00:38:13.420 align:middle line:84% We define v-- or not, I don't think I use v. Yeah I do. 00:38:13.420 --> 00:38:14.670 align:middle line:90% Well, let's not do that. 00:38:14.670 --> 00:38:22.160 align:middle line:90% 00:38:22.160 --> 00:38:25.280 align:middle line:84% So I mean, I shouldn't have used v1 and v2 00:38:25.280 --> 00:38:29.660 align:middle line:84% when talking about the subspace business. 00:38:29.660 --> 00:38:33.430 align:middle line:84% That was poor choice of notation. 00:38:33.430 --> 00:38:37.050 align:middle line:90% 00:38:37.050 --> 00:38:41.670 align:middle line:90% So let's make that x1, x2. 00:38:41.670 --> 00:38:56.300 align:middle line:84% And the chain, blah, blah, blah, x1, x2, and therefore x1, x2 00:38:56.300 --> 00:38:56.840 align:middle line:90% is in-- 00:38:56.840 --> 00:38:57.870 align:middle line:90% OK. 00:38:57.870 --> 00:39:01.850 align:middle line:84% So I just don't want to mix up the elements of the vector 00:39:01.850 --> 00:39:07.710 align:middle line:84% space with these functionals, which I'm labeling by v. 00:39:07.710 --> 00:39:19.310 align:middle line:84% So now, we define a function v from N to C by the following-- 00:39:19.310 --> 00:39:23.390 align:middle line:90% 00:39:23.390 --> 00:39:26.930 align:middle line:84% if t is an element of this union, 00:39:26.930 --> 00:39:32.350 align:middle line:84% it has to be an element of one of these N sub i's. 00:39:32.350 --> 00:39:38.200 align:middle line:84% Then I define v of t to be simply v 00:39:38.200 --> 00:39:41.335 align:middle line:84% sub i, which is defined on this linear subspace. 00:39:41.335 --> 00:39:46.590 align:middle line:90% 00:39:46.590 --> 00:39:51.180 align:middle line:84% And so I take v of t to be the value of v sub i evaluated 00:39:51.180 --> 00:39:51.690 align:middle line:90% at t. 00:39:51.690 --> 00:39:54.720 align:middle line:84% So now, one question is, is this well defined? 00:39:54.720 --> 00:40:00.880 align:middle line:84% Because an element t in N sub i could also 00:40:00.880 --> 00:40:03.055 align:middle line:84% have been an element of a different N sub i. 00:40:03.055 --> 00:40:12.900 align:middle line:90% 00:40:12.900 --> 00:40:14.500 align:middle line:90% So is this well defined? 00:40:14.500 --> 00:40:15.390 align:middle line:90% So we have to check. 00:40:15.390 --> 00:40:23.562 align:middle line:90% 00:40:23.562 --> 00:40:29.630 align:middle line:84% If it's in two of these, does this 00:40:29.630 --> 00:40:38.080 align:middle line:84% imply, question mark, v i1 of t equals v i2 of t? 00:40:38.080 --> 00:40:42.400 align:middle line:84% And again, we're going to use the fact that this is a chain 00:40:42.400 --> 00:40:45.050 align:middle line:90% to verify this. 00:40:45.050 --> 00:40:55.900 align:middle line:84% So suppose t is N sub i1 intersect N sub i2. 00:40:55.900 --> 00:41:01.990 align:middle line:84% And again, for any two elements of this chain, 00:41:01.990 --> 00:41:03.110 align:middle line:90% we can compare them. 00:41:03.110 --> 00:41:06.826 align:middle line:84% So let's assume i2 corresponds to the bigger index. 00:41:06.826 --> 00:41:16.550 align:middle line:84% So v i1 N i1 is less than or equal to v i2, N i2. 00:41:16.550 --> 00:41:23.380 align:middle line:90% 00:41:23.380 --> 00:41:27.940 align:middle line:84% Now, this order not only is defined 00:41:27.940 --> 00:41:29.680 align:middle line:84% in terms of the subspaces, remember, 00:41:29.680 --> 00:41:32.890 align:middle line:84% but in terms of the functionals defined on these subspaces. 00:41:32.890 --> 00:41:35.650 align:middle line:84% And it's defined by the fact that this functional 00:41:35.650 --> 00:41:37.990 align:middle line:84% is an extension of this functional. 00:41:37.990 --> 00:41:53.060 align:middle line:84% And therefore, since v i2 is the bigger one, 00:41:53.060 --> 00:41:59.440 align:middle line:90% it extends the smaller one. 00:41:59.440 --> 00:42:05.230 align:middle line:84% This implies v i2 of t equals v i1 of t. 00:42:05.230 --> 00:42:16.320 align:middle line:84% And therefore, v is well defined. 00:42:16.320 --> 00:42:19.740 align:middle line:84% And not only that, you can also-- 00:42:19.740 --> 00:42:25.890 align:middle line:84% so I wrote this out carefully showing it's well defined, 00:42:25.890 --> 00:42:28.320 align:middle line:84% but by a similar argument, you can then 00:42:28.320 --> 00:42:31.180 align:middle line:90% show that v is, in fact linear. 00:42:31.180 --> 00:42:36.780 align:middle line:84% And let's see, do I do that or do I stop there? 00:42:36.780 --> 00:42:39.970 align:middle line:90% 00:42:39.970 --> 00:42:45.220 align:middle line:84% So it's well defined on N. It's also 00:42:45.220 --> 00:42:49.660 align:middle line:84% an extension of every single functional defined on each N 00:42:49.660 --> 00:42:50.590 align:middle line:90% sub i. 00:42:50.590 --> 00:42:54.730 align:middle line:84% So the last thing to check is that it's linear 00:42:54.730 --> 00:42:59.770 align:middle line:84% and it's a continuous extension of all these v i's, meaning 00:42:59.770 --> 00:43:01.810 align:middle line:90% it satisfies the same bound. 00:43:01.810 --> 00:43:05.380 align:middle line:84% But all of the v i's satisfy that bound. 00:43:05.380 --> 00:43:10.570 align:middle line:84% V is defined in terms of the v i's, so we can just 00:43:10.570 --> 00:43:13.990 align:middle line:84% read off from here that v will satisfy that same bound. 00:43:13.990 --> 00:43:21.740 align:middle line:90% 00:43:21.740 --> 00:43:29.640 align:middle line:84% So I will leave it to you that you can check that v is 00:43:29.640 --> 00:43:36.080 align:middle line:84% an element of the dual space of this subspace N, 00:43:36.080 --> 00:43:38.880 align:middle line:84% so it's a bounded linear functional on N 00:43:38.880 --> 00:43:49.680 align:middle line:84% and a continuous extension of all v i's. 00:43:49.680 --> 00:44:11.540 align:middle line:90% 00:44:11.540 --> 00:44:22.720 align:middle line:84% So for all i, for little i and capital I, we conclude v i, 00:44:22.720 --> 00:44:26.770 align:middle line:84% N i is less than or equal to V, N, 00:44:26.770 --> 00:44:47.130 align:middle line:84% and therefore, v, N is an upper bound of C. 00:44:47.130 --> 00:44:51.360 align:middle line:84% So we've verified the hypotheses of Zorn's Lemma 00:44:51.360 --> 00:44:53.220 align:middle line:90% which I just erased. 00:44:53.220 --> 00:44:57.640 align:middle line:84% So that means that set E has a maximal element. 00:44:57.640 --> 00:45:16.650 align:middle line:84% So by Zorn, the set E has the maximal element capital U, N. 00:45:16.650 --> 00:45:27.610 align:middle line:84% So I claim N equals V. And therefore, capital U 00:45:27.610 --> 00:45:28.240 align:middle line:90% does the job. 00:45:28.240 --> 00:45:34.340 align:middle line:84% Because remember, since capital U, 00:45:34.340 --> 00:45:36.350 align:middle line:84% capital N is an element of E, that 00:45:36.350 --> 00:45:41.240 align:middle line:84% means capital U is a continuous extension of little u. 00:45:41.240 --> 00:45:44.510 align:middle line:84% And now we just want to conclude that for this maximal element, 00:45:44.510 --> 00:45:51.790 align:middle line:84% the subspace on which it's defined is the entire space V. 00:45:51.790 --> 00:45:53.815 align:middle line:90% So suppose not. 00:45:53.815 --> 00:45:58.760 align:middle line:90% 00:45:58.760 --> 00:46:14.730 align:middle line:84% Let x be an element not in N. By the lemma, 00:46:14.730 --> 00:46:26.370 align:middle line:84% there exists a continuous extension of capital 00:46:26.370 --> 00:46:37.440 align:middle line:84% U to the subspace N plus the span of x. 00:46:37.440 --> 00:46:43.370 align:middle line:90% 00:46:43.370 --> 00:46:45.800 align:middle line:84% And this is a continuous extension 00:46:45.800 --> 00:46:47.990 align:middle line:84% of capital U, which is a continuous extension 00:46:47.990 --> 00:46:48.800 align:middle line:90% of little u. 00:46:48.800 --> 00:46:54.650 align:middle line:84% And therefore, it's a continuous extension of little u. 00:46:54.650 --> 00:47:01.630 align:middle line:84% so continuous extension, let's call this something. 00:47:01.630 --> 00:47:03.475 align:middle line:90% Let's call it little v-- 00:47:03.475 --> 00:47:16.450 align:middle line:90% 00:47:16.450 --> 00:47:18.830 align:middle line:90% v N plus. 00:47:18.830 --> 00:47:25.190 align:middle line:90% 00:47:25.190 --> 00:47:31.910 align:middle line:84% So if the subspace that capital U is defined on 00:47:31.910 --> 00:47:34.100 align:middle line:84% is not all of v, then by the lemma, 00:47:34.100 --> 00:47:37.430 align:middle line:84% we can extend capital U continuously to N 00:47:37.430 --> 00:47:40.910 align:middle line:90% plus the span of x. 00:47:40.910 --> 00:47:43.820 align:middle line:84% And therefore, this element will be a continuous extension 00:47:43.820 --> 00:47:46.130 align:middle line:90% of little u. 00:47:46.130 --> 00:47:51.030 align:middle line:84% And therefore, it's an element of E. 00:47:51.030 --> 00:48:05.350 align:middle line:84% But then, U, N is smaller than v N plus the span of x. 00:48:05.350 --> 00:48:08.410 align:middle line:84% This is a bigger subspace than this 00:48:08.410 --> 00:48:15.510 align:middle line:84% because x is not in N, which implies u, 00:48:15.510 --> 00:48:21.390 align:middle line:90% N is not a maximal element. 00:48:21.390 --> 00:48:22.860 align:middle line:90% And that's a contradiction. 00:48:22.860 --> 00:48:26.190 align:middle line:90% And what did we contradict? 00:48:26.190 --> 00:48:28.830 align:middle line:84% Or what was the assumption that led us astray 00:48:28.830 --> 00:48:32.490 align:middle line:84% is the fact that we assumed that this maximal element is not 00:48:32.490 --> 00:48:36.495 align:middle line:84% defined on all of the entire normed space. 00:48:36.495 --> 00:48:39.350 align:middle line:90% 00:48:39.350 --> 00:48:50.080 align:middle line:84% Thus, U is defined on the entire normed space 00:48:50.080 --> 00:49:00.510 align:middle line:84% and it's a continuous of little u. 00:49:00.510 --> 00:49:02.790 align:middle line:84% And that's the proof of Hahn-Banach. 00:49:02.790 --> 00:49:05.370 align:middle line:90% 00:49:05.370 --> 00:49:10.050 align:middle line:84% So I hope that the proof was clear. 00:49:10.050 --> 00:49:12.900 align:middle line:84% If you followed the Hamel basis argument, 00:49:12.900 --> 00:49:19.910 align:middle line:84% this should be reasonable to expect, too. 00:49:19.910 --> 00:49:22.970 align:middle line:84% This argument is almost the same as the Hamel basis argument, 00:49:22.970 --> 00:49:26.270 align:middle line:84% except now, instead of the Hamel basis where 00:49:26.270 --> 00:49:30.140 align:middle line:84% the elements of our partially ordered set are just subsets, 00:49:30.140 --> 00:49:34.490 align:middle line:84% we also have two pieces of data for our partially ordered 00:49:34.490 --> 00:49:39.350 align:middle line:84% set here, one being the subspace and the second 00:49:39.350 --> 00:49:41.990 align:middle line:84% being the functional that's defined on that subspace, that 00:49:41.990 --> 00:49:46.310 align:middle line:84% extends the original continuous linear functional that we 00:49:46.310 --> 00:49:47.750 align:middle line:90% wanted to extend. 00:49:47.750 --> 00:49:52.920 align:middle line:90% 00:49:52.920 --> 00:49:57.605 align:middle line:84% So let's prove the lemma, and that 00:49:57.605 --> 00:49:59.730 align:middle line:84% will conclude the proof of the Hahn-Banach theorem. 00:49:59.730 --> 00:50:03.130 align:middle line:90% 00:50:03.130 --> 00:50:12.810 align:middle line:84% So now, we're going to proof a lemma. 00:50:12.810 --> 00:50:15.690 align:middle line:84% So that lemma up there, if V is a normed space 00:50:15.690 --> 00:50:21.490 align:middle line:84% and you take something that's not in the subspace, 00:50:21.490 --> 00:50:24.750 align:middle line:84% then you can extend U continuously 00:50:24.750 --> 00:50:27.045 align:middle line:90% to this bigger subspace. 00:50:27.045 --> 00:50:30.690 align:middle line:90% 00:50:30.690 --> 00:50:36.240 align:middle line:84% So first off, even though I keep saying that this is a subspace, 00:50:36.240 --> 00:50:38.993 align:middle line:84% this is kind of something that needs to be-- 00:50:38.993 --> 00:50:41.160 align:middle line:84% I mean it's not difficult to check, but still check. 00:50:41.160 --> 00:50:44.670 align:middle line:84% But also, one thing we need is that every element 00:50:44.670 --> 00:50:51.150 align:middle line:84% in a subspace plus this constant times x 00:50:51.150 --> 00:50:52.230 align:middle line:90% can be written uniquely. 00:50:52.230 --> 00:51:18.200 align:middle line:84% So we first note if t prime is in M prime, which 00:51:18.200 --> 00:51:25.810 align:middle line:84% remember is M plus a constant times x, 00:51:25.810 --> 00:51:34.440 align:middle line:84% then there exists unique t and M and a in complex numbers such 00:51:34.440 --> 00:51:38.190 align:middle line:84% that t prime equals t plus a times x. 00:51:38.190 --> 00:51:42.150 align:middle line:90% 00:51:42.150 --> 00:51:45.360 align:middle line:90% So why is that? 00:51:45.360 --> 00:51:48.060 align:middle line:84% Why can an element of this space not 00:51:48.060 --> 00:51:49.690 align:middle line:84% have two different representations, 00:51:49.690 --> 00:51:52.920 align:middle line:84% not be written as two different elements of M plus two 00:51:52.920 --> 00:51:54.720 align:middle line:90% different scalar times x? 00:51:54.720 --> 00:52:02.910 align:middle line:84% Well, if t plus ax equals t tilde plus a tilde x, then 00:52:02.910 --> 00:52:09.180 align:middle line:84% this implies that a minus a tilde times x is 00:52:09.180 --> 00:52:16.690 align:middle line:84% equal to t tilde minus t, which is in M. M is a subspace. 00:52:16.690 --> 00:52:26.240 align:middle line:84% So the difference of two elements of M is in M, 00:52:26.240 --> 00:52:32.000 align:middle line:84% and therefore, if a does not equal a tilde, 00:52:32.000 --> 00:52:35.380 align:middle line:84% then that means x is equal to a constant multiple of something 00:52:35.380 --> 00:52:39.310 align:middle line:84% in M. And therefore, x is in M. So we 00:52:39.310 --> 00:52:44.040 align:middle line:84% conclude that a must equal a tilde, which 00:52:44.040 --> 00:52:50.820 align:middle line:84% implies from assuming they're equal that t equals t tilde. 00:52:50.820 --> 00:52:54.270 align:middle line:84% So every element in this larger subspace 00:52:54.270 --> 00:52:58.230 align:middle line:84% can be written uniquely as an element of M plus a scalar 00:52:58.230 --> 00:52:59.025 align:middle line:90% multiple of x. 00:52:59.025 --> 00:53:06.940 align:middle line:90% 00:53:06.940 --> 00:53:09.980 align:middle line:90% Now, why do we need this fact? 00:53:09.980 --> 00:53:12.670 align:middle line:84% We need this fact to be able to say 00:53:12.670 --> 00:53:16.090 align:middle line:84% that this linear functional that we're going to define, 00:53:16.090 --> 00:53:21.100 align:middle line:84% that we hope to say extends u continuously, 00:53:21.100 --> 00:53:23.120 align:middle line:90% is in fact well defined. 00:53:23.120 --> 00:53:39.430 align:middle line:84% So thus, upon choosing a number lambda in the complex numbers, 00:53:39.430 --> 00:53:57.280 align:middle line:84% the map u prime of t plus ax given by u of t plus a lambda 00:53:57.280 --> 00:54:10.320 align:middle line:84% is well defined on M prime because we've 00:54:10.320 --> 00:54:11.910 align:middle line:84% shown every element of M prime can be 00:54:11.910 --> 00:54:14.640 align:middle line:90% written uniquely in this way. 00:54:14.640 --> 00:54:25.860 align:middle line:84% It's well defined on M prime and u prime going from M prime to C 00:54:25.860 --> 00:54:26.370 align:middle line:90% is linear. 00:54:26.370 --> 00:54:30.790 align:middle line:90% 00:54:30.790 --> 00:54:38.280 align:middle line:84% So if the original functional little u-- 00:54:38.280 --> 00:54:40.410 align:middle line:84% if that constant C is equal to 0, 00:54:40.410 --> 00:54:42.240 align:middle line:84% then little u is just identically 0. 00:54:42.240 --> 00:54:44.680 align:middle line:84% And we know how to extend the zero functional. 00:54:44.680 --> 00:54:50.170 align:middle line:84% So let's suppose capital C is nonzero. 00:54:50.170 --> 00:54:53.350 align:middle line:84% And if I divide little u by capital C, 00:54:53.350 --> 00:54:59.560 align:middle line:84% I can then extend that functional with the capital 00:54:59.560 --> 00:55:04.090 align:middle line:84% C equals 1 and obtain an extension satisfying 00:55:04.090 --> 00:55:06.490 align:middle line:90% the bound I want. 00:55:06.490 --> 00:55:12.800 align:middle line:84% And so what I'm saying kind of poorly is that-- 00:55:12.800 --> 00:55:16.830 align:middle line:90% 00:55:16.830 --> 00:55:18.940 align:middle line:84% I'll leave you a second to think about it. 00:55:18.940 --> 00:55:22.230 align:middle line:84% It's not difficult to understand why, 00:55:22.230 --> 00:55:29.470 align:middle line:84% without loss of generality, we can assume C equals 1. 00:55:29.470 --> 00:55:32.420 align:middle line:90% 00:55:32.420 --> 00:55:36.110 align:middle line:84% So if we do the C equals 1 case, then for the case C not 00:55:36.110 --> 00:55:41.000 align:middle line:84% equal to 1, we extend u over capital C using the C equals 1 00:55:41.000 --> 00:55:41.630 align:middle line:90% case. 00:55:41.630 --> 00:55:44.280 align:middle line:90% And then the result follows. 00:55:44.280 --> 00:55:51.030 align:middle line:84% So we'll just do the case capital C equals 1. 00:55:51.030 --> 00:55:59.580 align:middle line:84% So now, our one free parameter is lambda. 00:55:59.580 --> 00:56:02.460 align:middle line:90% 00:56:02.460 --> 00:56:03.255 align:middle line:90% And this is for-- 00:56:03.255 --> 00:56:11.110 align:middle line:90% 00:56:11.110 --> 00:56:16.030 align:middle line:84% and we want to be able to choose lambda so that-- 00:56:16.030 --> 00:56:19.450 align:middle line:84% so this is already an extension of little u 00:56:19.450 --> 00:56:21.190 align:middle line:90% if I take a equals 0. 00:56:21.190 --> 00:56:23.380 align:middle line:84% So I'm just taking elements in M. 00:56:23.380 --> 00:56:27.520 align:middle line:84% This is u prime of t is equal to u of t 00:56:27.520 --> 00:56:31.000 align:middle line:84% So it's already an extension of little u 00:56:31.000 --> 00:56:32.590 align:middle line:90% to this bigger subspace. 00:56:32.590 --> 00:56:35.810 align:middle line:84% And now we want to be able to choose lambda 00:56:35.810 --> 00:56:38.860 align:middle line:84% so that it extends it in a continuous way 00:56:38.860 --> 00:56:43.330 align:middle line:84% with the same constant being 1 and that inequality up there. 00:56:43.330 --> 00:56:45.580 align:middle line:90% So I keep pointing up there. 00:56:45.580 --> 00:56:48.520 align:middle line:84% I don't think the camera's looking up there, 00:56:48.520 --> 00:56:56.590 align:middle line:84% so you have to hopefully interpret my meaning correctly. 00:56:56.590 --> 00:57:01.410 align:middle line:84% So we want to choose lambda, complex number, 00:57:01.410 --> 00:57:05.850 align:middle line:90% such that the following holds-- 00:57:05.850 --> 00:57:19.170 align:middle line:84% for all t in M, a in C, u prime of t plus ax, which is just 00:57:19.170 --> 00:57:22.790 align:middle line:84% u of t plus a lambda in absolute value 00:57:22.790 --> 00:57:29.960 align:middle line:84% is less than or equal to the norm of t plus a lambda. 00:57:29.960 --> 00:57:32.090 align:middle line:84% If we're able to do that, then u prime 00:57:32.090 --> 00:57:35.690 align:middle line:84% is then our continuous extension that we're looking for. 00:57:35.690 --> 00:57:58.570 align:middle line:90% 00:57:58.570 --> 00:58:05.570 align:middle line:84% So all that we need to do now is find a lambda 00:58:05.570 --> 00:58:08.730 align:middle line:90% so that this holds. 00:58:08.730 --> 00:58:10.855 align:middle line:84% And once we've done that, we've finished the proof. 00:58:10.855 --> 00:58:19.330 align:middle line:90% 00:58:19.330 --> 00:58:23.740 align:middle line:84% Now, that thing I boxed in a minute ago, 00:58:23.740 --> 00:58:26.440 align:middle line:84% we're going to reduce it to a simplest form. 00:58:26.440 --> 00:58:33.570 align:middle line:90% 00:58:33.570 --> 00:58:41.280 align:middle line:84% Now, what's in the box holds regardless what 00:58:41.280 --> 00:58:45.390 align:middle line:90% lambda is when a is equal to 0. 00:58:45.390 --> 00:58:48.300 align:middle line:84% And now what I'm going to do is basically remove the fact 00:58:48.300 --> 00:58:50.560 align:middle line:90% that a can change. 00:58:50.560 --> 00:59:02.140 align:middle line:84% So the estimate u of t plus a lambda less than 00:59:02.140 --> 00:59:14.080 align:middle line:84% or equal to t plus a lambda holds 00:59:14.080 --> 00:59:20.060 align:middle line:84% when a equals 0 regardless of how we've chosen lambda. 00:59:20.060 --> 00:59:22.240 align:middle line:84% So we just need to be able to choose lambda so 00:59:22.240 --> 00:59:25.150 align:middle line:90% that this holds for a nonzero. 00:59:25.150 --> 00:59:35.110 align:middle line:90% 00:59:35.110 --> 00:59:38.740 align:middle line:84% Consider trying to choose lambda so that this 00:59:38.740 --> 00:59:43.790 align:middle line:90% holds for all a nonzero. 00:59:43.790 --> 00:59:50.230 align:middle line:84% Now, let me take this inequality here and divide it by 00:59:50.230 --> 00:59:51.655 align:middle line:90% the absolute value of a. 00:59:51.655 --> 00:59:54.390 align:middle line:90% 00:59:54.390 --> 01:00:03.990 align:middle line:84% Then that inequality for a not equal to 0, 01:00:03.990 --> 01:00:10.245 align:middle line:84% this is equivalent to if I divide by the norm of a 01:00:10.245 --> 01:00:22.800 align:middle line:84% and bring this inside the norm, u of t minus a minus lambda 01:00:22.800 --> 01:00:26.210 align:middle line:90% is less than or equal to-- 01:00:26.210 --> 01:00:28.823 align:middle line:84% I should say this is an absolute value 01:00:28.823 --> 01:00:30.115 align:middle line:90% and that's the absolute value-- 01:00:30.115 --> 01:00:36.380 align:middle line:90% 01:00:36.380 --> 01:00:43.790 align:middle line:84% less than or equal to t over minus a minus lambda 01:00:43.790 --> 01:00:52.300 align:middle line:84% for all t in M. Now, if t is in M, t over minus a is also in M. 01:00:52.300 --> 01:00:58.450 align:middle line:84% So this bound here is equivalent to showing 01:00:58.450 --> 01:01:04.360 align:middle line:84% that we can choose lambda so that u of t minus lambda 01:01:04.360 --> 01:01:07.960 align:middle line:84% is less than or equal to the norm of t 01:01:07.960 --> 01:01:15.340 align:middle line:84% minus lambda for all t in M. So since t is a subspace proving 01:01:15.340 --> 01:01:19.870 align:middle line:84% this bound, or choosing lambda so that this bound holds, 01:01:19.870 --> 01:01:23.740 align:middle line:84% is equivalent to choosing lambda so that this bound holds. 01:01:23.740 --> 01:01:30.470 align:middle line:90% So in sum or in summary-- 01:01:30.470 --> 01:01:32.920 align:middle line:90% I should say in sum. 01:01:32.920 --> 01:01:40.500 align:middle line:84% I'm thinking in terms of notation. 01:01:40.500 --> 01:01:43.220 align:middle line:84% So in summary, the thing that we wanted to show originally, 01:01:43.220 --> 01:01:46.190 align:middle line:84% that we can choose lambda so that for all t in M 01:01:46.190 --> 01:01:50.420 align:middle line:84% a complex number, that inequality holds, 01:01:50.420 --> 01:01:53.250 align:middle line:84% is equivalent to showing that we can choose lambda 01:01:53.250 --> 01:01:57.092 align:middle line:90% so that this inequality holds. 01:01:57.092 --> 01:01:57.800 align:middle line:90% That's the point. 01:01:57.800 --> 01:02:01.953 align:middle line:90% 01:02:01.953 --> 01:02:03.370 align:middle line:84% And now, what we're going to do is 01:02:03.370 --> 01:02:08.530 align:middle line:84% we're going to choose the real and imaginary parts of lambda. 01:02:08.530 --> 01:02:12.100 align:middle line:84% And we'll choose the real part of lambda first. 01:02:12.100 --> 01:02:13.180 align:middle line:90% So we first prove-- 01:02:13.180 --> 01:02:19.050 align:middle line:90% 01:02:19.050 --> 01:02:20.050 align:middle line:90% piece of chalk is weird. 01:02:20.050 --> 01:02:21.383 align:middle line:90% I've got to get a different one. 01:02:21.383 --> 01:02:25.500 align:middle line:90% 01:02:25.500 --> 01:02:31.060 align:middle line:84% So we first prove that there exists 01:02:31.060 --> 01:02:41.950 align:middle line:84% an alpha in R such that w of t minus alpha in absolute value 01:02:41.950 --> 01:02:48.460 align:middle line:84% is less than or equal to t minus alpha for all t in M. 01:02:48.460 --> 01:02:49.885 align:middle line:90% And what is w of t? 01:02:49.885 --> 01:02:53.610 align:middle line:90% 01:02:53.610 --> 01:02:57.450 align:middle line:84% This is equal to the real part of u of t. 01:02:57.450 --> 01:03:01.130 align:middle line:90% 01:03:01.130 --> 01:03:03.920 align:middle line:84% Which let me remind you, this is just 01:03:03.920 --> 01:03:09.860 align:middle line:84% equal to u of t plus u of t complex conjugate over 2. 01:03:09.860 --> 01:03:19.520 align:middle line:90% 01:03:19.520 --> 01:03:22.760 align:middle line:84% Now, the real part of any complex number 01:03:22.760 --> 01:03:26.750 align:middle line:84% is always less than or equal to the absolute value 01:03:26.750 --> 01:03:29.100 align:middle line:90% of that complex number. 01:03:29.100 --> 01:03:41.405 align:middle line:84% So we haven't chosen what alpha is yet. 01:03:41.405 --> 01:03:43.530 align:middle line:84% I'll show you how to choose alpha in just a minute. 01:03:43.530 --> 01:03:50.560 align:middle line:84% Note that for all t in M w of t, an absolute value, 01:03:50.560 --> 01:03:54.390 align:middle line:84% which remember is defined to be the real part of u of t-- 01:03:54.390 --> 01:04:02.460 align:middle line:84% this is less than or equal to the absolute value of u of t. 01:04:02.460 --> 01:04:04.500 align:middle line:84% I should say the modulus of u of t. 01:04:04.500 --> 01:04:08.340 align:middle line:84% The modulus of a complex number is the square root 01:04:08.340 --> 01:04:10.710 align:middle line:84% of the sum of the real part squared 01:04:10.710 --> 01:04:12.160 align:middle line:90% and the imaginary part squared. 01:04:12.160 --> 01:04:15.040 align:middle line:84% So that's always less than or equal to that. 01:04:15.040 --> 01:04:22.810 align:middle line:84% And by assumption, this is less than or equal to the norm of t. 01:04:22.810 --> 01:04:29.350 align:middle line:84% Now, we're going to use the fact that w is real valued. 01:04:29.350 --> 01:04:34.240 align:middle line:90% Then w of t1 minus w of t2-- 01:04:34.240 --> 01:04:37.990 align:middle line:84% so let me say for all t1, t2 in M-- 01:04:37.990 --> 01:04:42.620 align:middle line:84% if I look at w of t1 minus w of t2, this is equal to-- 01:04:42.620 --> 01:04:47.740 align:middle line:84% and since u is linear and taking the real part is linear, 01:04:47.740 --> 01:04:53.050 align:middle line:84% this is equal to w of t1 minus w of t2. 01:04:53.050 --> 01:04:55.730 align:middle line:84% This is less than or equal to its absolute value. 01:04:55.730 --> 01:04:58.255 align:middle line:84% This is where I'm using the fact that it's real value. 01:04:58.255 --> 01:05:03.690 align:middle line:90% 01:05:03.690 --> 01:05:07.320 align:middle line:84% And as we've just shown that this is always less than 01:05:07.320 --> 01:05:13.330 align:middle line:84% or equal to the absolute value, we have that. 01:05:13.330 --> 01:05:19.000 align:middle line:90% 01:05:19.000 --> 01:05:21.220 align:middle line:84% And now, I'm going to do one more thing 01:05:21.220 --> 01:05:22.480 align:middle line:90% and add and subtract x. 01:05:22.480 --> 01:05:26.660 align:middle line:90% So in the end, we want-- 01:05:26.660 --> 01:05:29.150 align:middle line:90% whoa, whoa, whoa. 01:05:29.150 --> 01:05:31.263 align:middle line:90% This should have been x. 01:05:31.263 --> 01:05:32.180 align:middle line:90% It should have been x. 01:05:32.180 --> 01:05:33.110 align:middle line:90% Sorry, sorry, sorry. 01:05:33.110 --> 01:05:37.070 align:middle line:84% I hope you fast forwarded to this point 01:05:37.070 --> 01:05:41.360 align:middle line:84% and saw that this should have been x and x. 01:05:41.360 --> 01:05:46.950 align:middle line:84% So in the end, we want to somehow connect this to-- 01:05:46.950 --> 01:05:48.720 align:middle line:90% and I did it again here-- 01:05:48.720 --> 01:05:51.540 align:middle line:90% x, to the norm of t minus x. 01:05:51.540 --> 01:05:54.270 align:middle line:84% So what I'm going to do is add and subtract x and use 01:05:54.270 --> 01:05:55.620 align:middle line:90% the triangle inequality. 01:05:55.620 --> 01:06:03.570 align:middle line:84% And this is less than or equal to t1 minus x plus t2 minus x. 01:06:03.570 --> 01:06:17.780 align:middle line:84% And therefore, w of t1 minus the norm of t1 minus x 01:06:17.780 --> 01:06:27.560 align:middle line:84% is less than or equal to w of t2 plus norm of t minus t2 minus x 01:06:27.560 --> 01:06:44.530 align:middle line:84% for all t1 and t2 in M. So this holds for all t1 and t2. 01:06:44.530 --> 01:06:52.850 align:middle line:84% So I could fix t2 and take the sup over all t1's, 01:06:52.850 --> 01:06:55.760 align:middle line:90% which implies that sup of-- 01:06:55.760 --> 01:07:06.030 align:middle line:90% 01:07:06.030 --> 01:07:09.015 align:middle line:84% so let's not make that a t1, I could just say t-- 01:07:09.015 --> 01:07:13.330 align:middle line:90% 01:07:13.330 --> 01:07:18.880 align:middle line:84% is always less than or equal to w of t2 plus t2 minus x. 01:07:18.880 --> 01:07:22.660 align:middle line:90% And this holds for all t2 in M. 01:07:22.660 --> 01:07:25.480 align:middle line:84% So the fact that this holds for all t1 tells you 01:07:25.480 --> 01:07:30.460 align:middle line:84% this thing is an upper bound of this for all t in M. 01:07:30.460 --> 01:07:31.990 align:middle line:84% And therefore, its supremum is less 01:07:31.990 --> 01:07:35.260 align:middle line:84% than or equal to this thing for all t2 in M. 01:07:35.260 --> 01:07:37.600 align:middle line:84% And therefore, this quantity on the left 01:07:37.600 --> 01:07:44.620 align:middle line:84% is a lower bound for this thing on the right for all t2 in M. 01:07:44.620 --> 01:07:49.270 align:middle line:84% And therefore, we can conclude that the sup of t 01:07:49.270 --> 01:07:56.500 align:middle line:84% in M of w of t minus norm of t minus x is less than 01:07:56.500 --> 01:08:07.970 align:middle line:84% or equal to the inf over all t in M w of t plus t minus x. 01:08:07.970 --> 01:08:14.570 align:middle line:90% 01:08:14.570 --> 01:08:16.050 align:middle line:90% How do I choose alpha? 01:08:16.050 --> 01:08:19.414 align:middle line:84% So I have these two numbers here which are related in this way. 01:08:19.414 --> 01:08:22.939 align:middle line:90% 01:08:22.939 --> 01:08:30.640 align:middle line:84% I choose alpha between these two numbers-- between this number 01:08:30.640 --> 01:08:31.433 align:middle line:90% and this number. 01:08:31.433 --> 01:08:33.100 align:middle line:84% So there's a less than or equal to sign, 01:08:33.100 --> 01:08:35.950 align:middle line:84% so I can pick some number in between them. 01:08:35.950 --> 01:08:37.390 align:middle line:84% Maybe these two things are equal, 01:08:37.390 --> 01:08:40.270 align:middle line:84% and therefore, alpha is equal to both of them. 01:08:40.270 --> 01:08:42.200 align:middle line:84% Or I could just choose alpha to be this one. 01:08:42.200 --> 01:08:42.950 align:middle line:90% It doesn't matter. 01:08:42.950 --> 01:08:53.279 align:middle line:90% 01:08:53.279 --> 01:08:55.979 align:middle line:84% And the proof is just about completed. 01:08:55.979 --> 01:09:06.569 align:middle line:90% 01:09:06.569 --> 01:09:10.899 align:middle line:84% Now, I'm going to show that this alpha works. 01:09:10.899 --> 01:09:18.569 align:middle line:84% So then for all t in capital M, I 01:09:18.569 --> 01:09:26.040 align:middle line:84% have w of t minus norm of t minus x 01:09:26.040 --> 01:09:29.040 align:middle line:84% is less than or equal to alpha is less than 01:09:29.040 --> 01:09:37.770 align:middle line:84% or equal to w of t plus norm of t minus x. 01:09:37.770 --> 01:09:45.660 align:middle line:84% And therefore, it's less than or equal to alpha minus w of t 01:09:45.660 --> 01:09:47.630 align:middle line:84% is less than or equal to norm of t 01:09:47.630 --> 01:09:56.060 align:middle line:84% minus x, which is the same as alpha minus w of t, 01:09:56.060 --> 01:09:56.705 align:middle line:90% I should say. 01:09:56.705 --> 01:10:07.530 align:middle line:90% 01:10:07.530 --> 01:10:10.890 align:middle line:84% All right, so we showed how to do it for-- 01:10:10.890 --> 01:10:13.260 align:middle line:84% we were able to choose an alpha so that essentially, 01:10:13.260 --> 01:10:15.480 align:middle line:84% this inequality holds for the real part. 01:10:15.480 --> 01:10:18.890 align:middle line:84% We can do that also for the imaginary part. 01:10:18.890 --> 01:10:24.000 align:middle line:90%