WEBVTT 00:00:00.000 --> 00:00:19.380 align:middle line:84% [SQUEAKING] [RUSTLING] [CLICKING] 00:00:19.380 --> 00:00:30.090 align:middle line:90% PROFESSOR: OK, so ortho-- 00:00:30.090 --> 00:00:33.260 align:middle line:90% 00:00:33.260 --> 00:00:36.590 align:middle line:84% today, we're going to be discussing orthonormal bases 00:00:36.590 --> 00:00:39.090 align:middle line:90% of a Hilbert space. 00:00:39.090 --> 00:00:43.140 align:middle line:84% So let me just recall what we did at the end of last time. 00:00:43.140 --> 00:00:50.420 align:middle line:84% We introduced maximal orthonormal sets, so 00:00:50.420 --> 00:00:55.610 align:middle line:90% a collection of vectors in a-- 00:00:55.610 --> 00:00:57.590 align:middle line:84% we could say a pre-Hilbert space, 00:00:57.590 --> 00:00:59.540 align:middle line:84% but let's say in a Hilbert space. 00:00:59.540 --> 00:01:04.860 align:middle line:90% 00:01:04.860 --> 00:01:25.560 align:middle line:84% So this is maximal if u in H and u, 00:01:25.560 --> 00:01:31.740 align:middle line:84% e lambda equals 0 for all lambda in the index 00:01:31.740 --> 00:01:36.630 align:middle line:90% implies that u equals 0. 00:01:36.630 --> 00:01:39.940 align:middle line:90% So a collection of-- 00:01:39.940 --> 00:01:42.250 align:middle line:84% so I should say that a collection of orthonormal-- 00:01:42.250 --> 00:01:50.330 align:middle line:90% 00:01:50.330 --> 00:01:53.450 align:middle line:84% orthonormal vectors is maximal if the only thing 00:01:53.450 --> 00:01:58.400 align:middle line:84% that's orthogonal to all of them is the zero vector. 00:01:58.400 --> 00:02:01.310 align:middle line:90% And last time, we proved that-- 00:02:01.310 --> 00:02:08.699 align:middle line:90% 00:02:08.699 --> 00:02:17.080 align:middle line:84% let H be a separable Hilbert space, 00:02:17.080 --> 00:02:19.510 align:middle line:84% meaning it has a countable dense subset . 00:02:19.510 --> 00:02:23.590 align:middle line:84% Most-- I mean, I think all of the Hilbert spaces 00:02:23.590 --> 00:02:27.850 align:middle line:84% you've come into contact with are separable. 00:02:27.850 --> 00:02:36.147 align:middle line:90% Cn, Rn, also little l2, big L2-- 00:02:36.147 --> 00:02:37.980 align:middle line:84% this is what you're doing in the assignment. 00:02:37.980 --> 00:03:02.540 align:middle line:84% That's also separable-- then H has a countable maximal 00:03:02.540 --> 00:03:10.912 align:middle line:90% orthonormal subset, e sub n. 00:03:10.912 --> 00:03:12.620 align:middle line:84% So we proved this at the end of last time 00:03:12.620 --> 00:03:15.290 align:middle line:90% via the Gram-Schmidt process. 00:03:15.290 --> 00:03:20.291 align:middle line:84% We took a countable collection of-- 00:03:20.291 --> 00:03:24.750 align:middle line:84% a countable collection of elements in the Hilbert space, 00:03:24.750 --> 00:03:25.800 align:middle line:90% which were dense. 00:03:25.800 --> 00:03:33.300 align:middle line:84% And then we formed this countable orthonormal subset. 00:03:33.300 --> 00:03:34.920 align:middle line:84% And I shouldn't put N in, capital 00:03:34.920 --> 00:03:37.350 align:middle line:84% N because this could be a finite collection 00:03:37.350 --> 00:03:39.630 align:middle line:84% or it could be countably infinite collection. 00:03:39.630 --> 00:03:43.560 align:middle line:84% But then we applied the Gram-Schmidt process 00:03:43.560 --> 00:03:47.280 align:middle line:84% to that collection of dense elements 00:03:47.280 --> 00:03:52.590 align:middle line:84% and came up with this maximal orthonormal subset. 00:03:52.590 --> 00:03:59.770 align:middle line:84% Now, we have a special name for countable maximal orthonormal 00:03:59.770 --> 00:04:06.750 align:middle line:84% subsets for a good reason as we'll see. 00:04:06.750 --> 00:04:09.840 align:middle line:90% 00:04:09.840 --> 00:04:19.785 align:middle line:90% So let H be a Hilbert space. 00:04:19.785 --> 00:04:27.490 align:middle line:90% 00:04:27.490 --> 00:04:45.585 align:middle line:84% An orthonormal basis of H is a countable maximal orthonormal 00:04:45.585 --> 00:04:46.085 align:middle line:90% subset-- 00:04:46.085 --> 00:04:55.180 align:middle line:90% 00:04:55.180 --> 00:04:56.920 align:middle line:90% I shouldn't say n and N again-- 00:04:56.920 --> 00:05:05.430 align:middle line:84% of H. So an orthonormal basis is just a special name 00:05:05.430 --> 00:05:08.820 align:middle line:84% that we give to those maximal orthonormal subsets which 00:05:08.820 --> 00:05:11.790 align:middle line:84% are countable, meaning they can be either finite or countably 00:05:11.790 --> 00:05:14.740 align:middle line:90% infinite. 00:05:14.740 --> 00:05:17.980 align:middle line:84% So again, all of the Hilbert spaces we've encountered 00:05:17.980 --> 00:05:22.090 align:middle line:84% in practice-- for example, Cn, little l2, big L2, 00:05:22.090 --> 00:05:26.830 align:middle line:84% which are all separable, meaning they do have countable maximal 00:05:26.830 --> 00:05:31.310 align:middle line:84% orthonormal subset, meaning they have an orthonormal basis-- 00:05:31.310 --> 00:05:34.630 align:middle line:90% so they have these objects. 00:05:34.630 --> 00:05:35.740 align:middle line:90% Now, why do we call-- 00:05:35.740 --> 00:05:38.390 align:middle line:90% 00:05:38.390 --> 00:05:42.551 align:middle line:84% why do we call a countable orthonormal-- 00:05:42.551 --> 00:05:45.320 align:middle line:84% or a countable maximal orthonormal subset an 00:05:45.320 --> 00:05:46.350 align:middle line:90% all orthonormal basis? 00:05:46.350 --> 00:05:48.620 align:middle line:84% So a basis is supposed to be something 00:05:48.620 --> 00:05:52.280 align:middle line:84% where you can write every vector in the space 00:05:52.280 --> 00:05:56.780 align:middle line:84% as some sort of linear combination of the elements 00:05:56.780 --> 00:05:58.160 align:middle line:90% of the basis, right? 00:05:58.160 --> 00:05:59.990 align:middle line:90% Now, we dealt with Hamel basis-- 00:05:59.990 --> 00:06:02.510 align:middle line:84% or bases, which are you can write every element 00:06:02.510 --> 00:06:11.320 align:middle line:84% as a finite linear combination of the elements in this subset. 00:06:11.320 --> 00:06:15.310 align:middle line:84% Now, in this setting, in what sense is this a basis? 00:06:15.310 --> 00:06:17.630 align:middle line:84% Or what is-- I shouldn't say, is it a basis? 00:06:17.630 --> 00:06:20.980 align:middle line:84% But in what sense does this connect 00:06:20.980 --> 00:06:25.030 align:middle line:84% to what we've encountered, say, in finite dimensions? 00:06:25.030 --> 00:06:33.450 align:middle line:84% An orthonormal basis-- so as stated in the next theorem, 00:06:33.450 --> 00:06:37.350 align:middle line:84% every element can be written now as 00:06:37.350 --> 00:06:39.750 align:middle line:84% an infinite linear combination of elements 00:06:39.750 --> 00:06:42.780 align:middle line:84% of an orthonormal basis, not necessarily finite. 00:06:42.780 --> 00:06:46.390 align:middle line:84% So the statement is the following-- 00:06:46.390 --> 00:07:07.280 align:middle line:84% if en is an orthonormal basis in the Hilbert space H, 00:07:07.280 --> 00:07:22.470 align:middle line:84% then for all u in H, if I look at the partial sum, un 00:07:22.470 --> 00:07:26.970 align:middle line:84% inner product e sub n, e sub n, this converges as n 00:07:26.970 --> 00:07:37.280 align:middle line:84% goes to infinity to the element u in the space H. 00:07:37.280 --> 00:07:45.450 align:middle line:84% Written in shorter form is that u is equal to sum from n 00:07:45.450 --> 00:07:52.460 align:middle line:84% equals 1 to infinity of u inner product e sub n, e sub n. 00:07:52.460 --> 00:07:56.435 align:middle line:84% Sometimes, this is referred to as a Fourier-Bessel series. 00:07:56.435 --> 00:07:59.470 align:middle line:90% 00:07:59.470 --> 00:08:02.080 align:middle line:84% So just like in finite dimensions, 00:08:02.080 --> 00:08:06.400 align:middle line:84% if you have an orthonormal basis consisting 00:08:06.400 --> 00:08:09.010 align:middle line:84% of only finitely many vectors, then you 00:08:09.010 --> 00:08:14.710 align:middle line:84% can expand every vector in that, say, Cn in this way. 00:08:14.710 --> 00:08:17.020 align:middle line:84% Now, the statement is, in a Hilbert space, 00:08:17.020 --> 00:08:19.750 align:middle line:84% if you have an orthonormal basis in this sense, 00:08:19.750 --> 00:08:22.990 align:middle line:84% meaning it's a countable maximal or orthonormal subset, 00:08:22.990 --> 00:08:31.740 align:middle line:84% then you can expand every element 00:08:31.740 --> 00:08:36.030 align:middle line:84% as an infinite linear combination of these elements. 00:08:36.030 --> 00:08:47.420 align:middle line:90% 00:08:47.420 --> 00:09:02.460 align:middle line:84% OK, so to prove this, we will use Bessel's inequality 00:09:02.460 --> 00:09:04.410 align:middle line:84% and the completeness of H. This is 00:09:04.410 --> 00:09:07.680 align:middle line:84% where we're using the fact that H is not a pre-Hilbert space, 00:09:07.680 --> 00:09:09.518 align:middle line:90% but a Hilbert space, all right? 00:09:09.518 --> 00:09:12.060 align:middle line:84% What we're going to do is we're going to be able to show that 00:09:12.060 --> 00:09:16.300 align:middle line:84% the right-hand side of this equality-- 00:09:16.300 --> 00:09:21.780 align:middle line:84% so we're going to be able to show that this series here 00:09:21.780 --> 00:09:26.820 align:middle line:84% converges in H. That uses Bessel's 00:09:26.820 --> 00:09:30.180 align:middle line:84% inequality and the fact that H is a Hilbert space 00:09:30.180 --> 00:09:31.410 align:middle line:90% converges to some element. 00:09:31.410 --> 00:09:35.220 align:middle line:84% And then we're going to use the fact that this collection is 00:09:35.220 --> 00:09:39.450 align:middle line:84% maximal to show that the inner product of whatever this 00:09:39.450 --> 00:09:44.330 align:middle line:84% defines with each element e sub n is 00:09:44.330 --> 00:09:49.790 align:middle line:84% the same as the inner product of this thing with each element e 00:09:49.790 --> 00:09:53.600 align:middle line:84% sub n, and then conclude that this thing on the right side 00:09:53.600 --> 00:09:54.860 align:middle line:90% has to equal this thing. 00:09:54.860 --> 00:09:58.660 align:middle line:90% 00:09:58.660 --> 00:10:09.200 align:middle line:84% All right, so if we first prove that 00:10:09.200 --> 00:10:10.895 align:middle line:90% the sequence of partial sums-- 00:10:10.895 --> 00:10:16.900 align:middle line:90% 00:10:16.900 --> 00:10:19.890 align:middle line:90% so m is Cauchy. 00:10:19.890 --> 00:10:29.290 align:middle line:90% 00:10:29.290 --> 00:10:32.020 align:middle line:90% So let epsilon be positive. 00:10:32.020 --> 00:10:36.180 align:middle line:90% 00:10:36.180 --> 00:10:46.580 align:middle line:84% So by Bessel's inequality, we have that the sum of n 00:10:46.580 --> 00:10:52.290 align:middle line:84% equals 1 to infinity of u squared-- 00:10:52.290 --> 00:10:53.380 align:middle line:90% this thing converges. 00:10:53.380 --> 00:10:54.930 align:middle line:84% So the only thing we have to check 00:10:54.930 --> 00:10:58.020 align:middle line:84% is that, since this is a series involving 00:10:58.020 --> 00:11:01.120 align:middle line:84% non-negative terms, that this thing is bounded above. 00:11:01.120 --> 00:11:02.700 align:middle line:84% And we have by Bessel's inequality 00:11:02.700 --> 00:11:06.640 align:middle line:84% that this is always bounded above by the norm of u squared. 00:11:06.640 --> 00:11:12.030 align:middle line:84% So this sum, this series converges to a finite number. 00:11:12.030 --> 00:11:15.650 align:middle line:90% 00:11:15.650 --> 00:11:18.140 align:middle line:90% In particular, the series-- 00:11:18.140 --> 00:11:20.750 align:middle line:84% the partial sums corresponding to this series 00:11:20.750 --> 00:11:22.707 align:middle line:84% is a Cauchy sequence of real numbers, 00:11:22.707 --> 00:11:23.915 align:middle line:90% of non-negative real numbers. 00:11:23.915 --> 00:11:27.990 align:middle line:90% 00:11:27.990 --> 00:11:39.480 align:middle line:84% Thus, there exists an N natural number such that for all-- 00:11:39.480 --> 00:11:44.650 align:middle line:84% let's see-- all N bigger than or equal to M, 00:11:44.650 --> 00:11:50.340 align:middle line:84% we have that the tail is small, meaning n 00:11:50.340 --> 00:11:59.170 align:middle line:84% equals N plus 1 infinity of less than epsilon. 00:11:59.170 --> 00:12:04.810 align:middle line:84% You can just look back into your 18.100 notes or whatever that 00:12:04.810 --> 00:12:07.510 align:middle line:84% if I have a series of non-negative terms which 00:12:07.510 --> 00:12:10.150 align:middle line:84% converges, then for every epsilon, 00:12:10.150 --> 00:12:13.360 align:middle line:84% I can find a capital N so that the tail, 00:12:13.360 --> 00:12:18.770 align:middle line:84% so starting the sum at a large enough entry, will be small. 00:12:18.770 --> 00:12:23.180 align:middle line:90% 00:12:23.180 --> 00:12:32.920 align:middle line:84% Then for all m greater than l greater than this capital M, 00:12:32.920 --> 00:12:39.325 align:middle line:84% if I compute the norm of n equals 1 to m u 00:12:39.325 --> 00:12:48.400 align:middle line:84% inner product e sub n, e sub n minus N equals 1 to l u 00:12:48.400 --> 00:12:54.400 align:middle line:90% e sub n, e sub n squared. 00:12:54.400 --> 00:12:56.460 align:middle line:90% And let's make this a square. 00:12:56.460 --> 00:12:59.220 align:middle line:90% I can do that. 00:12:59.220 --> 00:13:01.635 align:middle line:90% This you can just compute. 00:13:01.635 --> 00:13:04.460 align:middle line:90% 00:13:04.460 --> 00:13:08.330 align:middle line:84% Expanding this out as the inner product of this whole thing 00:13:08.330 --> 00:13:15.860 align:middle line:84% with itself, this gives you sum from n equals l plus 1 to m 00:13:15.860 --> 00:13:20.460 align:middle line:90% of u en squared. 00:13:20.460 --> 00:13:25.610 align:middle line:84% And since l is bigger than or equal to m, 00:13:25.610 --> 00:13:29.300 align:middle line:84% this is certainly less than or equal to-- 00:13:29.300 --> 00:13:32.630 align:middle line:84% well, I can put infinity here. n equals l 00:13:32.630 --> 00:13:38.210 align:middle line:84% plus 1 u inner product e sub n squared. 00:13:38.210 --> 00:13:40.970 align:middle line:84% And that's less than-- since l is 00:13:40.970 --> 00:13:44.850 align:middle line:84% bigger than or equal to capital M, we can use this inequality. 00:13:44.850 --> 00:13:48.330 align:middle line:84% And that's less than epsilon squared. 00:13:48.330 --> 00:13:52.400 align:middle line:84% So we've shown that for n and l bigger than or equal to capital 00:13:52.400 --> 00:13:56.630 align:middle line:84% M, the norm squared of the difference 00:13:56.630 --> 00:13:58.192 align:middle line:90% is less than epsilon squared. 00:13:58.192 --> 00:13:59.900 align:middle line:84% And therefore, the norm of the difference 00:13:59.900 --> 00:14:04.610 align:middle line:84% is less than epsilon, proving that the sequence 00:14:04.610 --> 00:14:06.785 align:middle line:90% of partial sums is Cauchy. 00:14:06.785 --> 00:14:10.264 align:middle line:90% 00:14:10.264 --> 00:14:13.540 align:middle line:84% OK, so we've proven that the sequence of partial sums 00:14:13.540 --> 00:14:14.320 align:middle line:90% is Cauchy. 00:14:14.320 --> 00:14:26.651 align:middle line:84% So since H is complete, there exists 00:14:26.651 --> 00:14:32.990 align:middle line:84% an element u in H such that u-bar is 00:14:32.990 --> 00:14:36.620 align:middle line:84% equal to limit as n goes to infinity 00:14:36.620 --> 00:14:44.660 align:middle line:84% equals 1 m of these partial sums, where this limit means 00:14:44.660 --> 00:14:47.705 align:middle line:84% that the norm of the difference in H converges to 0. 00:14:47.705 --> 00:14:51.690 align:middle line:90% 00:14:51.690 --> 00:14:55.110 align:middle line:84% And so just in shorthand, again, this means-- 00:14:55.110 --> 00:15:02.140 align:middle line:90% 00:15:02.140 --> 00:15:06.310 align:middle line:84% now, what I'd like to do is show that u-bar equals u. 00:15:06.310 --> 00:15:07.750 align:middle line:90% I don't mean complex conjugate. 00:15:07.750 --> 00:15:14.500 align:middle line:90% I just mean an element of H. 00:15:14.500 --> 00:15:17.230 align:middle line:84% And now, how we're going to do that is show 00:15:17.230 --> 00:15:21.400 align:middle line:84% that the inner product of u minus u-bar 00:15:21.400 --> 00:15:25.600 align:middle line:84% against every element of this maximal orthonormal subset 00:15:25.600 --> 00:15:29.650 align:middle line:84% is 0, and therefore conclude u equals u-bar. 00:15:29.650 --> 00:15:33.550 align:middle line:84% Now, we have this result, that the inner product 00:15:33.550 --> 00:15:38.450 align:middle line:84% is continuous with respect to the entry. 00:15:38.450 --> 00:15:43.250 align:middle line:84% So by continuity of the inner product, 00:15:43.250 --> 00:15:47.590 align:middle line:84% we have that the inner product of u-- 00:15:47.590 --> 00:15:51.670 align:middle line:84% so let me-- so then for all l natural number, 00:15:51.670 --> 00:15:57.820 align:middle line:84% the inner product of u minus u-bar with el, 00:15:57.820 --> 00:16:00.730 align:middle line:90% this is equal to-- 00:16:00.730 --> 00:16:02.620 align:middle line:84% u-bar is equal to the limit as m goes 00:16:02.620 --> 00:16:05.200 align:middle line:84% to infinity of this partial sum in H. 00:16:05.200 --> 00:16:09.580 align:middle line:84% So this inner product is equal to the limit 00:16:09.580 --> 00:16:17.580 align:middle line:84% as m goes to infinity of u minus sum from n equals 1 to m of u 00:16:17.580 --> 00:16:23.110 align:middle line:90% en, en inner product el. 00:16:23.110 --> 00:16:26.140 align:middle line:90% And this is also-- 00:16:26.140 --> 00:16:28.140 align:middle line:84% I mean, I'm sure you've seen this at some point. 00:16:28.140 --> 00:16:30.350 align:middle line:84% But this is also why the coefficients appearing 00:16:30.350 --> 00:16:34.770 align:middle line:84% in front of the en's are the way they are. 00:16:34.770 --> 00:16:38.270 align:middle line:84% So this is equal to the limit as m 00:16:38.270 --> 00:16:45.860 align:middle line:84% goes to infinity of u inner product el minus the sum from n 00:16:45.860 --> 00:16:54.230 align:middle line:84% equals 1 to m of u en, en inner product el. 00:16:54.230 --> 00:16:58.440 align:middle line:90% 00:16:58.440 --> 00:17:02.280 align:middle line:84% And when n does not equal l-- so remember, 00:17:02.280 --> 00:17:04.230 align:middle line:90% these are orthonormal. 00:17:04.230 --> 00:17:06.240 align:middle line:84% When n does not equal l, I get 0. 00:17:06.240 --> 00:17:08.010 align:middle line:90% And when n equals l, I get 1. 00:17:08.010 --> 00:17:09.869 align:middle line:90% So all I pick up is u-- 00:17:09.869 --> 00:17:12.670 align:middle line:90% 00:17:12.670 --> 00:17:16.750 align:middle line:84% this just reduces to u inner product el, which 00:17:16.750 --> 00:17:18.326 align:middle line:90% cancels with that one. 00:17:18.326 --> 00:17:19.284 align:middle line:90% And therefore, I get 0. 00:17:19.284 --> 00:17:22.800 align:middle line:90% 00:17:22.800 --> 00:17:26.250 align:middle line:84% All right, so I've shown that for every l, u 00:17:26.250 --> 00:17:29.670 align:middle line:84% minus u-bar inner product with el is 0. 00:17:29.670 --> 00:17:46.350 align:middle line:84% And since this collection is maximal, anything 00:17:46.350 --> 00:17:48.780 align:middle line:90% that inner product with-- 00:17:48.780 --> 00:17:53.250 align:middle line:84% so the only element that's orthogonal to every element 00:17:53.250 --> 00:17:55.680 align:middle line:90% in this collection is 0. 00:17:55.680 --> 00:17:59.490 align:middle line:84% That implies that u minus u-bar equals 0. 00:17:59.490 --> 00:18:04.410 align:middle line:84% And therefore, u equals u-bar, which means 00:18:04.410 --> 00:18:09.345 align:middle line:90% u is equal to this series. 00:18:09.345 --> 00:18:21.300 align:middle line:90% 00:18:21.300 --> 00:18:35.420 align:middle line:84% OK, so we've shown that if a Hilbert space has 00:18:35.420 --> 00:18:38.750 align:middle line:84% an orthonormal basis, then every element 00:18:38.750 --> 00:18:42.800 align:middle line:84% can be expanded in what's typically 00:18:42.800 --> 00:18:46.520 align:middle line:84% referred to as a Fourier-Bessel series in this way, 00:18:46.520 --> 00:18:51.850 align:middle line:84% in terms of the elements of the orthonormal basis. 00:18:51.850 --> 00:19:05.420 align:middle line:84% Now, let me just tie one thing up. 00:19:05.420 --> 00:19:08.150 align:middle line:84% So we know that every separable Hilbert space 00:19:08.150 --> 00:19:12.788 align:middle line:84% does have an orthonormal basis, a countable maximal 00:19:12.788 --> 00:19:13.580 align:middle line:90% orthonormal subset. 00:19:13.580 --> 00:19:17.120 align:middle line:90% 00:19:17.120 --> 00:19:20.000 align:middle line:84% And so if H is separable, that implies 00:19:20.000 --> 00:19:21.710 align:middle line:90% H has an orthonormal basis. 00:19:21.710 --> 00:19:23.570 align:middle line:84% Now, what this theorem also proves 00:19:23.570 --> 00:19:27.710 align:middle line:84% is that if H has a orthonormal basis, then H is separable. 00:19:27.710 --> 00:19:32.390 align:middle line:84% So let me just state that as a simple theorem that 00:19:32.390 --> 00:19:33.290 align:middle line:90% follows from this. 00:19:33.290 --> 00:19:35.840 align:middle line:90% 00:19:35.840 --> 00:19:37.520 align:middle line:90% So we've shown that if H-- 00:19:37.520 --> 00:19:39.770 align:middle line:84% that was the first theorem my stated at the beginning, 00:19:39.770 --> 00:19:43.340 align:middle line:84% that if H is separable, then it has an orthonormal basis. 00:19:43.340 --> 00:19:51.645 align:middle line:84% If H has an orthonormal basis, H is a Hilbert space. 00:19:51.645 --> 00:19:54.150 align:middle line:90% 00:19:54.150 --> 00:19:56.070 align:middle line:84% So I should have said that at the beginning. 00:19:56.070 --> 00:19:57.240 align:middle line:90% But H is a Hilbert space. 00:19:57.240 --> 00:20:07.100 align:middle line:84% Then H is separable, meaning it has a countably-- 00:20:07.100 --> 00:20:10.420 align:middle line:90% a countable dense subset. 00:20:10.420 --> 00:20:12.045 align:middle line:84% So what is this countable dense subset? 00:20:12.045 --> 00:20:14.808 align:middle line:90% 00:20:14.808 --> 00:20:16.350 align:middle line:84% I'm just going to give you the subset 00:20:16.350 --> 00:20:27.570 align:middle line:84% and then talk through why this subset works. 00:20:27.570 --> 00:20:46.430 align:middle line:84% So suppose en is an orthonormal basis for H, again 00:20:46.430 --> 00:20:49.550 align:middle line:90% a Hilbert space. 00:20:49.550 --> 00:20:56.120 align:middle line:84% Then if I define S to be the union over all, 00:20:56.120 --> 00:21:02.240 align:middle line:84% lets say, m natural number of elements of the form sum from n 00:21:02.240 --> 00:21:10.490 align:middle line:84% equals 1 to m of qn en where q1-- 00:21:10.490 --> 00:21:15.930 align:middle line:90% these are just rational numbers. 00:21:15.930 --> 00:21:27.080 align:middle line:84% So first off, this is a countable subset of H. 00:21:27.080 --> 00:21:28.050 align:middle line:90% Why is that? 00:21:28.050 --> 00:21:37.300 align:middle line:84% Well, each of these is in one to one correspondence 00:21:37.300 --> 00:21:41.980 align:middle line:84% with the m-fold Cartesian product 00:21:41.980 --> 00:21:43.240 align:middle line:90% of the rational numbers. 00:21:43.240 --> 00:21:45.970 align:middle line:84% And now, the rational numbers are countable. 00:21:45.970 --> 00:21:51.544 align:middle line:84% And you proved back in 18.100 that any-- 00:21:51.544 --> 00:21:55.940 align:middle line:84% in fact, well, any Cartesian product, 00:21:55.940 --> 00:22:00.470 align:middle line:84% finite Cartesian product of a countable set 00:22:00.470 --> 00:22:01.820 align:middle line:90% is, again, countable. 00:22:01.820 --> 00:22:03.560 align:middle line:84% And then we're taking a countable union. 00:22:03.560 --> 00:22:07.400 align:middle line:84% Another thing you proved in 18.100 is that a countable 00:22:07.400 --> 00:22:09.410 align:middle line:84% union of countable subsets is countable. 00:22:09.410 --> 00:22:13.980 align:middle line:90% So that's why S is countable. 00:22:13.980 --> 00:22:19.650 align:middle line:84% And then so now, I'm just going to state 00:22:19.650 --> 00:22:29.540 align:middle line:84% by the previous theorem S is dense in H. 00:22:29.540 --> 00:22:31.920 align:middle line:84% And then I'm going to put a box here and explain why. 00:22:31.920 --> 00:22:39.970 align:middle line:84% So every element-- so if H has an orthonormal basis, 00:22:39.970 --> 00:22:44.020 align:middle line:84% by the theorem we proved, every element can be expanded in one 00:22:44.020 --> 00:22:47.440 align:middle line:84% of these Fourier-Bessel series as coefficients times-- 00:22:47.440 --> 00:22:49.090 align:middle line:84% as an infinite linear combination 00:22:49.090 --> 00:22:53.170 align:middle line:84% of orthonormal vectors, of these orthogonal vectors. 00:22:53.170 --> 00:22:56.160 align:middle line:90% 00:22:56.160 --> 00:22:59.990 align:middle line:84% Now, this means the partial sums are converging 00:22:59.990 --> 00:23:03.410 align:middle line:90% to a given element u. 00:23:03.410 --> 00:23:06.200 align:middle line:84% So what we have to show for this thing 00:23:06.200 --> 00:23:08.670 align:middle line:84% to be dense-- we have to show for every epsilon, 00:23:08.670 --> 00:23:12.560 align:middle line:84% there exists something from this set within epsilon distance 00:23:12.560 --> 00:23:14.270 align:middle line:90% of that given vector. 00:23:14.270 --> 00:23:17.480 align:middle line:90% Now, give yourself a vector. 00:23:17.480 --> 00:23:20.270 align:middle line:84% Its Fourier-Bessel series converges to it. 00:23:20.270 --> 00:23:24.780 align:middle line:90% So we can cut off a-- 00:23:24.780 --> 00:23:26.940 align:middle line:84% we can cut the series off at a certain point 00:23:26.940 --> 00:23:31.260 align:middle line:84% and still be within epsilon over 2 00:23:31.260 --> 00:23:33.750 align:middle line:84% to that element we're trying to get close to. 00:23:33.750 --> 00:23:41.943 align:middle line:84% Then that finite-- that finite sum will be in one of these-- 00:23:41.943 --> 00:23:43.360 align:middle line:84% well, it won't be in one of these. 00:23:43.360 --> 00:23:47.080 align:middle line:84% But it will just be the sum from n equals 1 to m of some 00:23:47.080 --> 00:23:49.000 align:middle line:90% coefficients time the e sub n's. 00:23:49.000 --> 00:23:51.220 align:middle line:90% Now, the rational numbers-- 00:23:51.220 --> 00:23:58.490 align:middle line:84% I should say-- there, now that's correct. 00:23:58.490 --> 00:24:03.260 align:middle line:84% Now, any complex number here can be approximated 00:24:03.260 --> 00:24:07.670 align:middle line:84% by a rational number here plus i times the rational number here. 00:24:07.670 --> 00:24:14.370 align:middle line:84% This is still in one to one correspondence with q squared, 00:24:14.370 --> 00:24:17.130 align:middle line:84% the Cartesian product of with itself. 00:24:17.130 --> 00:24:18.900 align:middle line:84% And therefore, this is still countable-- 00:24:18.900 --> 00:24:22.992 align:middle line:84% and so I should have said 2m here. 00:24:22.992 --> 00:24:24.450 align:middle line:84% I was thinking about real numbers-- 00:24:24.450 --> 00:24:27.040 align:middle line:90% 00:24:27.040 --> 00:24:31.740 align:middle line:84% so that you can get close to the actual Fourier-Bessel 00:24:31.740 --> 00:24:34.080 align:middle line:84% coefficients that appear in that sum 00:24:34.080 --> 00:24:39.450 align:middle line:84% by now just rational or complex numbers with rational 00:24:39.450 --> 00:24:43.300 align:middle line:90% real and imaginary part. 00:24:43.300 --> 00:24:45.000 align:middle line:84% So I hope that explanation was clear. 00:24:45.000 --> 00:24:46.650 align:middle line:84% Maybe sit down, and think about it, 00:24:46.650 --> 00:24:51.190 align:middle line:84% and actually write down carefully the epsilon argument. 00:24:51.190 --> 00:24:55.890 align:middle line:84% But that's essentially why this is true. 00:24:55.890 --> 00:25:01.800 align:middle line:84% So what we've shown is that a Hilbert space is separable if 00:25:01.800 --> 00:25:04.510 align:middle line:84% and only if it has an orthonormal basis. 00:25:04.510 --> 00:25:07.620 align:middle line:84% So let me make a remark that what we've shown up 00:25:07.620 --> 00:25:19.810 align:middle line:84% to this point is that if H is a Hilbert space, 00:25:19.810 --> 00:25:36.660 align:middle line:84% then H is separable if and only if H has an orthonormal basis. 00:25:36.660 --> 00:25:47.262 align:middle line:90% 00:25:47.262 --> 00:25:54.420 align:middle line:84% All right, so we've proven that, in the Hilbert space, 00:25:54.420 --> 00:25:56.760 align:middle line:84% if a Hilbert space has an orthonormal basis, 00:25:56.760 --> 00:25:58.470 align:middle line:84% then every element can be expanded 00:25:58.470 --> 00:26:00.750 align:middle line:84% in this kind of infinite series involving 00:26:00.750 --> 00:26:02.190 align:middle line:90% or the orthonormal vectors. 00:26:02.190 --> 00:26:03.300 align:middle line:90% I say infinite. 00:26:03.300 --> 00:26:08.220 align:middle line:84% That's only if the orthonormal basis is countably infinite. 00:26:08.220 --> 00:26:12.220 align:middle line:84% If it was finite, then that's actually not an infinite sum. 00:26:12.220 --> 00:26:13.020 align:middle line:90% It's a finite sum. 00:26:13.020 --> 00:26:15.830 align:middle line:90% 00:26:15.830 --> 00:26:23.090 align:middle line:84% Now, what follows from that is that-- 00:26:23.090 --> 00:26:24.890 align:middle line:84% so we have Bessel's inequality, which 00:26:24.890 --> 00:26:30.140 align:middle line:84% says that the sum of squares of the coefficients appearing 00:26:30.140 --> 00:26:33.345 align:middle line:84% in this Fourier-Bessel series is always less 00:26:33.345 --> 00:26:34.970 align:middle line:84% than or equal to the norm of u squared. 00:26:34.970 --> 00:26:39.050 align:middle line:84% That's no matter what you assume about the orthonormal subset. 00:26:39.050 --> 00:26:42.170 align:middle line:84% That's for any orthonormal subset. 00:26:42.170 --> 00:26:47.120 align:middle line:84% But now, we have that if it's an orthonormal basis, then, 00:26:47.120 --> 00:26:48.350 align:middle line:90% in fact, we have equality. 00:26:48.350 --> 00:27:15.780 align:middle line:84% So if H is the Hilbert space and this is a countable orthonormal 00:27:15.780 --> 00:27:25.580 align:middle line:84% basis, then for all u in H, we have that the sum n equals 1-- 00:27:25.580 --> 00:27:28.970 align:middle line:84% I'll just put sum over n because this may be a finite sum-- 00:27:28.970 --> 00:27:33.858 align:middle line:84% u en, which by Bessel's inequality, 00:27:33.858 --> 00:27:35.900 align:middle line:84% we always had less than or equal to the norm of u 00:27:35.900 --> 00:27:38.900 align:middle line:84% squared, in fact, equals norm of u squared. 00:27:38.900 --> 00:27:42.260 align:middle line:84% And this is sometimes referred to as Parseval's identity. 00:27:42.260 --> 00:27:48.140 align:middle line:90% 00:27:48.140 --> 00:27:50.375 align:middle line:84% All right, so what's the proof of that? 00:27:50.375 --> 00:27:54.690 align:middle line:90% 00:27:54.690 --> 00:28:03.350 align:middle line:84% We have that u equals the sum of u, en en. 00:28:03.350 --> 00:28:09.780 align:middle line:84% And therefore, immediately, by the continuity 00:28:09.780 --> 00:28:17.580 align:middle line:84% of the inner product, we can write norm of u squared as-- 00:28:17.580 --> 00:28:21.600 align:middle line:84% so this is for the case that we have countably infinite. 00:28:21.600 --> 00:28:24.608 align:middle line:84% If it's finite, if it's a finite orthonormal basis, 00:28:24.608 --> 00:28:25.650 align:middle line:90% this follows immediately. 00:28:25.650 --> 00:28:28.680 align:middle line:90% But sum from m equals-- 00:28:28.680 --> 00:28:41.630 align:middle line:84% or the sum as m goes to infinity of n equals 1 to m u, en en sum 00:28:41.630 --> 00:28:51.260 align:middle line:84% from, let's say, l equals 1 to m u, el el. 00:28:51.260 --> 00:28:55.490 align:middle line:84% And this is equal to the limit as m 00:28:55.490 --> 00:29:00.720 align:middle line:84% goes to infinity of sum of n equals-- 00:29:00.720 --> 00:29:06.440 align:middle line:90% say nl equals 1 to m of u, en. 00:29:06.440 --> 00:29:08.390 align:middle line:84% And now, this constant here comes out 00:29:08.390 --> 00:29:10.190 align:middle line:90% with a complex conjugate-- 00:29:10.190 --> 00:29:16.400 align:middle line:90% u, el times en inner product el. 00:29:16.400 --> 00:29:18.390 align:middle line:90% And now, we only pick up-- 00:29:18.390 --> 00:29:20.945 align:middle line:84% so n and l are going from 1 to m. 00:29:20.945 --> 00:29:23.860 align:middle line:90% 00:29:23.860 --> 00:29:26.470 align:middle line:84% For a fixed n, we only pick up the l 00:29:26.470 --> 00:29:31.720 align:middle line:84% where l equals n because these vectors are orthonormal. 00:29:31.720 --> 00:29:34.300 align:middle line:84% And when n equals l, this gives me 1. 00:29:34.300 --> 00:29:35.380 align:middle line:90% So I only pick up-- 00:29:35.380 --> 00:29:40.290 align:middle line:90% 00:29:40.290 --> 00:29:48.330 align:middle line:84% when l equals n, I get u, en times the complex conjugate 00:29:48.330 --> 00:29:53.190 align:middle line:84% of u inner product en, which is just the norm squared 00:29:53.190 --> 00:29:54.720 align:middle line:90% when I multiply by that. 00:29:54.720 --> 00:29:57.190 align:middle line:84% And I'm taking the limit as m goes to infinity. 00:29:57.190 --> 00:30:06.080 align:middle line:84% So I pick up the whole sum, the infinite sum, if it's infinite. 00:30:06.080 --> 00:30:20.570 align:middle line:90% 00:30:20.570 --> 00:30:27.385 align:middle line:84% All right, so we've seen a couple of simple applications 00:30:27.385 --> 00:30:28.010 align:middle line:90% of the theorem. 00:30:28.010 --> 00:30:35.270 align:middle line:84% We proved that if I have a normal basis, then 00:30:35.270 --> 00:30:38.510 align:middle line:84% the Hilbert space must be separable. 00:30:38.510 --> 00:30:43.050 align:middle line:84% And also, we have equality in Bessel's inequality for the-- 00:30:43.050 --> 00:30:44.780 align:middle line:84% in terms of the orthonormal basis. 00:30:44.780 --> 00:30:50.380 align:middle line:90% 00:30:50.380 --> 00:30:53.950 align:middle line:84% What's more is that the previous actually gives us 00:30:53.950 --> 00:30:56.860 align:middle line:84% a way to identify every separable Hilbert 00:30:56.860 --> 00:30:59.320 align:middle line:84% space with the one you were introduced 00:30:59.320 --> 00:31:03.590 align:middle line:84% to in the first week, although I didn't call it a Hilbert space. 00:31:03.590 --> 00:31:11.230 align:middle line:84% So this is the following theorem, that if H-- 00:31:11.230 --> 00:31:14.950 align:middle line:84% so if H is a finite dimensional Hilbert space, 00:31:14.950 --> 00:31:19.390 align:middle line:84% that means I can find finitely many orthonormal vectors that 00:31:19.390 --> 00:31:20.380 align:middle line:90% span the space. 00:31:20.380 --> 00:31:24.910 align:middle line:84% And it's quite easy to show that then that is isomorphic to Cn 00:31:24.910 --> 00:31:26.600 align:middle line:90% in an isometric way. 00:31:26.600 --> 00:31:30.340 align:middle line:84% So I'm just going to state the infinite dimensional version. 00:31:30.340 --> 00:31:52.130 align:middle line:84% If H is an infinite dimensional separable Hilbert space, 00:31:52.130 --> 00:31:54.350 align:middle line:84% then H is isometrically isomorphic-- 00:31:54.350 --> 00:32:04.600 align:middle line:90% 00:32:04.600 --> 00:32:07.405 align:middle line:84% and I'll spell out what these words mean-- 00:32:07.405 --> 00:32:12.030 align:middle line:90% 00:32:12.030 --> 00:32:15.750 align:middle line:90% to l2, little l2, meaning what? 00:32:15.750 --> 00:32:24.250 align:middle line:84% Meaning there exists a bijective map 00:32:24.250 --> 00:32:28.780 align:middle line:90% T going from H to little l2-- 00:32:28.780 --> 00:32:31.000 align:middle line:90% so it's one to on and on to-- 00:32:31.000 --> 00:32:42.040 align:middle line:84% so that for all u, v in H, we have that the norm of Tu-- 00:32:42.040 --> 00:32:53.080 align:middle line:84% I should say bijective linear map, bijective linear map. 00:32:53.080 --> 00:32:57.200 align:middle line:84% Or I should even say, I mean, bijective linear operator. 00:32:57.200 --> 00:33:00.520 align:middle line:84% So it's going to be a bounded linear map. 00:33:00.520 --> 00:33:02.290 align:middle line:84% And this follows immediately from what 00:33:02.290 --> 00:33:03.850 align:middle line:90% I'm about to write down-- 00:33:03.850 --> 00:33:06.760 align:middle line:84% such that if I take the norm of the image in little l2, 00:33:06.760 --> 00:33:12.730 align:middle line:84% this is equal to the norm of the vector in H. 00:33:12.730 --> 00:33:18.520 align:middle line:84% So that's the isometric part here, 00:33:18.520 --> 00:33:22.390 align:middle line:84% is that the map is not changing lengths. 00:33:22.390 --> 00:33:26.230 align:middle line:84% And basically, if I have two maps between-- 00:33:26.230 --> 00:33:28.120 align:middle line:84% or if I have a map between two Hilbert spaces 00:33:28.120 --> 00:33:29.650 align:middle line:84% that preserve the lengths, then it 00:33:29.650 --> 00:33:31.240 align:middle line:84% also preserves inner products by-- 00:33:31.240 --> 00:33:35.330 align:middle line:90% 00:33:35.330 --> 00:33:39.470 align:middle line:84% not the parallelogram law, it's the polarization identity, 00:33:39.470 --> 00:33:41.450 align:middle line:90% which you see in the assignment. 00:33:41.450 --> 00:33:48.752 align:middle line:84% But anyways-- and also, we have Tu, Tv-- 00:33:48.752 --> 00:33:51.210 align:middle line:84% taking this inner product and l2 because these are elements 00:33:51.210 --> 00:33:53.040 align:middle line:84% now of little l2, this is the same 00:33:53.040 --> 00:33:59.580 align:middle line:84% as the inner product in H of u and v. 00:33:59.580 --> 00:34:03.000 align:middle line:84% And it really just follows kind of immediately 00:34:03.000 --> 00:34:08.037 align:middle line:84% from what we've done, once I just write down the map. 00:34:08.037 --> 00:34:08.870 align:middle line:90% So what's the proof? 00:34:08.870 --> 00:34:11.929 align:middle line:90% 00:34:11.929 --> 00:34:28.989 align:middle line:84% So since H is a separable Hilbert space, 00:34:28.989 --> 00:34:40.153 align:middle line:84% it has an orthonormal basis, en, which 00:34:40.153 --> 00:34:41.570 align:middle line:84% is countably infinite, since we're 00:34:41.570 --> 00:34:44.644 align:middle line:84% in the infinite dimensional setting. 00:34:44.644 --> 00:34:50.409 align:middle line:90% 00:34:50.409 --> 00:34:57.240 align:middle line:84% And by the previous theorem, we have for all u in H, 00:34:57.240 --> 00:35:01.510 align:middle line:84% u is equal to sum from n equals 1 00:35:01.510 --> 00:35:07.750 align:middle line:84% to infinity u inner product en, en. 00:35:07.750 --> 00:35:17.753 align:middle line:84% And with the norm of u squared equals-- 00:35:17.753 --> 00:35:19.420 align:middle line:84% if you'd like, let me remove the squares 00:35:19.420 --> 00:35:22.810 align:middle line:84% and I can write this as sum from n 00:35:22.810 --> 00:35:32.610 align:middle line:84% equals 1 to infinity of u en norm squared 1/2. 00:35:32.610 --> 00:35:35.910 align:middle line:84% So now, this should just jump at you. 00:35:35.910 --> 00:35:42.950 align:middle line:84% How do I define my map from H to little l2? 00:35:42.950 --> 00:35:52.310 align:middle line:84% Then I define T of u to simply be the sequence of coefficients 00:35:52.310 --> 00:35:53.150 align:middle line:90% appearing here. 00:35:53.150 --> 00:36:05.050 align:middle line:90% 00:36:05.050 --> 00:36:14.620 align:middle line:84% And this is an element of little l2 by this identity here, 00:36:14.620 --> 00:36:15.790 align:middle line:90% Parseval's identity. 00:36:15.790 --> 00:36:22.990 align:middle line:90% 00:36:22.990 --> 00:36:30.115 align:middle line:90% Then T does the job. 00:36:30.115 --> 00:36:33.870 align:middle line:90% 00:36:33.870 --> 00:36:37.680 align:middle line:84% So I didn't go through and check all the properties 00:36:37.680 --> 00:36:38.880 align:middle line:90% that I needed to check. 00:36:38.880 --> 00:36:41.340 align:middle line:84% I mean, it's clear that it's going to be linear in u 00:36:41.340 --> 00:36:43.800 align:middle line:84% because these coefficients are linear in u. 00:36:43.800 --> 00:36:47.610 align:middle line:90% And that's clear. 00:36:47.610 --> 00:36:49.140 align:middle line:90% It's one to one. 00:36:49.140 --> 00:36:54.090 align:middle line:84% It follows from the fact that every u is expanded this way. 00:36:54.090 --> 00:36:56.730 align:middle line:84% And therefore, if the coefficients 00:36:56.730 --> 00:37:02.010 align:middle line:84% are the same for two different u's, then those two u's 00:37:02.010 --> 00:37:03.880 align:middle line:90% have to be the same. 00:37:03.880 --> 00:37:06.780 align:middle line:90% So that makes it injective. 00:37:06.780 --> 00:37:08.150 align:middle line:90% The fact that it's surjective-- 00:37:08.150 --> 00:37:11.480 align:middle line:90% 00:37:11.480 --> 00:37:13.820 align:middle line:84% again, it doesn't take too much to prove, 00:37:13.820 --> 00:37:21.680 align:middle line:84% although you just show that for every choice of sequence 00:37:21.680 --> 00:37:25.822 align:middle line:84% in little l2, that forming such a sum-- 00:37:25.822 --> 00:37:27.930 align:middle line:90% so let's label these Cn. 00:37:27.930 --> 00:37:32.400 align:middle line:84% So now, you have just the element of little l2. 00:37:32.400 --> 00:37:37.320 align:middle line:84% Now, if I put Cn here en, now, you 00:37:37.320 --> 00:37:40.560 align:middle line:84% can argue, as we did before, that this series is, in fact, 00:37:40.560 --> 00:37:43.980 align:middle line:84% Cauchy in H and therefore converges to some element in H. 00:37:43.980 --> 00:37:50.640 align:middle line:84% And then you can prove that T takes that element that you 00:37:50.640 --> 00:37:53.520 align:middle line:84% have to the sequence that you started with, 00:37:53.520 --> 00:37:56.610 align:middle line:90% proving that T is surjective. 00:37:56.610 --> 00:37:59.110 align:middle line:84% So I'm not going to go through all the details. 00:37:59.110 --> 00:38:02.650 align:middle line:84% But it should be kind of clear that this-- 00:38:02.650 --> 00:38:07.900 align:middle line:84% at least based on this identity here, what the map should be. 00:38:07.900 --> 00:38:28.170 align:middle line:90% 00:38:28.170 --> 00:38:30.360 align:middle line:84% All right, so we've seen some applications 00:38:30.360 --> 00:38:36.060 align:middle line:84% of this general theory of orthonormal bases 00:38:36.060 --> 00:38:43.030 align:middle line:84% of countable maximal subsets in the Hilbert space applications, 00:38:43.030 --> 00:38:47.040 align:middle line:84% meaning if it has a basis, orthonormal basis, 00:38:47.040 --> 00:38:50.100 align:middle line:84% then the space has to be separable. 00:38:50.100 --> 00:38:54.210 align:middle line:84% Every separable infinite dimensional Hilbert space 00:38:54.210 --> 00:38:55.420 align:middle line:90% is basically the same. 00:38:55.420 --> 00:39:00.090 align:middle line:84% They're all isometrically isomorphic to little l2. 00:39:00.090 --> 00:39:03.870 align:middle line:84% But how can we use this in a more concrete setting, whatever 00:39:03.870 --> 00:39:04.830 align:middle line:90% concrete means? 00:39:04.830 --> 00:39:07.410 align:middle line:84% I mean, concrete kind of is by taste. 00:39:07.410 --> 00:39:12.180 align:middle line:84% So I thought we'd pause here on the general theory for Hilbert 00:39:12.180 --> 00:39:14.400 align:middle line:84% spaces that we've been doing and do something 00:39:14.400 --> 00:39:20.430 align:middle line:84% a little more specific and look at Fourier series, which 00:39:20.430 --> 00:39:22.860 align:middle line:84% will connect this general stuff that we've been doing 00:39:22.860 --> 00:39:31.890 align:middle line:84% with Hilbert spaces to more of the concrete bit of producing 00:39:31.890 --> 00:39:35.720 align:middle line:84% Lebesgue integration in these big LP spaces, which we proved 00:39:35.720 --> 00:39:41.750 align:middle line:84% are complete spaces involving, in some sense, 00:39:41.750 --> 00:39:45.290 align:middle line:90% integrable functions. 00:39:45.290 --> 00:39:58.620 align:middle line:84% So let's take a pause from general theory 00:39:58.620 --> 00:40:01.620 align:middle line:84% and talk a little bit about Fourier series. 00:40:01.620 --> 00:40:09.770 align:middle line:90% 00:40:09.770 --> 00:40:17.990 align:middle line:84% OK, Fourier series, which was the reason why 00:40:17.990 --> 00:40:20.690 align:middle line:84% a lot of especially the integration theory 00:40:20.690 --> 00:40:22.760 align:middle line:84% was created in the first place, was 00:40:22.760 --> 00:40:25.250 align:middle line:84% to understand a certain question, which 00:40:25.250 --> 00:40:28.320 align:middle line:90% we'll get to in a minute. 00:40:28.320 --> 00:40:32.300 align:middle line:84% Let me just start off with a very simple theorem, 00:40:32.300 --> 00:40:47.380 align:middle line:84% that the subset of functions, e to the inx over root 2 00:40:47.380 --> 00:40:59.670 align:middle line:84% pi in now an integer is an orthonormal-- 00:40:59.670 --> 00:41:10.830 align:middle line:84% orthonormal subset of l2 minus pi to pi. 00:41:10.830 --> 00:41:15.690 align:middle line:90% 00:41:15.690 --> 00:41:17.370 align:middle line:90% And here, let me-- 00:41:17.370 --> 00:41:23.700 align:middle line:84% just for a quick refresher, if t is a real number, 00:41:23.700 --> 00:41:25.740 align:middle line:84% e to the it, this is simply defined 00:41:25.740 --> 00:41:33.720 align:middle line:84% to be the complex number of cosine t plus i sine t. 00:41:33.720 --> 00:41:38.170 align:middle line:84% And it satisfies all the things that you know and love 00:41:38.170 --> 00:41:39.400 align:middle line:90% about the exponential. 00:41:39.400 --> 00:41:42.800 align:middle line:90% 00:41:42.800 --> 00:41:45.420 align:middle line:84% If I multiply e to the it times e to the i tau, 00:41:45.420 --> 00:41:48.480 align:middle line:84% that's equal to e to the it plus tau just by referring 00:41:48.480 --> 00:41:56.810 align:middle line:84% to the definition and using angle sum formulas and so on. 00:41:56.810 --> 00:41:59.840 align:middle line:90% 00:41:59.840 --> 00:42:02.960 align:middle line:84% All right, so this is not too difficult to prove. 00:42:02.960 --> 00:42:04.970 align:middle line:90% What's the proof of this? 00:42:04.970 --> 00:42:13.330 align:middle line:84% If I take e to the inx and inner product it with e to the inx, 00:42:13.330 --> 00:42:16.180 align:middle line:84% now this inner product is in big L2. 00:42:16.180 --> 00:42:19.450 align:middle line:84% This is, by definition, equal to the integral from minus pi 00:42:19.450 --> 00:42:27.250 align:middle line:84% to pi e to the inx times e to the imx complex conjugate. 00:42:27.250 --> 00:42:32.790 align:middle line:84% And again, from the definition, when 00:42:32.790 --> 00:42:36.590 align:middle line:84% I take a complex conjugate, that flips this i to a minus i. 00:42:36.590 --> 00:42:39.390 align:middle line:90% 00:42:39.390 --> 00:42:43.020 align:middle line:84% And then I can take that minus that sits out here 00:42:43.020 --> 00:42:45.840 align:middle line:84% and put it in, and make this a minus t here and a minus t 00:42:45.840 --> 00:42:48.750 align:middle line:84% there, since cosine t is even and sine t is odd. 00:42:48.750 --> 00:42:56.250 align:middle line:84% So then this becomes e to the inx e to the minus imx x dx. 00:42:56.250 --> 00:43:07.680 align:middle line:84% And again, you can just go from the definition of what this is. 00:43:07.680 --> 00:43:10.470 align:middle line:84% You don't need any fancy complex analysis. 00:43:10.470 --> 00:43:14.850 align:middle line:84% This is i to the n minus m times x dx. 00:43:14.850 --> 00:43:19.004 align:middle line:90% 00:43:19.004 --> 00:43:23.560 align:middle line:90% Now, this quantity here-- 00:43:23.560 --> 00:43:27.350 align:middle line:84% when n equals m, I just get 0 here. 00:43:27.350 --> 00:43:28.690 align:middle line:90% And therefore, this is just 1. 00:43:28.690 --> 00:43:32.500 align:middle line:84% And when I integrate that, I get 2 pi. 00:43:32.500 --> 00:43:37.330 align:middle line:90% So this equals 2 pi. 00:43:37.330 --> 00:43:40.960 align:middle line:84% And now, when n does not equal m, 00:43:40.960 --> 00:43:44.270 align:middle line:84% this has an antiderivative that you expect it to be. 00:43:44.270 --> 00:43:50.170 align:middle line:84% And therefore, this is equal to e to the i n minus m 00:43:50.170 --> 00:43:59.680 align:middle line:84% x over n minus m times i evaluated at pi n minus pi. 00:43:59.680 --> 00:44:04.170 align:middle line:84% And this is if n equals m, n does not equal m. 00:44:04.170 --> 00:44:09.060 align:middle line:90% 00:44:09.060 --> 00:44:14.040 align:middle line:84% Now, what is it about e to the it? 00:44:14.040 --> 00:44:16.980 align:middle line:90% This thing is 2 pi periodic. 00:44:16.980 --> 00:44:18.930 align:middle line:84% And it's the same if I have an n here. 00:44:18.930 --> 00:44:24.310 align:middle line:84% Then that's going to be 2 pi over n periodic. 00:44:24.310 --> 00:44:30.120 align:middle line:84% So if I stick in pi here, the value I get 00:44:30.120 --> 00:44:33.330 align:middle line:84% is going to be the same as when I stick in minus pi 00:44:33.330 --> 00:44:36.810 align:middle line:84% here because the difference between these two values is 2 00:44:36.810 --> 00:44:37.650 align:middle line:90% pi. 00:44:37.650 --> 00:44:38.940 align:middle line:90% This is 2 pi periodic. 00:44:38.940 --> 00:44:41.940 align:middle line:84% And I'm sticking in two numbers separated by 2 pi. 00:44:41.940 --> 00:44:45.480 align:middle line:84% So this will be 0 when n does not equal m. 00:44:45.480 --> 00:44:51.640 align:middle line:90% 00:44:51.640 --> 00:44:55.690 align:middle line:84% And therefore, that proves the claim 00:44:55.690 --> 00:44:57.190 align:middle line:84% that this is an orthonormal subset. 00:44:57.190 --> 00:44:59.020 align:middle line:84% We divide by square root of 2 pi so 00:44:59.020 --> 00:45:03.130 align:middle line:84% that when I take the inner product of this with itself 00:45:03.130 --> 00:45:05.050 align:middle line:84% or the element with itself, I get 1. 00:45:05.050 --> 00:45:07.630 align:middle line:84% Here, I just did e to the inx, so I got 2 pi. 00:45:07.630 --> 00:45:10.810 align:middle line:84% So I divide by the square root of that 00:45:10.810 --> 00:45:15.105 align:middle line:90% to get the orthonormal thing. 00:45:15.105 --> 00:45:18.050 align:middle line:90% 00:45:18.050 --> 00:45:21.620 align:middle line:90% So make a definition. 00:45:21.620 --> 00:45:24.480 align:middle line:90% 00:45:24.480 --> 00:45:30.335 align:middle line:84% So let f be in l2 of minus pi to pi. 00:45:30.335 --> 00:45:35.310 align:middle line:90% 00:45:35.310 --> 00:45:41.870 align:middle line:90% The n-th Fourier coefficient-- 00:45:41.870 --> 00:45:44.210 align:middle line:84% this is the new bit of terminology-- 00:45:44.210 --> 00:45:47.680 align:middle line:90% 00:45:47.680 --> 00:45:55.520 align:middle line:84% of f is the complex number f-hat of n equals 1 over 2 pi inner 00:45:55.520 --> 00:45:56.240 align:middle line:90% product-- 00:45:56.240 --> 00:45:57.830 align:middle line:84% I mean-- inner product-- integral 00:45:57.830 --> 00:46:07.370 align:middle line:84% of minus pi to pi of f of t e to the minus int dt. 00:46:07.370 --> 00:46:28.230 align:middle line:84% And the N-th partial Fourier sum of f 00:46:28.230 --> 00:46:34.830 align:middle line:84% is denoted by capital S sub N of f of x. 00:46:34.830 --> 00:46:39.750 align:middle line:84% This is equal to the sum for n in absolute value less than 00:46:39.750 --> 00:46:45.480 align:middle line:84% or equal to capital N. So little n is an integer of f-hat of n 00:46:45.480 --> 00:46:54.030 align:middle line:84% times e to the inx, which I can-- 00:46:54.030 --> 00:47:01.415 align:middle line:84% is the same as f inner product with in-- 00:47:01.415 --> 00:47:04.140 align:middle line:90% 00:47:04.140 --> 00:47:09.120 align:middle line:84% let's say t over square root of 2 pi e to the i. 00:47:09.120 --> 00:47:15.510 align:middle line:90% 00:47:15.510 --> 00:47:18.960 align:middle line:84% All I did with my definition of the Fourier coefficient 00:47:18.960 --> 00:47:21.240 align:middle line:84% is I combined these two square root of 2 pi's 00:47:21.240 --> 00:47:23.460 align:middle line:90% in my definition of f-hat of n. 00:47:23.460 --> 00:47:30.240 align:middle line:84% But this partial sum is just equal to a partial sum in terms 00:47:30.240 --> 00:47:32.070 align:middle line:90% of this orthonormal subset. 00:47:32.070 --> 00:47:45.292 align:middle line:90% 00:47:45.292 --> 00:47:59.850 align:middle line:84% OK, and we also associate to function f a formal object. 00:47:59.850 --> 00:48:24.430 align:middle line:84% The Fourier series of f is the formal series, 00:48:24.430 --> 00:48:27.520 align:middle line:84% because we are not making any claims about its convergence 00:48:27.520 --> 00:48:40.715 align:middle line:84% at all, summed from n in z f-hat of n e to the inx. 00:48:40.715 --> 00:48:46.220 align:middle line:90% 00:48:46.220 --> 00:48:58.190 align:middle line:84% So the question that we're going to answer-- 00:48:58.190 --> 00:49:04.280 align:middle line:84% and like I said, what motivated all of this to begin with 00:49:04.280 --> 00:49:05.280 align:middle line:90% is the following. 00:49:05.280 --> 00:49:10.220 align:middle line:84% So back when Fourier was studying heat conduction, 00:49:10.220 --> 00:49:12.080 align:middle line:84% he made a claim that every function can 00:49:12.080 --> 00:49:14.720 align:middle line:84% be expanded in terms of-- essentially, 00:49:14.720 --> 00:49:16.280 align:middle line:90% he was doing cosines and sines. 00:49:16.280 --> 00:49:20.720 align:middle line:84% But in terms of this, he said every function 00:49:20.720 --> 00:49:23.850 align:middle line:90% can be expanded-- 00:49:23.850 --> 00:49:26.115 align:middle line:90% is equal to its Fourier series. 00:49:26.115 --> 00:49:28.720 align:middle line:90% 00:49:28.720 --> 00:49:32.110 align:middle line:84% Now, at the time, people said, no, that's not the case. 00:49:32.110 --> 00:49:33.460 align:middle line:90% Not every function is periodic. 00:49:33.460 --> 00:49:35.560 align:middle line:90% And these are 2 pi-- 00:49:35.560 --> 00:49:37.160 align:middle line:90% each of these is 2 pi periodic. 00:49:37.160 --> 00:49:40.070 align:middle line:90% So what are you talking about? 00:49:40.070 --> 00:49:43.540 align:middle line:84% And then he said, maybe if we restrict 00:49:43.540 --> 00:49:46.600 align:middle line:84% to continuous functions, maybe it's 00:49:46.600 --> 00:49:48.730 align:middle line:90% equal to its Fourier series. 00:49:48.730 --> 00:49:49.600 align:middle line:90% That's not true. 00:49:49.600 --> 00:49:52.240 align:middle line:84% There's continuous functions that have-- 00:49:52.240 --> 00:49:58.000 align:middle line:84% where the Fourier series actually diverges at a point 00:49:58.000 --> 00:49:59.860 align:middle line:84% and doesn't converge back to the function. 00:49:59.860 --> 00:50:02.810 align:middle line:90% 00:50:02.810 --> 00:50:06.080 align:middle line:84% But Fourier series, now if you're 00:50:06.080 --> 00:50:09.500 align:middle line:84% asking about pointwise convergence, 00:50:09.500 --> 00:50:13.100 align:middle line:84% that's a very tricky and delicate issue. 00:50:13.100 --> 00:50:20.630 align:middle line:84% But Fourier series has a very nice and beautiful answer 00:50:20.630 --> 00:50:25.670 align:middle line:84% when you look in terms of the space that you're supposed to. 00:50:25.670 --> 00:50:27.710 align:middle line:84% As I started off this discussion, 00:50:27.710 --> 00:50:31.160 align:middle line:84% I said these elements are orthonormal with respect 00:50:31.160 --> 00:50:32.660 align:middle line:90% to this inner product. 00:50:32.660 --> 00:50:36.620 align:middle line:84% And this inner product lives, as far as in a Hilbert space, 00:50:36.620 --> 00:50:38.780 align:middle line:90% in the space big L2. 00:50:38.780 --> 00:50:42.140 align:middle line:84% So the question that one should ask then 00:50:42.140 --> 00:50:46.910 align:middle line:84% is-- you have this or the normal subset of L2. 00:50:46.910 --> 00:51:05.220 align:middle line:84% So the question is, do we have for all f in L2 f of x 00:51:05.220 --> 00:51:15.000 align:middle line:90% equals f-hat of n e to the inx. 00:51:15.000 --> 00:51:17.100 align:middle line:90% So this is an infinite sum. 00:51:17.100 --> 00:51:21.230 align:middle line:84% So I should be talking about in what sense is this series 00:51:21.230 --> 00:51:22.070 align:middle line:90% converging. 00:51:22.070 --> 00:51:25.640 align:middle line:84% And I mean in the sense of in the space 00:51:25.640 --> 00:51:29.510 align:middle line:84% that we're asking this question, in L2, i.e. 00:51:29.510 --> 00:51:32.030 align:middle line:90% 00:51:32.030 --> 00:51:46.030 align:middle line:84% do I have that the partial sums converge to f in the L2 norm? 00:51:46.030 --> 00:51:54.210 align:middle line:90% 00:51:54.210 --> 00:51:57.165 align:middle line:84% So I'm not going to write the argument. 00:51:57.165 --> 00:51:58.380 align:middle line:90% OK, maybe I will. 00:51:58.380 --> 00:52:05.700 align:middle line:90% 00:52:05.700 --> 00:52:09.550 align:middle line:90% Does this equal-- does this-- 00:52:09.550 --> 00:52:11.620 align:middle line:90% OK, now, I'm butchering this. 00:52:11.620 --> 00:52:15.160 align:middle line:84% Instead of writing that limit, let's say, 00:52:15.160 --> 00:52:23.200 align:middle line:84% does this norm converge to 0 as capital N goes to infinity? 00:52:23.200 --> 00:52:26.710 align:middle line:90% 00:52:26.710 --> 00:52:28.380 align:middle line:90% So that's the question. 00:52:28.380 --> 00:52:33.130 align:middle line:84% Do we have convergence of this series, 00:52:33.130 --> 00:52:37.280 align:middle line:84% meaning the partial sums, do they converge to the function f 00:52:37.280 --> 00:52:39.020 align:middle line:90% in the L2 norm? 00:52:39.020 --> 00:52:41.090 align:middle line:84% So this is something weaker than-- 00:52:41.090 --> 00:52:44.210 align:middle line:84% or maybe not even comparable to pointwise convergence. 00:52:44.210 --> 00:52:47.120 align:middle line:84% So sometimes, this is referred to as convergence in the mean. 00:52:47.120 --> 00:52:49.860 align:middle line:90% 00:52:49.860 --> 00:52:55.410 align:middle line:84% And so what is the way to phrase this question based 00:52:55.410 --> 00:52:56.265 align:middle line:90% on what we've done? 00:52:56.265 --> 00:53:00.090 align:middle line:90% 00:53:00.090 --> 00:53:16.900 align:middle line:84% So this question is equivalent to the question, 00:53:16.900 --> 00:53:37.424 align:middle line:84% is this collection of orthonormal elements in big L2 00:53:37.424 --> 00:53:45.000 align:middle line:90% a maximal subset of big L2? i.e. 00:53:45.000 --> 00:53:52.620 align:middle line:84% and so now, let's really put this question into equivalence 00:53:52.620 --> 00:53:56.640 align:middle line:84% with just a pure statement-- i.e. 00:53:56.640 --> 00:54:09.260 align:middle line:84% does the vanishing of all the Fourier coefficients 00:54:09.260 --> 00:54:11.075 align:middle line:90% imply that the function is 0? 00:54:11.075 --> 00:54:15.880 align:middle line:90% 00:54:15.880 --> 00:54:19.300 align:middle line:84% So we proved that this statement here 00:54:19.300 --> 00:54:26.770 align:middle line:84% is equivalent to e to the inx being an orthonormal basis. 00:54:26.770 --> 00:54:28.000 align:middle line:90% So we proved one direction. 00:54:28.000 --> 00:54:30.130 align:middle line:84% We proved that if I have a collection 00:54:30.130 --> 00:54:34.750 align:middle line:84% of orthonormal vectors, then that's an orthonormal basis 00:54:34.750 --> 00:54:38.530 align:middle line:84% implies that every function can be expanded 00:54:38.530 --> 00:54:40.360 align:middle line:90% in this infinite series. 00:54:40.360 --> 00:54:42.280 align:middle line:84% But the converse clearly holds, too. 00:54:42.280 --> 00:54:45.970 align:middle line:84% If I can expand every function in an infinite series 00:54:45.970 --> 00:54:47.740 align:middle line:84% with those coefficients, then that 00:54:47.740 --> 00:54:50.410 align:middle line:84% implies that the subset has to be ortho-- 00:54:50.410 --> 00:54:53.590 align:middle line:84% has to be maximal because if something's 00:54:53.590 --> 00:54:56.705 align:middle line:84% orthogonal to everything in the collection, 00:54:56.705 --> 00:54:59.080 align:middle line:84% then all those coefficients appearing in the series are 0 00:54:59.080 --> 00:55:00.580 align:middle line:90% and therefore the function is 0. 00:55:00.580 --> 00:55:05.860 align:middle line:84% So this question here is equivalent to asking 00:55:05.860 --> 00:55:08.890 align:middle line:84% whether or not this collection is maximal, which 00:55:08.890 --> 00:55:13.090 align:middle line:84% means if I have an element in L2 that 00:55:13.090 --> 00:55:16.450 align:middle line:84% is orthogonal to everything in here, 00:55:16.450 --> 00:55:19.810 align:middle line:84% then it has to be 0, which is this statement here. 00:55:19.810 --> 00:55:27.160 align:middle line:90% So let me put a box around it. 00:55:27.160 --> 00:55:30.970 align:middle line:84% Based on what we've done, of the question of convergence 00:55:30.970 --> 00:55:35.020 align:middle line:90% of Fourier series in big L2-- 00:55:35.020 --> 00:55:38.200 align:middle line:84% and what I'm using here really as well 00:55:38.200 --> 00:55:40.510 align:middle line:84% is the fact that big L2 is complete. 00:55:40.510 --> 00:55:43.090 align:middle line:84% This all only works in a Hilbert space. 00:55:43.090 --> 00:55:46.120 align:middle line:84% At certain points, we relied on the fact 00:55:46.120 --> 00:55:49.690 align:middle line:84% that Cauchy sequences converge to something in this space. 00:55:49.690 --> 00:55:55.300 align:middle line:84% So the fact that I can cleanly reduce this question down 00:55:55.300 --> 00:55:57.820 align:middle line:84% to what's in yellow relies on the fact 00:55:57.820 --> 00:56:02.290 align:middle line:84% that big L2 is complete, which we did a lot of work 00:56:02.290 --> 00:56:06.250 align:middle line:90% to show and to construct. 00:56:06.250 --> 00:56:19.810 align:middle line:90% 00:56:19.810 --> 00:56:24.190 align:middle line:84% So this is the question that we're going to try to answer. 00:56:24.190 --> 00:56:30.370 align:middle line:90% And let me go ahead and answer-- 00:56:30.370 --> 00:56:32.630 align:middle line:84% I asked a question, so I should give the answer. 00:56:32.630 --> 00:56:36.430 align:middle line:84% The answer is yes, but it's going to take some work. 00:56:36.430 --> 00:56:37.660 align:middle line:90% This is a non-trivial matter. 00:56:37.660 --> 00:56:41.730 align:middle line:90% 00:56:41.730 --> 00:56:45.470 align:middle line:84% So let's see, how much time do I have? 00:56:45.470 --> 00:56:47.450 align:middle line:84% OK, so how we're going to proceed 00:56:47.450 --> 00:56:52.170 align:middle line:84% is via what may be referred to as Fejer's method. 00:56:52.170 --> 00:56:54.560 align:middle line:84% So again, our goal is to show this, 00:56:54.560 --> 00:56:57.710 align:middle line:84% to answer the question about convergence of Fourier series 00:56:57.710 --> 00:56:59.240 align:middle line:90% in L2. 00:56:59.240 --> 00:57:00.050 align:middle line:90% Why in L2? 00:57:00.050 --> 00:57:04.550 align:middle line:90% Because that's the complete-- 00:57:04.550 --> 00:57:12.540 align:middle line:84% that's a complete Hilbert space that we're working in, 00:57:12.540 --> 00:57:15.600 align:middle line:84% that we're going to apply our general framework to. 00:57:15.600 --> 00:57:25.000 align:middle line:84% So let me start off with a following simple calculation, 00:57:25.000 --> 00:57:40.870 align:middle line:84% that for all f in L2 minus pi to pi and for all natural numbers 00:57:40.870 --> 00:57:45.070 align:middle line:84% n, including 0, if I want to look 00:57:45.070 --> 00:57:53.760 align:middle line:84% at the n-th partial sum of f, I can 00:57:53.760 --> 00:57:55.980 align:middle line:84% write this as the integral from minus pi 00:57:55.980 --> 00:58:03.510 align:middle line:84% to pi of a function evaluated at x minus t times 00:58:03.510 --> 00:58:06.420 align:middle line:90% f of t integrated dt. 00:58:06.420 --> 00:58:10.050 align:middle line:84% And again, this should be interpreted as a Lesbegue 00:58:10.050 --> 00:58:13.080 align:middle line:84% integral, even though I'm writing it using 00:58:13.080 --> 00:58:16.090 align:middle line:90% the notation that you use-- 00:58:16.090 --> 00:58:24.233 align:middle line:90% Lord, sorry, that e was awful. 00:58:24.233 --> 00:58:25.650 align:middle line:84% I just couldn't let it stay there. 00:58:25.650 --> 00:58:29.350 align:middle line:90% 00:58:29.350 --> 00:58:32.590 align:middle line:84% Even though I was using this notation at one point to denote 00:58:32.590 --> 00:58:35.950 align:middle line:84% the Riemann integral, I'm now using this notation 00:58:35.950 --> 00:58:38.710 align:middle line:84% for the Lebesgue integral if the function's-- 00:58:38.710 --> 00:58:41.980 align:middle line:84% if I'm talking about Lebesgue integration, 00:58:41.980 --> 00:58:43.740 align:middle line:84% which is not such an abuse of notation 00:58:43.740 --> 00:58:45.900 align:middle line:84% because we found out that the Lebesgue 00:58:45.900 --> 00:58:49.870 align:middle line:84% integral of continuous functions is the Riemann integral. 00:58:49.870 --> 00:58:51.340 align:middle line:90% So let me finish the statement. 00:58:51.340 --> 00:58:56.070 align:middle line:84% So we can write it as some function of x minus t times 00:58:56.070 --> 00:59:07.960 align:middle line:84% f of t integrate from minus pi to pi where dn of x, this 00:59:07.960 --> 00:59:14.440 align:middle line:84% is equal to the function, which is 2n plus 1 over 2 pi when 00:59:14.440 --> 00:59:29.290 align:middle line:84% x equals 0 and sine of n plus 1/2 00:59:29.290 --> 00:59:40.360 align:middle line:84% of x over 2 pi sine x over 2 when x does not equal 0. 00:59:40.360 --> 00:59:42.940 align:middle line:84% And this function here-- so first off, 00:59:42.940 --> 00:59:46.810 align:middle line:84% note that this is a continuous function. 00:59:46.810 --> 00:59:51.190 align:middle line:84% As x converges to 0, using L'Hopital's rule, if you like, 00:59:51.190 --> 00:59:53.967 align:middle line:84% this thing converges to 2n plus 1 over 2 pi. 00:59:53.967 --> 00:59:55.300 align:middle line:90% In fact, it's a smooth function. 00:59:55.300 --> 00:59:57.830 align:middle line:90% 00:59:57.830 --> 01:00:00.845 align:middle line:84% This function here is referred to as the Dirichlet kernel. 01:00:00.845 --> 01:00:34.160 align:middle line:90% 01:00:34.160 --> 01:00:38.760 align:middle line:84% As a first step, I said, we're going to look at-- 01:00:38.760 --> 01:00:41.850 align:middle line:84% we're going to rewrite the partial sums in this way. 01:00:41.850 --> 01:00:43.110 align:middle line:90% I'll tell you why in a minute. 01:00:43.110 --> 01:00:46.560 align:middle line:90% 01:00:46.560 --> 01:00:49.760 align:middle line:84% But let's just take this as a warm-up calculation 01:00:49.760 --> 01:00:51.665 align:middle line:90% of some calculations to come. 01:00:51.665 --> 01:01:13.720 align:middle line:90% 01:01:13.720 --> 01:01:16.010 align:middle line:90% OK, so what's the proof? 01:01:16.010 --> 01:01:32.410 align:middle line:84% We have that the n-th partial Fourier sum of f, 01:01:32.410 --> 01:01:37.990 align:middle line:84% this is equal to n less than or equal to N of-- 01:01:37.990 --> 01:01:42.490 align:middle line:84% let me just write out here what the Fourier coefficient is-- 01:01:42.490 --> 01:01:52.960 align:middle line:84% 1 over 2 pi minus pi to pi of e to the minus i of f of t, e 01:01:52.960 --> 01:02:00.040 align:middle line:84% tot he minus int dt e to the inx. 01:02:00.040 --> 01:02:02.990 align:middle line:90% And this is just a finite sum. 01:02:02.990 --> 01:02:05.560 align:middle line:84% So I can bring this inside the integral 01:02:05.560 --> 01:02:06.700 align:middle line:90% and combine everything. 01:02:06.700 --> 01:02:08.770 align:middle line:84% And I get this is equal to minus pi 01:02:08.770 --> 01:02:21.725 align:middle line:84% to phi of f of t times dn of x minus t, where dn of-- 01:02:21.725 --> 01:02:26.100 align:middle line:90% 01:02:26.100 --> 01:02:30.150 align:middle line:84% so in fact, let me-- let me not jump ahead. 01:02:30.150 --> 01:02:32.910 align:middle line:84% I'll go ahead and write this out. 01:02:32.910 --> 01:02:48.170 align:middle line:84% This is equal to now sum times e to the in x minus t dt. 01:02:48.170 --> 01:02:51.430 align:middle line:90% So this is dn, all right? 01:02:51.430 --> 01:02:56.010 align:middle line:90% 01:02:56.010 --> 01:03:05.160 align:middle line:84% So dn-- so call this thing, dn of x minus t, x 01:03:05.160 --> 01:03:06.480 align:middle line:90% minus t appearing there. 01:03:06.480 --> 01:03:13.420 align:middle line:90% 01:03:13.420 --> 01:03:17.410 align:middle line:90% And now, let's compute dn of x. 01:03:17.410 --> 01:03:23.260 align:middle line:84% So let me just rewrite what it is, sum in less than 01:03:23.260 --> 01:03:24.880 align:middle line:90% or equal to n e to the inx. 01:03:24.880 --> 01:03:28.010 align:middle line:90% 01:03:28.010 --> 01:03:28.720 align:middle line:90% Where'd the t go? 01:03:28.720 --> 01:03:31.690 align:middle line:84% Again, again this is dn of x minus t. 01:03:31.690 --> 01:03:33.160 align:middle line:90% So the argument's there. 01:03:33.160 --> 01:03:35.470 align:middle line:84% So dn of x is-- there's the argument. 01:03:35.470 --> 01:03:37.930 align:middle line:84% Maybe I should have put a y or something like that, 01:03:37.930 --> 01:03:42.420 align:middle line:84% but I didn't think that far ahead. 01:03:42.420 --> 01:03:46.610 align:middle line:84% So this is the d capital N. And now, let's 01:03:46.610 --> 01:03:50.990 align:middle line:90% just massage this a little bit. 01:03:50.990 --> 01:03:58.610 align:middle line:84% We can write as e to the minus inx times now the sum from n 01:03:58.610 --> 01:04:02.345 align:middle line:90% equals 0 to 2n e to the inx. 01:04:02.345 --> 01:04:05.410 align:middle line:90% 01:04:05.410 --> 01:04:11.440 align:middle line:84% This is a sum from, if you'd like, n equals minus capital 01:04:11.440 --> 01:04:14.230 align:middle line:90% N to n. 01:04:14.230 --> 01:04:17.140 align:middle line:84% All right, so if I factor out an e to the minus inx, 01:04:17.140 --> 01:04:19.110 align:middle line:84% I can write this sum as this sum. 01:04:19.110 --> 01:04:24.550 align:middle line:90% 01:04:24.550 --> 01:04:31.370 align:middle line:90% Now, this is a geometric sum. 01:04:31.370 --> 01:04:33.680 align:middle line:84% This e to the inx, again, this is something 01:04:33.680 --> 01:04:35.600 align:middle line:84% that you can just check from the definition. 01:04:35.600 --> 01:04:41.330 align:middle line:84% This is also equal to e to the ix raised to the n-th power. 01:04:41.330 --> 01:04:44.390 align:middle line:84% And I know how to sum things that involves something being 01:04:44.390 --> 01:04:46.130 align:middle line:90% raised to the n-th power-- 01:04:46.130 --> 01:04:48.590 align:middle line:90% finite sums, that is. 01:04:48.590 --> 01:04:51.260 align:middle line:84% This is equal to 1 over 2 pi times e 01:04:51.260 --> 01:05:07.430 align:middle line:84% to the minus inx times 1 minus e to the i 2n plus 1 x 01:05:07.430 --> 01:05:13.580 align:middle line:84% over 1 minus e to the ix, the thing that's appearing here. 01:05:13.580 --> 01:05:16.820 align:middle line:84% 2n plus 1 the thing that appears on top plus 1. 01:05:16.820 --> 01:05:23.440 align:middle line:90% 01:05:23.440 --> 01:05:31.390 align:middle line:84% Now, multiplying through by this e to the inx, pulling out a 1/2 01:05:31.390 --> 01:05:34.090 align:middle line:84% e to the ix over 2 and distributing that 01:05:34.090 --> 01:05:37.960 align:middle line:84% to the bottom, we get that the previous 01:05:37.960 --> 01:05:39.730 align:middle line:90% is equal to 1 over 2 pi. 01:05:39.730 --> 01:05:43.800 align:middle line:90% 01:05:43.800 --> 01:05:48.250 align:middle line:84% So first off, this is valid only when x-- 01:05:48.250 --> 01:05:49.530 align:middle line:90% what? 01:05:49.530 --> 01:05:53.530 align:middle line:90% When x is not equal to 0. 01:05:53.530 --> 01:05:58.480 align:middle line:84% When x equals 0, I just get 1 here and I get 1 here. 01:05:58.480 --> 01:06:04.010 align:middle line:84% And then I get the sum from 0 to 2 n, 01:06:04.010 --> 01:06:05.630 align:middle line:90% which is equal to 2n plus 1. 01:06:05.630 --> 01:06:07.940 align:middle line:84% That's why I get what I get when x equals 0. 01:06:07.940 --> 01:06:12.140 align:middle line:84% So this is valid for x not equal to 0. 01:06:12.140 --> 01:06:17.730 align:middle line:84% And so I get e to the in plus 1 x. 01:06:17.730 --> 01:06:20.310 align:middle line:90% 01:06:20.310 --> 01:06:24.210 align:middle line:84% I should say, after I've taken away 01:06:24.210 --> 01:06:28.110 align:middle line:84% that 1/2 x that appeared with this 1x 01:06:28.110 --> 01:06:39.770 align:middle line:84% and distributed it down on the bottom e to the ix over 2 01:06:39.770 --> 01:06:44.930 align:middle line:90% minus e to the minus ix over 2. 01:06:44.930 --> 01:06:49.910 align:middle line:84% Now, if I have e to the i times something 01:06:49.910 --> 01:06:53.390 align:middle line:84% minus e to the minus i times that something 01:06:53.390 --> 01:06:57.380 align:middle line:84% and I subtract those two, I pick up 2i times sine 01:06:57.380 --> 01:06:59.060 align:middle line:90% of whatever's in the-- 01:06:59.060 --> 01:07:00.450 align:middle line:90% whatever this real number is. 01:07:00.450 --> 01:07:12.620 align:middle line:84% So I pick up 1 over 2 pi times 2i sine of n plus 1/2 x 01:07:12.620 --> 01:07:18.230 align:middle line:90% over 2i sine of x over 2. 01:07:18.230 --> 01:07:20.510 align:middle line:90% And these 2i's cancel. 01:07:20.510 --> 01:07:31.010 align:middle line:84% And that equals 1 over 2 pi sine of n plus 1/2 x 01:07:31.010 --> 01:07:35.420 align:middle line:84% over sine x over 2, which is what I wanted. 01:07:35.420 --> 01:07:49.660 align:middle line:90% 01:07:49.660 --> 01:07:59.020 align:middle line:84% So what's the idea of trying to prove this? 01:07:59.020 --> 01:08:03.760 align:middle line:84% In the end-- I mean, in the beginning, I should say, 01:08:03.760 --> 01:08:05.860 align:middle line:84% we were asking about the convergence 01:08:05.860 --> 01:08:08.740 align:middle line:84% of the partial Fourier sums to f. 01:08:08.740 --> 01:08:11.610 align:middle line:90% 01:08:11.610 --> 01:08:13.180 align:middle line:90% And we don't know that going in. 01:08:13.180 --> 01:08:15.530 align:middle line:84% That's what we're trying to prove. 01:08:15.530 --> 01:08:19.149 align:middle line:84% So maybe working with the partial sums 01:08:19.149 --> 01:08:20.260 align:middle line:90% is not the best thing. 01:08:20.260 --> 01:08:21.399 align:middle line:90% What am I getting at here? 01:08:21.399 --> 01:08:25.330 align:middle line:84% How about I introduce the next bit, and then 01:08:25.330 --> 01:08:30.060 align:middle line:84% I'll explain why we're interested in it. 01:08:30.060 --> 01:08:37.520 align:middle line:84% So if f is in l2 from minus pi to pi, 01:08:37.520 --> 01:09:00.410 align:middle line:84% we define the N-th Cesaro-Fourier mean of f-- 01:09:00.410 --> 01:09:03.590 align:middle line:84% so this is the new bit of terminology-- 01:09:03.590 --> 01:09:06.260 align:middle line:90% 01:09:06.260 --> 01:09:10.415 align:middle line:84% to be the average of the partial sums of f. 01:09:10.415 --> 01:09:15.060 align:middle line:90% 01:09:15.060 --> 01:09:18.270 align:middle line:84% This we denote by sigma n of f of x. 01:09:18.270 --> 01:09:32.859 align:middle line:84% This is equal to the average of the first n partial sums of f. 01:09:32.859 --> 01:09:44.080 align:middle line:90% 01:09:44.080 --> 01:09:48.370 align:middle line:84% In the end, what we'd like to do is 01:09:48.370 --> 01:09:54.620 align:middle line:84% we're trying to establish that this claim in the yellow box-- 01:09:54.620 --> 01:09:58.020 align:middle line:84% if the Fourier coefficients are all zero, 01:09:58.020 --> 01:10:00.270 align:middle line:84% then the function has to be zero. 01:10:00.270 --> 01:10:04.980 align:middle line:84% So you may think, well, let's prove that the partial sums 01:10:04.980 --> 01:10:06.690 align:middle line:90% converge to the function f. 01:10:06.690 --> 01:10:08.395 align:middle line:84% And that would give us immediately 01:10:08.395 --> 01:10:10.770 align:middle line:84% that the function is 0 because the partial sums would all 01:10:10.770 --> 01:10:13.680 align:middle line:84% be 0 because they involve the coefficients of f. 01:10:13.680 --> 01:10:17.610 align:middle line:84% But that's ridiculous because that's actually 01:10:17.610 --> 01:10:18.930 align:middle line:90% what we're trying to prove. 01:10:18.930 --> 01:10:23.040 align:middle line:84% That's equivalent to the question that we're trying to-- 01:10:23.040 --> 01:10:25.290 align:middle line:84% that's equivalent to what's in the yellow box. 01:10:25.290 --> 01:10:30.510 align:middle line:84% We're not making our life any easier by doing that. 01:10:30.510 --> 01:10:38.550 align:middle line:84% So now, what one can do instead is 01:10:38.550 --> 01:10:43.398 align:middle line:84% instead of trying to prove what's in the yellow box, 01:10:43.398 --> 01:10:44.940 align:middle line:84% that if all the coefficients are zero 01:10:44.940 --> 01:10:49.440 align:middle line:84% the function is zero, what if we can prove that this object now 01:10:49.440 --> 01:10:52.830 align:middle line:90% converges to f? 01:10:52.830 --> 01:10:59.780 align:middle line:84% This object here, which is this mean of the partial sums maybe 01:10:59.780 --> 01:11:03.530 align:middle line:84% has better properties than the partial sums 01:11:03.530 --> 01:11:06.980 align:middle line:84% originally that we were trying to study. 01:11:06.980 --> 01:11:13.840 align:middle line:84% Remember if you look back to 18.100, 01:11:13.840 --> 01:11:17.350 align:middle line:84% if you have just say a sequence of real numbers which 01:11:17.350 --> 01:11:21.550 align:middle line:84% converges, you can define a Cesaro mean of that 01:11:21.550 --> 01:11:25.090 align:middle line:84% by averaging the first n terms, just like we did here. 01:11:25.090 --> 01:11:28.000 align:middle line:84% There's an n plus 1 here because we're starting at 0 01:11:28.000 --> 01:11:30.280 align:middle line:84% and going up to n rather than going from 1 to n. 01:11:30.280 --> 01:11:33.040 align:middle line:84% But if you have a sequence of real numbers, 01:11:33.040 --> 01:11:36.880 align:middle line:84% you can look at its Cesaro sums or Cesaro means. 01:11:36.880 --> 01:11:41.630 align:middle line:84% And what's great about the Cesaro means is that you 01:11:41.630 --> 01:11:44.080 align:middle line:90% don't-- 01:11:44.080 --> 01:11:47.950 align:middle line:84% it kind of behaves a little bit better than the sequence 01:11:47.950 --> 01:11:48.820 align:middle line:90% you start with. 01:11:48.820 --> 01:11:50.480 align:middle line:84% But you don't lose any information. 01:11:50.480 --> 01:11:52.670 align:middle line:84% So if the original sequence converges, 01:11:52.670 --> 01:11:55.570 align:middle line:84% then the Cesaro mean also converges. 01:11:55.570 --> 01:12:00.910 align:middle line:84% So if you're expecting the partial sums to converge to f, 01:12:00.910 --> 01:12:04.480 align:middle line:84% then the Cesaro means should converge to f. 01:12:04.480 --> 01:12:10.750 align:middle line:84% But the Cesaro means have an even better quality. 01:12:10.750 --> 01:12:15.100 align:middle line:84% Let's go back to sequences of real numbers. 01:12:15.100 --> 01:12:18.520 align:middle line:84% You could have sequences of real numbers that don't converge 01:12:18.520 --> 01:12:20.770 align:middle line:90% whose Cesaro means do converge. 01:12:20.770 --> 01:12:25.540 align:middle line:84% So take the sequence 1, minus 1, 1 minus, 1, 1 minus 1, 01:12:25.540 --> 01:12:28.870 align:middle line:84% and so on, that sequence doesn't converge. 01:12:28.870 --> 01:12:31.120 align:middle line:84% But the Cesaro means do converge. 01:12:31.120 --> 01:12:41.910 align:middle line:84% The Cesaro mean is 1, 0, 1/3, 0, 1/5, 0, 1/7, 0. 01:12:41.910 --> 01:12:43.710 align:middle line:84% So the Cesaro means converge to 0. 01:12:43.710 --> 01:12:48.700 align:middle line:84% So all of that is to say that this object here, 01:12:48.700 --> 01:12:51.570 align:middle line:84% which is the average of the partial sums, 01:12:51.570 --> 01:12:55.200 align:middle line:84% we expect to behave better than just 01:12:55.200 --> 01:12:56.670 align:middle line:90% how the partial sums converge. 01:12:56.670 --> 01:13:03.650 align:middle line:84% We expect it to have better convergence properties. 01:13:03.650 --> 01:13:07.180 align:middle line:84% And what does better convergence properties mean? 01:13:07.180 --> 01:13:08.980 align:middle line:84% Quite honestly, it means we should 01:13:08.980 --> 01:13:13.090 align:middle line:84% be able to show it converges to f in hopefully 01:13:13.090 --> 01:13:14.740 align:middle line:90% some straightforward way-- 01:13:14.740 --> 01:13:16.690 align:middle line:84% or maybe not straightforward way, but in a-- 01:13:16.690 --> 01:13:22.810 align:middle line:90% 01:13:22.810 --> 01:13:26.510 align:middle line:84% yeah, I don't know how to describe that. 01:13:26.510 --> 01:13:28.120 align:middle line:90% And that's what we'll do. 01:13:28.120 --> 01:13:33.650 align:middle line:84% That's the plan, is what we're going to show is that for every 01:13:33.650 --> 01:13:39.500 align:middle line:84% f in l2, the Cesaro means converge to f in l2, 01:13:39.500 --> 01:13:41.870 align:middle line:84% meaning the limit as capital N-- so let me-- 01:13:41.870 --> 01:13:44.930 align:middle line:90% 01:13:44.930 --> 01:13:49.970 align:middle line:84% so what's the goal that we'll show 01:13:49.970 --> 01:13:54.470 align:middle line:84% is we'll show that the Cesaro means-- 01:13:54.470 --> 01:14:01.835 align:middle line:84% Cesaro-Fourier means of f converge to f in L2. 01:14:01.835 --> 01:14:06.670 align:middle line:90% 01:14:06.670 --> 01:14:12.340 align:middle line:84% Now, if we can do this, then we will have answered 01:14:12.340 --> 01:14:15.870 align:middle line:84% and we will have shown what's in the yellow box 01:14:15.870 --> 01:14:21.030 align:middle line:84% because if you assume all of the Fourier coefficients are 0, 01:14:21.030 --> 01:14:24.300 align:middle line:84% then all of the partial sums are 0. 01:14:24.300 --> 01:14:28.380 align:middle line:84% And therefore, all the Cesaro means are 0. 01:14:28.380 --> 01:14:31.980 align:middle line:84% And by this thing that we are hoping to prove, 01:14:31.980 --> 01:14:35.590 align:middle line:84% this will prove that 0 converges to f. 01:14:35.590 --> 01:14:39.320 align:middle line:84% And therefore, f is 0, which is what we wanted to show. 01:14:39.320 --> 01:14:42.190 align:middle line:90% So one more time-- 01:14:42.190 --> 01:14:45.640 align:middle line:84% we want to prove that if all the Fourier coefficients are 0, 01:14:45.640 --> 01:14:47.920 align:middle line:90% then the function has to be 0. 01:14:47.920 --> 01:14:49.960 align:middle line:84% Now, if all the Fourier coefficients are 0, 01:14:49.960 --> 01:14:51.430 align:middle line:90% then all the partial sums are 0. 01:14:51.430 --> 01:14:54.010 align:middle line:84% And therefore, all the 0 means are 0. 01:14:54.010 --> 01:15:00.400 align:middle line:84% And if we're able to prove this holds for all f in l2, 01:15:00.400 --> 01:15:03.100 align:middle line:84% then this would tell us that 0, which 01:15:03.100 --> 01:15:06.250 align:middle line:84% is what all of these Cesaro-Fourier means are, 01:15:06.250 --> 01:15:09.200 align:middle line:90% converges to f. 01:15:09.200 --> 01:15:13.970 align:middle line:84% And therefore, f is 0, our desired conclusion. 01:15:13.970 --> 01:15:19.550 align:middle line:84% And from that, we conclude that the collection 01:15:19.550 --> 01:15:22.370 align:middle line:84% e to the inx over square root of 2 pi is a maximal-- 01:15:22.370 --> 01:15:24.320 align:middle line:90% is a orthonormal basis for l2. 01:15:24.320 --> 01:15:26.600 align:middle line:84% And therefore, the Fourier series 01:15:26.600 --> 01:15:36.020 align:middle line:84% of a given l2 function converges to the function in l2, meaning 01:15:36.020 --> 01:15:39.140 align:middle line:84% f minus the partial sum converges to 0 01:15:39.140 --> 01:15:42.710 align:middle line:90% as n goes to infinity. 01:15:42.710 --> 01:15:45.730 align:middle line:90% 01:15:45.730 --> 01:15:49.350 align:middle line:84% All right, and so that'll be what we do next time, 01:15:49.350 --> 01:15:54.800 align:middle line:90% is prove this claim right here. 01:15:54.800 --> 01:16:01.000 align:middle line:90%