WEBVTT 00:00:00.000 --> 00:00:02.435 align:middle line:90% [SQUEAKING] 00:00:02.435 --> 00:00:03.409 align:middle line:90% [RUSTLING] 00:00:03.409 --> 00:00:04.383 align:middle line:90% 00:00:04.383 --> 00:00:07.792 align:middle line:90% [CLICKING] 00:00:07.792 --> 00:00:18.530 align:middle line:90% 00:00:18.530 --> 00:00:22.830 align:middle line:84% CASEY RODRIGUEZ: OK, so let's continue our discussion 00:00:22.830 --> 00:00:27.240 align:middle line:84% about spectral theory for self adjoint compact operators. 00:00:27.240 --> 00:00:32.130 align:middle line:84% So let me just briefly recall the spectrum 00:00:32.130 --> 00:00:33.840 align:middle line:84% of a bounded operator, which was supposed 00:00:33.840 --> 00:00:38.550 align:middle line:84% to be a generalization of the eigenvalues of a matrix. 00:00:38.550 --> 00:00:42.660 align:middle line:84% So we defined the resolvent set of A 00:00:42.660 --> 00:00:52.210 align:middle line:84% to be those complex numbers such that A minus 00:00:52.210 --> 00:00:55.510 align:middle line:84% lambda times the identity, which I just write is A minus lambda, 00:00:55.510 --> 00:01:00.220 align:middle line:84% is an invertible, bounded linear operator, 00:01:00.220 --> 00:01:05.110 align:middle line:84% meaning it is bijective and, which by the open mapping 00:01:05.110 --> 00:01:07.810 align:middle line:84% theorem, tells you that the inverse is also continuous. 00:01:07.810 --> 00:01:10.750 align:middle line:90% 00:01:10.750 --> 00:01:16.060 align:middle line:84% And the spectrum of A is simply those lambda so 00:01:16.060 --> 00:01:17.980 align:middle line:84% that A minus lambda is not invertible, 00:01:17.980 --> 00:01:26.660 align:middle line:84% so the complement of resolvent set of A. 00:01:26.660 --> 00:01:40.450 align:middle line:84% So from linear algebra, you have the following characterization 00:01:40.450 --> 00:01:48.440 align:middle line:84% of what the spectrum is, that if H is CN, 00:01:48.440 --> 00:01:52.430 align:middle line:84% and A is just therefore matrix on CN, or RN if you like, 00:01:52.430 --> 00:02:01.320 align:middle line:84% then the spectrum is just simply the set of eigenvalues of A. 00:02:01.320 --> 00:02:09.130 align:middle line:84% And if we restrict our attention to Hermitian matrices, 00:02:09.130 --> 00:02:10.750 align:middle line:90% which are referred to as-- 00:02:10.750 --> 00:02:15.240 align:middle line:84% which are also referred to as self adjoint also 00:02:15.240 --> 00:02:18.330 align:middle line:84% in linear algebra, or symmetric if we're just 00:02:18.330 --> 00:02:27.640 align:middle line:84% looking at real vector spaces Rn then the eigenvalues are real. 00:02:27.640 --> 00:02:33.550 align:middle line:84% And you can find an orthonormal basis of the space CN or RN, 00:02:33.550 --> 00:02:39.590 align:middle line:84% depending on what you're looking at for this symmetric matrix 00:02:39.590 --> 00:02:46.210 align:middle line:84% so that in that basis, the matrix is completely 00:02:46.210 --> 00:02:47.770 align:middle line:90% diagonalized. 00:02:47.770 --> 00:02:48.280 align:middle line:90% and. 00:02:48.280 --> 00:02:49.822 align:middle line:84% What are the diagonal elements there? 00:02:49.822 --> 00:02:52.180 align:middle line:90% The eigenvalues of the matrix. 00:02:52.180 --> 00:02:55.330 align:middle line:84% And what we're going to end up proving is that that picture, 00:02:55.330 --> 00:03:03.710 align:middle line:84% this picture where for a self adjoint matrix on CN, 00:03:03.710 --> 00:03:09.060 align:middle line:84% that the spectrum is given by the eigenvalues. 00:03:09.060 --> 00:03:13.640 align:middle line:84% And you can diagonalize, essentially diagonalize 00:03:13.640 --> 00:03:16.340 align:middle line:84% this operator, meaning you can find 00:03:16.340 --> 00:03:19.850 align:middle line:84% an orthonormal basis consisting entirely of eigenvectors 00:03:19.850 --> 00:03:26.480 align:middle line:84% of the operator is also true for self-adjoint compact operators. 00:03:26.480 --> 00:03:28.520 align:middle line:84% This shouldn't come too much of a surprise, 00:03:28.520 --> 00:03:33.170 align:middle line:84% as too much of a surprise, since compact operators are limits 00:03:33.170 --> 00:03:39.560 align:middle line:84% of finite rank, i.e. matrices in the space 00:03:39.560 --> 00:03:42.180 align:middle line:90% of bounded linear operators. 00:03:42.180 --> 00:03:44.990 align:middle line:90% So that's where we're headed. 00:03:44.990 --> 00:03:52.610 align:middle line:84% Now, so just to follow up on this and for a little bit 00:03:52.610 --> 00:03:57.510 align:middle line:84% of review, so in the finite dimensional case, 00:03:57.510 --> 00:03:59.690 align:middle line:90% the spectrum can be-- 00:03:59.690 --> 00:04:01.910 align:middle line:84% or the spectrum is always just the eigenvalues 00:04:01.910 --> 00:04:03.770 align:middle line:90% of the matrix A-- 00:04:03.770 --> 00:04:08.390 align:middle line:84% not so in the infinite dimensional setting, 00:04:08.390 --> 00:04:11.300 align:middle line:84% for example, if we're looking at little l2. 00:04:11.300 --> 00:04:17.540 align:middle line:84% And then A times a is equal, to let's say, a1 over 1, 00:04:17.540 --> 00:04:24.410 align:middle line:84% a2 over 2, and so on, for a, a sequence in l2. 00:04:24.410 --> 00:04:33.950 align:middle line:84% Then what you can prove is that 0 is in the spectrum of A. One 00:04:33.950 --> 00:04:42.540 align:middle line:84% way to see that is that each of the basis vectors of H 00:04:42.540 --> 00:04:46.590 align:middle line:84% given where you just have 1 in the n-th slot and 0 otherwise, 00:04:46.590 --> 00:04:50.310 align:middle line:84% each of those is an eigenvector of this operator 00:04:50.310 --> 00:04:53.940 align:middle line:84% A with eigenvalue 1 over n, where the n tells you 00:04:53.940 --> 00:04:56.350 align:middle line:90% where the 1 is and 0 otherwise. 00:04:56.350 --> 00:05:04.330 align:middle line:84% So 1 over n is an eigenvalue of this operator for each n. 00:05:04.330 --> 00:05:07.450 align:middle line:90% And 1 over n convergence to 0. 00:05:07.450 --> 00:05:09.610 align:middle line:84% And since the spectrum of a bounded linear operator 00:05:09.610 --> 00:05:11.710 align:middle line:84% is a compact set, in particular closed, 00:05:11.710 --> 00:05:13.720 align:middle line:84% 0, which is the limit of that sequence, 00:05:13.720 --> 00:05:16.130 align:middle line:90% has to also be in the spectrum. 00:05:16.130 --> 00:05:18.760 align:middle line:84% And, in fact, what we're going to show is that 00:05:18.760 --> 00:05:23.290 align:middle line:84% in the nondegenerate case, what we see here is what in general 00:05:23.290 --> 00:05:25.570 align:middle line:84% happens for compact self-adjoint operators, 00:05:25.570 --> 00:05:30.110 align:middle line:84% that if it's not a finite rank operator, 00:05:30.110 --> 00:05:34.730 align:middle line:90% then it has countably many-- 00:05:34.730 --> 00:05:36.800 align:middle line:90% or countably infinite many-- 00:05:36.800 --> 00:05:38.750 align:middle line:84% that's not a very good string of words-- 00:05:38.750 --> 00:05:43.370 align:middle line:84% countably infinite eigenvalues which converge to 0. 00:05:43.370 --> 00:05:45.740 align:middle line:84% And 0 may be an eigenvalue, may not be. 00:05:45.740 --> 00:05:50.820 align:middle line:90% 00:05:50.820 --> 00:06:00.490 align:middle line:84% And what's more, that's going to be the only that completely 00:06:00.490 --> 00:06:01.880 align:middle line:90% characterizes the spectrum. 00:06:01.880 --> 00:06:05.960 align:middle line:84% So let me just write here for this example that, in fact, 00:06:05.960 --> 00:06:10.750 align:middle line:84% the spectrum of A is equal to the point 0 union 00:06:10.750 --> 00:06:15.220 align:middle line:90% 1 over n in a natural number. 00:06:15.220 --> 00:06:18.070 align:middle line:84% And these are-- again for this operator here, 00:06:18.070 --> 00:06:20.260 align:middle line:90% these are the eigenvalues. 00:06:20.260 --> 00:06:22.630 align:middle line:90% And this is just-- well, 0 is 0. 00:06:22.630 --> 00:06:29.090 align:middle line:90% But 0 is not an eigenvalue, OK. 00:06:29.090 --> 00:06:32.178 align:middle line:90% 00:06:32.178 --> 00:06:33.720 align:middle line:84% Now, this is not the general picture, 00:06:33.720 --> 00:06:36.870 align:middle line:84% meaning that for a compact self-adjoint operator, 00:06:36.870 --> 00:06:39.870 align:middle line:84% you'll have infinitely many eigenvalues. 00:06:39.870 --> 00:06:41.720 align:middle line:84% And then 0 will not be an eigenvalue. 00:06:41.720 --> 00:06:43.720 align:middle line:84% You could have 0 in eigenvalue as well. 00:06:43.720 --> 00:06:46.170 align:middle line:84% But in general what the picture is 00:06:46.170 --> 00:06:49.110 align:middle line:84% is that anything in the spectrum that's not 0 00:06:49.110 --> 00:06:51.800 align:middle line:90% has to be an eigenvalue. 00:06:51.800 --> 00:06:54.920 align:middle line:84% And that's essentially what will prove 00:06:54.920 --> 00:07:04.170 align:middle line:84% our first result of this lecture is the following. 00:07:04.170 --> 00:07:17.900 align:middle line:84% So this is the Fredholm alternative. 00:07:17.900 --> 00:07:33.720 align:middle line:84% So let A be a self-adjoint compact operator 00:07:33.720 --> 00:07:42.180 align:middle line:84% and lambda be a nonzero real number. 00:07:42.180 --> 00:07:49.060 align:middle line:90% 00:07:49.060 --> 00:07:58.770 align:middle line:84% Then the range of A minus lambda is closed. 00:07:58.770 --> 00:08:00.270 align:middle line:90% So this is the conclusion. 00:08:00.270 --> 00:08:05.970 align:middle line:90% 00:08:05.970 --> 00:08:11.860 align:middle line:84% And thus the range of A minus lambda 00:08:11.860 --> 00:08:13.930 align:middle line:84% is equal to the orthogonal complement 00:08:13.930 --> 00:08:24.090 align:middle line:84% of the orthogonal complement of itself, which-- 00:08:24.090 --> 00:08:30.150 align:middle line:90% 00:08:30.150 --> 00:08:32.760 align:middle line:84% the orthogonal complement of the range of A minus lambda 00:08:32.760 --> 00:08:35.400 align:middle line:84% is equal to the null space of the adjoint. 00:08:35.400 --> 00:08:37.830 align:middle line:84% And since A is self-adjoint and lambda is a real number, 00:08:37.830 --> 00:08:39.599 align:middle line:84% the adjoint is just A minus lambda. 00:08:39.599 --> 00:08:42.370 align:middle line:90% 00:08:42.370 --> 00:08:46.060 align:middle line:84% So this right here is the main conclusion 00:08:46.060 --> 00:08:47.320 align:middle line:90% from which you get this. 00:08:47.320 --> 00:08:53.200 align:middle line:84% And therefore just to spell out the conclusion 00:08:53.200 --> 00:09:02.010 align:middle line:84% of this equality, therefore either one of two alternatives 00:09:02.010 --> 00:09:02.850 align:middle line:90% happens. 00:09:02.850 --> 00:09:09.400 align:middle line:84% So that's the name, alternative-- either A minus 00:09:09.400 --> 00:09:20.550 align:middle line:84% lambda is bijective, or the null space 00:09:20.550 --> 00:09:25.440 align:middle line:84% of A minus lambda, which is just the eigenspace of A minus 00:09:25.440 --> 00:09:30.190 align:middle line:84% lambda, or I should say the eigenspace corresponding 00:09:30.190 --> 00:09:49.110 align:middle line:84% to lambda, is non-trivial and finite dimensional. 00:09:49.110 --> 00:09:59.660 align:middle line:90% 00:09:59.660 --> 00:10:01.550 align:middle line:84% Moreover, this equality here tells you 00:10:01.550 --> 00:10:06.260 align:middle line:84% when you can solve the equation A minus lambda u equals f. 00:10:06.260 --> 00:10:08.930 align:middle line:84% You can solve a minus lambda u equals f if 00:10:08.930 --> 00:10:11.990 align:middle line:84% and only if f is in the range of A minus lambda, which is if 00:10:11.990 --> 00:10:15.490 align:middle line:84% and only if f is orthogonal to the null space of A minus 00:10:15.490 --> 00:10:15.990 align:middle line:90% lambda. 00:10:15.990 --> 00:10:21.070 align:middle line:84% So let me make a little remark here. 00:10:21.070 --> 00:10:26.130 align:middle line:84% So first off, the fact that this space has to be finite 00:10:26.130 --> 00:10:27.870 align:middle line:90% dimensional we proved last time. 00:10:27.870 --> 00:10:33.840 align:middle line:84% We proved at the end of lecture, the last lecture, 00:10:33.840 --> 00:10:37.950 align:middle line:84% that for a compact self-adjoint operator, 00:10:37.950 --> 00:10:40.320 align:middle line:84% the null space, the eigenspace corresponding 00:10:40.320 --> 00:10:45.480 align:middle line:84% to a given nonzero eigenvalue, is finite dimensional. 00:10:45.480 --> 00:10:48.330 align:middle line:84% And then we also prove the eigenspaces corresponding 00:10:48.330 --> 00:10:51.450 align:middle line:84% to two different eigenvalues are orthogonal. 00:10:51.450 --> 00:10:53.160 align:middle line:84% And we also prove that the eigenvalues 00:10:53.160 --> 00:10:56.280 align:middle line:84% that are nonzero, or any eigenvalue, has to be real. 00:10:56.280 --> 00:11:02.910 align:middle line:90% 00:11:02.910 --> 00:11:08.470 align:middle line:84% So now let me make a couple of remarks. 00:11:08.470 --> 00:11:12.660 align:middle line:84% The first is just a rephrasing of what's in the theorem. 00:11:12.660 --> 00:11:20.030 align:middle line:84% Therefore f is in the range of A minus lambda, meaning you 00:11:20.030 --> 00:11:23.990 align:middle line:84% can solve the equation A minus lambda u equals f if 00:11:23.990 --> 00:11:29.870 align:middle line:84% and only if f is in the null space of-- 00:11:29.870 --> 00:11:33.080 align:middle line:84% or the orthogonal complement of A minus-- 00:11:33.080 --> 00:11:35.790 align:middle line:84% or the null space of A minus lambda. 00:11:35.790 --> 00:11:38.210 align:middle line:84% So what this says is that you can 00:11:38.210 --> 00:11:41.240 align:middle line:84% solve for f, given that f satisfies 00:11:41.240 --> 00:11:46.610 align:middle line:84% finitely many linear conditions because the null space 00:11:46.610 --> 00:11:49.310 align:middle line:84% of A minus lambda is finite dimensional. 00:11:49.310 --> 00:11:53.120 align:middle line:84% So f being orthogonal to that means pick a finite base, 00:11:53.120 --> 00:11:57.290 align:middle line:84% a finite orthonormal basis of A minus null 00:11:57.290 --> 00:11:58.730 align:middle line:90% space of A minus lambda. 00:11:58.730 --> 00:12:01.670 align:middle line:84% Then f in our product with those finitely many vectors 00:12:01.670 --> 00:12:02.390 align:middle line:90% has to be 0. 00:12:02.390 --> 00:12:05.660 align:middle line:84% So you have finitely many conditions on f 00:12:05.660 --> 00:12:07.340 align:middle line:90% to be able to solve for f. 00:12:07.340 --> 00:12:10.550 align:middle line:84% And not only that, this solution that you compute 00:12:10.550 --> 00:12:13.400 align:middle line:84% is also unique up to finitely many conditions 00:12:13.400 --> 00:12:18.150 align:middle line:84% because, again, the null space of A minus lambda is-- 00:12:18.150 --> 00:12:20.700 align:middle line:84% or unique up to a finite dimensional 00:12:20.700 --> 00:12:23.670 align:middle line:84% subspace, again, because the null space of A minus lambda 00:12:23.670 --> 00:12:24.870 align:middle line:90% is finite dimensional. 00:12:24.870 --> 00:12:36.120 align:middle line:84% And the second is that for a self-adjoint operator, we have 00:12:36.120 --> 00:12:44.145 align:middle line:84% that the spectrum is a subset of the real numbers. 00:12:44.145 --> 00:12:45.520 align:middle line:84% For a self-adjoint operator, this 00:12:45.520 --> 00:12:47.560 align:middle line:84% is something we proved last lecture. 00:12:47.560 --> 00:12:50.450 align:middle line:84% It doesn't have to be compact, just a self-adjoint operator. 00:12:50.450 --> 00:12:54.490 align:middle line:84% So since the spectrum is a subset of the reals, 00:12:54.490 --> 00:12:59.230 align:middle line:84% this proves that for a compact self-adjoint operator, 00:12:59.230 --> 00:13:08.400 align:middle line:84% the spectrum of A is equal to the set of eigenvalues of A, 00:13:08.400 --> 00:13:14.510 align:middle line:84% or I should say nonzero eigenvalues of A. 00:13:14.510 --> 00:13:18.370 align:middle line:90% Let's write it this way. 00:13:18.370 --> 00:13:23.920 align:middle line:84% If I look at what's in the spectrum other than 0 possibly, 00:13:23.920 --> 00:13:27.310 align:middle line:84% then the nonzero numbers that are in the spectrum 00:13:27.310 --> 00:13:28.810 align:middle line:90% have to be eigenvalues. 00:13:28.810 --> 00:13:34.120 align:middle line:90% 00:13:34.120 --> 00:13:37.050 align:middle line:84% So for a compact self-adjoint operator, 00:13:37.050 --> 00:13:42.780 align:middle line:84% and just in the case of matrices, the spectrum, 00:13:42.780 --> 00:13:45.390 align:middle line:90% the nonzero spectrum has to be-- 00:13:45.390 --> 00:13:47.790 align:middle line:90% are nothing but eigenvalues. 00:13:47.790 --> 00:13:49.380 align:middle line:84% And last time, remember, we proved 00:13:49.380 --> 00:13:53.700 align:middle line:84% that the eigenvalues are countably infinite, 00:13:53.700 --> 00:13:54.750 align:middle line:90% or countable. 00:13:54.750 --> 00:13:56.670 align:middle line:84% They're either finite or countably infinite. 00:13:56.670 --> 00:13:58.920 align:middle line:84% And if they're countable infinite, they converge to 0. 00:13:58.920 --> 00:14:04.560 align:middle line:90% 00:14:04.560 --> 00:14:12.460 align:middle line:84% So, again, by last lecture, we conclude 00:14:12.460 --> 00:14:20.040 align:middle line:84% that the spectrum of A take away 0 equals 00:14:20.040 --> 00:14:32.750 align:middle line:84% either finitely many eigenvalues or countably infinite 00:14:32.750 --> 00:14:38.790 align:middle line:84% eigenvalues that are converging to 0. 00:14:38.790 --> 00:14:44.690 align:middle line:90% 00:14:44.690 --> 00:14:48.373 align:middle line:84% So from the Fredholm alternative we get a lot of information 00:14:48.373 --> 00:14:49.790 align:middle line:84% about when we can solve equations. 00:14:49.790 --> 00:14:51.440 align:middle line:90% But it also tells us-- 00:14:51.440 --> 00:14:53.600 align:middle line:84% I mean from that ability to say when 00:14:53.600 --> 00:14:57.650 align:middle line:84% we can solve the equations, we can also 00:14:57.650 --> 00:15:00.170 align:middle line:84% characterize the nonzero spectrum 00:15:00.170 --> 00:15:03.080 align:middle line:84% of a self-adjoint compact operator. 00:15:03.080 --> 00:15:11.940 align:middle line:90% 00:15:11.940 --> 00:15:13.878 align:middle line:90% So we need to prove that-- 00:15:13.878 --> 00:15:15.420 align:middle line:84% so remember all of this just followed 00:15:15.420 --> 00:15:20.125 align:middle line:84% from stuff we had proven and the main conclusion 00:15:20.125 --> 00:15:22.500 align:middle line:84% of the theorem, which is that the range of A minus lambda 00:15:22.500 --> 00:15:23.310 align:middle line:90% is closed. 00:15:23.310 --> 00:15:26.250 align:middle line:84% So we need to prove that the range of A minus lambda 00:15:26.250 --> 00:15:29.080 align:middle line:84% is closed when lambda is a nonzero real number. 00:15:29.080 --> 00:15:36.210 align:middle line:84% So suppose that you have a sequence in the range, which 00:15:36.210 --> 00:15:39.900 align:middle line:84% I'll write as A minus lambda times un converging 00:15:39.900 --> 00:15:46.050 align:middle line:84% to some element f in H. So what we'd like to be able to show 00:15:46.050 --> 00:15:48.430 align:middle line:90% is that f is in the range. 00:15:48.430 --> 00:16:01.930 align:middle line:84% So we want to show f is in the range of H-- 00:16:01.930 --> 00:16:06.090 align:middle line:90% or not range of H, range of A-- 00:16:06.090 --> 00:16:08.020 align:middle line:90% minus lambda, sorry. 00:16:08.020 --> 00:16:11.395 align:middle line:84% Hopefully I didn't make that mistake elsewhere-- no, just-- 00:16:11.395 --> 00:16:11.895 align:middle line:90% OK. 00:16:11.895 --> 00:16:14.420 align:middle line:90% 00:16:14.420 --> 00:16:19.520 align:middle line:84% Now, we're only assuming that a minus lambda when 00:16:19.520 --> 00:16:21.230 align:middle line:90% it hits u sub n converges to f. 00:16:21.230 --> 00:16:24.950 align:middle line:84% We're not a priori assuming the use of n's converge. 00:16:24.950 --> 00:16:25.730 align:middle line:90% In fact, we can't. 00:16:25.730 --> 00:16:29.870 align:middle line:90% 00:16:29.870 --> 00:16:32.030 align:middle line:84% But in the end we would like to come up 00:16:32.030 --> 00:16:37.400 align:middle line:84% with maybe a subsequence or a part of the u sub n's 00:16:37.400 --> 00:16:39.410 align:middle line:84% up to a subsequence which does converge, 00:16:39.410 --> 00:16:42.380 align:middle line:84% and then we conclude that f is in the range. 00:16:42.380 --> 00:16:47.600 align:middle line:84% So first I want to get rid of the useless part of the u sub 00:16:47.600 --> 00:16:48.470 align:middle line:90% n's. 00:16:48.470 --> 00:16:53.420 align:middle line:90% So let W be-- 00:16:53.420 --> 00:16:56.990 align:middle line:90% 00:16:56.990 --> 00:16:59.060 align:middle line:84% well, I don't need to give it a name. 00:16:59.060 --> 00:17:01.655 align:middle line:90% Really, where is my eraser? 00:17:01.655 --> 00:17:11.119 align:middle line:90% 00:17:11.119 --> 00:17:19.819 align:middle line:84% So let vn be the projection onto orthogonal complement 00:17:19.819 --> 00:17:24.650 align:middle line:84% of the null space of A minus lambda of u sub n. 00:17:24.650 --> 00:17:30.440 align:middle line:90% 00:17:30.440 --> 00:17:38.150 align:middle line:90% So now this is just part of-- 00:17:38.150 --> 00:17:40.010 align:middle line:90% so every u sub n is written-- 00:17:40.010 --> 00:17:44.960 align:middle line:84% you can write as something in the null space of A minus 00:17:44.960 --> 00:17:47.660 align:middle line:90% lambda. 00:17:47.660 --> 00:17:49.610 align:middle line:84% So since null space of A minus lambda 00:17:49.610 --> 00:17:54.680 align:middle line:84% is a closed subspace of H, it has an orthogonal 00:17:54.680 --> 00:17:58.160 align:middle line:84% complement so that it and its orthogonal complement 00:17:58.160 --> 00:18:01.730 align:middle line:90% gives the direct product-- 00:18:01.730 --> 00:18:04.280 align:middle line:84% or when you take their direct product, gives you H. 00:18:04.280 --> 00:18:09.060 align:middle line:90% So why am I saying that? 00:18:09.060 --> 00:18:14.330 align:middle line:84% Because then if I take A minus lambda u sub n, 00:18:14.330 --> 00:18:18.200 align:middle line:84% this is equal to A minus lambda applied 00:18:18.200 --> 00:18:26.150 align:middle line:84% to pi, so the projection onto the null space of u sub 00:18:26.150 --> 00:18:29.300 align:middle line:84% n plus the projection onto the orthogonal complement 00:18:29.300 --> 00:18:33.200 align:middle line:84% of the null space, which I defined as v sub n. 00:18:33.200 --> 00:18:37.410 align:middle line:84% Now, this element here is in the null space of A minus lambda. 00:18:37.410 --> 00:18:39.500 align:middle line:84% So when A minus lambda hits it, I get 0. 00:18:39.500 --> 00:18:44.780 align:middle line:84% So I get A minus lambda applied to v sub n, all right. 00:18:44.780 --> 00:18:49.860 align:middle line:90% 00:18:49.860 --> 00:18:56.750 align:middle line:84% Then a minus lambda v sub n equals A minus lambda u sub 00:18:56.750 --> 00:18:58.310 align:middle line:90% n, which converges to f. 00:18:58.310 --> 00:19:04.010 align:middle line:84% So, basically, I've taken away some noise, all right, the part 00:19:04.010 --> 00:19:06.650 align:middle line:84% that when it hits A minus lambda, I get 0. 00:19:06.650 --> 00:19:09.260 align:middle line:84% So now I just have these v sub n's 00:19:09.260 --> 00:19:22.080 align:middle line:84% which lie in the orthogonal complement of the null space 00:19:22.080 --> 00:19:25.250 align:middle line:90% of A minus lambda. 00:19:25.250 --> 00:19:34.390 align:middle line:84% So my claim is, first, is that the sequence v sub 00:19:34.390 --> 00:19:41.350 align:middle line:90% n is bounded. 00:19:41.350 --> 00:19:43.870 align:middle line:84% Basically, once I can show this, then I'm 00:19:43.870 --> 00:19:46.960 align:middle line:84% done because if I can show v sub n is bounded, then 00:19:46.960 --> 00:19:51.790 align:middle line:84% since A is a compact operator, when A hits v sub n up 00:19:51.790 --> 00:19:54.530 align:middle line:84% to a subsequence, that converges. 00:19:54.530 --> 00:19:56.680 align:middle line:84% Now, this whole expression converges and therefore 00:19:56.680 --> 00:19:58.900 align:middle line:90% lambda times v sub n converges. 00:19:58.900 --> 00:20:01.900 align:middle line:84% Lambda is nonzero, so then v sub n converges up 00:20:01.900 --> 00:20:03.490 align:middle line:90% to a subsequence to something. 00:20:03.490 --> 00:20:05.290 align:middle line:84% And therefore A minus lambda v sub n 00:20:05.290 --> 00:20:08.320 align:middle line:84% then converges to A minus lambda v 00:20:08.320 --> 00:20:11.410 align:middle line:84% for some v, which shows that f is 00:20:11.410 --> 00:20:13.760 align:middle line:90% in the range of A minus lambda. 00:20:13.760 --> 00:20:17.290 align:middle line:84% So this is really the whole ball game. 00:20:17.290 --> 00:20:19.150 align:middle line:84% And we'll use here crucially as well 00:20:19.150 --> 00:20:21.820 align:middle line:84% that we threw away useless parts of u sub n, 00:20:21.820 --> 00:20:23.905 align:middle line:84% useless at least to this argument. 00:20:23.905 --> 00:20:27.030 align:middle line:90% 00:20:27.030 --> 00:20:31.330 align:middle line:84% So I claim this is bounded, so suppose not. 00:20:31.330 --> 00:20:36.000 align:middle line:90% 00:20:36.000 --> 00:20:43.970 align:middle line:84% Then there exists subsequence v sub n sub 00:20:43.970 --> 00:20:58.100 align:middle line:84% j such that v sub n sub j goes to infinity 00:20:58.100 --> 00:20:59.540 align:middle line:90% as j goes to infinity. 00:20:59.540 --> 00:21:04.430 align:middle line:90% 00:21:04.430 --> 00:21:11.290 align:middle line:84% All right, now if I look at A minus lambda applied 00:21:11.290 --> 00:21:19.100 align:middle line:84% to v sub n sub j over norm of v sub n sub j, this converges-- 00:21:19.100 --> 00:21:25.420 align:middle line:84% so first off, since is a linear operator, this is equal to 1 00:21:25.420 --> 00:21:30.730 align:middle line:84% over norm v n sub j times A minus lambda applied 00:21:30.730 --> 00:21:33.100 align:middle line:90% to v sub n sub j. 00:21:33.100 --> 00:21:39.580 align:middle line:84% And so this scalar, 1 over norm v n sub j converges to 0. 00:21:39.580 --> 00:21:41.710 align:middle line:90% This converges to f. 00:21:41.710 --> 00:21:45.400 align:middle line:90% So I get the 0 vector in H. 00:21:45.400 --> 00:21:57.330 align:middle line:84% So this thing converges to 0 in the Hilbert space H. Now, 00:21:57.330 --> 00:21:58.200 align:middle line:90% why is that bad? 00:21:58.200 --> 00:22:02.740 align:middle line:84% Because essentially what this is going to say 00:22:02.740 --> 00:22:05.530 align:middle line:84% is that there exists some element, 00:22:05.530 --> 00:22:08.530 align:middle line:84% or that this sequence converges at least up to a subsequence 00:22:08.530 --> 00:22:11.410 align:middle line:84% to an element v with norm 1 because they 00:22:11.410 --> 00:22:14.770 align:middle line:84% all have norm 1 so that A minus lambda v equals 0. 00:22:14.770 --> 00:22:18.790 align:middle line:84% But all of these are in the null space of A minus lambda 00:22:18.790 --> 00:22:20.080 align:middle line:90% and we get a contradiction. 00:22:20.080 --> 00:22:24.220 align:middle line:90% 00:22:24.220 --> 00:22:29.930 align:middle line:90% So we have that A minus lambda-- 00:22:29.930 --> 00:22:30.805 align:middle line:90% so we have that part. 00:22:30.805 --> 00:22:38.940 align:middle line:90% 00:22:38.940 --> 00:22:48.000 align:middle line:84% Since is A is a compact operator, 00:22:48.000 --> 00:22:51.000 align:middle line:84% there exists a subsequence, so a further subsequence-- 00:22:51.000 --> 00:22:56.742 align:middle line:84% I'm just going to call it n sub k instead of n sub j sub k-- 00:22:56.742 --> 00:23:07.430 align:middle line:84% v sub n sub k of v sub n sub j such 00:23:07.430 --> 00:23:14.860 align:middle line:84% that the sequence a sub v sub n sub k converges. 00:23:14.860 --> 00:23:18.880 align:middle line:90% 00:23:18.880 --> 00:23:23.080 align:middle line:84% But then I get that v sub n sub k-- 00:23:23.080 --> 00:23:38.520 align:middle line:84% or I should say v sub n sub k over norm of v sub n sub k. 00:23:38.520 --> 00:23:40.620 align:middle line:90% So these all have norm 1. 00:23:40.620 --> 00:23:46.530 align:middle line:84% And A applied to something that has unit length-- 00:23:46.530 --> 00:23:51.690 align:middle line:84% or the image by A of the closed unit ball is a precompact-- 00:23:51.690 --> 00:23:53.790 align:middle line:90% or the closure of it is compact. 00:23:53.790 --> 00:23:59.250 align:middle line:84% And therefore every sequence has a convergent subsequence. 00:23:59.250 --> 00:24:05.190 align:middle line:84% Then v sub n sub k over norm of v sub n sub K, , 00:24:05.190 --> 00:24:13.830 align:middle line:84% this is equal to 1 over lambda times A applied to v sub n sub 00:24:13.830 --> 00:24:22.370 align:middle line:84% k over norm of v sub n sub k minus A minus lambda applied 00:24:22.370 --> 00:24:23.820 align:middle line:90% to v sub n sub k. 00:24:23.820 --> 00:24:28.620 align:middle line:90% 00:24:28.620 --> 00:24:32.880 align:middle line:84% Now this sequence of elements converges to 0. 00:24:32.880 --> 00:24:35.430 align:middle line:84% That's, in fact, what we just proved. 00:24:35.430 --> 00:24:40.440 align:middle line:84% And this converges by how we've taken this subsequence 00:24:40.440 --> 00:24:42.130 align:middle line:90% because A is a compact operator. 00:24:42.130 --> 00:24:46.290 align:middle line:84% So I have this sequence of vectors is equal to-- 00:24:46.290 --> 00:24:50.580 align:middle line:84% and here we can divide by lambda because lambda is nonzero. 00:24:50.580 --> 00:24:55.350 align:middle line:84% It's equal to a linear combination of two sequences 00:24:55.350 --> 00:24:58.480 align:middle line:90% which converge. 00:24:58.480 --> 00:25:00.720 align:middle line:84% And therefore we get that v sub n 00:25:00.720 --> 00:25:16.320 align:middle line:84% sub k over norm of v sub n sub k, k converges to an element v. 00:25:16.320 --> 00:25:21.120 align:middle line:84% And now the null space or the orthogonal complement 00:25:21.120 --> 00:25:26.430 align:middle line:84% of the null space of A minus lambda, each of these is in-- 00:25:26.430 --> 00:25:31.550 align:middle line:84% remember the orthogonal complement of the null space. 00:25:31.550 --> 00:25:35.950 align:middle line:84% And since it's converging to an element and this is closed, 00:25:35.950 --> 00:25:39.330 align:middle line:84% this element has to be in the same set. 00:25:39.330 --> 00:25:46.600 align:middle line:90% 00:25:46.600 --> 00:25:49.660 align:middle line:90% So, again, this follows from-- 00:25:49.660 --> 00:25:53.680 align:middle line:84% the fact that v is in here is because this set is closed. 00:25:53.680 --> 00:25:57.305 align:middle line:84% The orthogonal complement of any subset of a Hilbert space 00:25:57.305 --> 00:25:57.805 align:middle line:90% is closed. 00:25:57.805 --> 00:26:09.310 align:middle line:90% 00:26:09.310 --> 00:26:13.270 align:middle line:84% Then by continuity of the norm, basically, the norm of v 00:26:13.270 --> 00:26:16.480 align:middle line:84% has to be equal to limit as k goes 00:26:16.480 --> 00:26:20.410 align:middle line:84% to infinity of the norm of the elements 00:26:20.410 --> 00:26:24.355 align:middle line:84% converging to it, which all equal 1. 00:26:24.355 --> 00:26:29.860 align:middle line:90% 00:26:29.860 --> 00:26:37.120 align:middle line:84% And if I compute A minus lambda applied to v, 00:26:37.120 --> 00:26:38.350 align:middle line:90% this is equal to-- 00:26:38.350 --> 00:26:42.040 align:middle line:84% since the v sub n sub k's over norm v sub n 00:26:42.040 --> 00:26:50.200 align:middle line:84% sub k's are converging to v A minus lambda applied to v sub n 00:26:50.200 --> 00:26:54.490 align:middle line:90% sub k over norm v sub n sub k. 00:26:54.490 --> 00:26:59.410 align:middle line:84% And, remember, this is a subsequence of v sub n sub j's. 00:26:59.410 --> 00:27:01.660 align:middle line:84% And when A minus lambda hits that, 00:27:01.660 --> 00:27:03.460 align:middle line:90% they're converging to 0, so-- 00:27:03.460 --> 00:27:07.790 align:middle line:90% 00:27:07.790 --> 00:27:10.610 align:middle line:84% all of that predicated upon the assumption 00:27:10.610 --> 00:27:15.200 align:middle line:84% that the norm of v sub n sub k's converges to infinity 00:27:15.200 --> 00:27:21.080 align:middle line:84% or that the sequence is unbounded. 00:27:21.080 --> 00:27:27.680 align:middle line:84% So we have this element in the null space that has-- 00:27:27.680 --> 00:27:30.620 align:middle line:84% or the orthogonal complement of the null space 00:27:30.620 --> 00:27:36.170 align:middle line:84% that has norm 1 but also gives you 0. 00:27:36.170 --> 00:27:46.500 align:middle line:84% And therefore we get that v is in the null space of A minus 00:27:46.500 --> 00:27:55.700 align:middle line:84% lambda from this computation and it's orthogonal complement. 00:27:55.700 --> 00:27:59.560 align:middle line:90% 00:27:59.560 --> 00:28:02.170 align:middle line:84% But the only possible vector that's 00:28:02.170 --> 00:28:06.460 align:middle line:84% in a space and it's orthogonal complement is the 0 vector, 00:28:06.460 --> 00:28:12.850 align:middle line:84% or the only vector in a subspace and its orthogonal complement 00:28:12.850 --> 00:28:16.060 align:middle line:90% is the zero vector. 00:28:16.060 --> 00:28:21.420 align:middle line:84% And therefore v equals 0, which this 00:28:21.420 --> 00:28:24.450 align:middle line:84% is a contradiction to the fact that the norm of v equals 1. 00:28:24.450 --> 00:28:27.700 align:middle line:90% 00:28:27.700 --> 00:28:31.570 align:middle line:84% So we started off with the sequence v sub n's. 00:28:31.570 --> 00:28:33.640 align:middle line:84% Assuming that they are unbounded, 00:28:33.640 --> 00:28:41.080 align:middle line:84% v sub n over norm v sub n is a sequence of essentially-- yeah. 00:28:41.080 --> 00:28:43.480 align:middle line:84% So maybe you got lost in the subsequences. 00:28:43.480 --> 00:28:45.280 align:middle line:84% But let's just assume I'm talking 00:28:45.280 --> 00:28:48.610 align:middle line:90% about the entire sequence. 00:28:48.610 --> 00:28:52.210 align:middle line:84% Then v sub n over norm v sub n, when a. 00:28:52.210 --> 00:28:55.090 align:middle line:84% Minus lambda hits it, converges to 0. 00:28:55.090 --> 00:28:56.770 align:middle line:84% Since A is a compact operator, we 00:28:56.770 --> 00:29:02.290 align:middle line:84% can show essentially that A applied to v sub n over norm v 00:29:02.290 --> 00:29:04.840 align:middle line:84% sub n, because those things have unit length, 00:29:04.840 --> 00:29:06.430 align:middle line:90% converges to something. 00:29:06.430 --> 00:29:08.740 align:middle line:84% And since lambda is nonzero, we can then 00:29:08.740 --> 00:29:14.330 align:middle line:84% conclude that those vectors converge, 00:29:14.330 --> 00:29:16.400 align:middle line:84% in fact, to something, not just their images 00:29:16.400 --> 00:29:19.740 align:middle line:84% by A or their images by A minus lambda, 00:29:19.740 --> 00:29:22.940 align:middle line:84% again, because lambda is nonzero. 00:29:22.940 --> 00:29:25.250 align:middle line:84% And since they all have unit length, 00:29:25.250 --> 00:29:28.020 align:middle line:84% their limit must have unit length. 00:29:28.020 --> 00:29:33.450 align:middle line:84% And since when a minus lambda hits these guys they go to 0, 00:29:33.450 --> 00:29:37.830 align:middle line:84% the limit must also, when A minus lambda hit it, equals 0. 00:29:37.830 --> 00:29:39.330 align:middle line:84% And that gives us our contradiction 00:29:39.330 --> 00:29:42.930 align:middle line:84% because this limit v had to be in the orthogonal complement 00:29:42.930 --> 00:29:45.690 align:middle line:84% of the null space, but then also in the null space, 00:29:45.690 --> 00:29:47.910 align:middle line:90% and have unit length. 00:29:47.910 --> 00:29:50.370 align:middle line:84% Those three things can all happen at once. 00:29:50.370 --> 00:29:53.010 align:middle line:90% 00:29:53.010 --> 00:30:05.630 align:middle line:84% So thus the sequence v sub n is bounded. 00:30:05.630 --> 00:30:09.940 align:middle line:84% So, remember, what were the v sub n's to start with? 00:30:09.940 --> 00:30:12.115 align:middle line:84% They were so that A minus lambda v 00:30:12.115 --> 00:30:14.500 align:middle line:84% sub n's converged to this element f. 00:30:14.500 --> 00:30:17.950 align:middle line:84% And we wanted to show that f is in the range. 00:30:17.950 --> 00:30:21.940 align:middle line:90% So we coming back over here. 00:30:21.940 --> 00:30:24.220 align:middle line:84% We had these v sub n's so that A minus lambda 00:30:24.220 --> 00:30:26.038 align:middle line:90% v sub n converges to f. 00:30:26.038 --> 00:30:27.580 align:middle line:84% And we want to show f is in the range 00:30:27.580 --> 00:30:32.870 align:middle line:84% to conclude that the range of A minus lambda is closed. 00:30:32.870 --> 00:30:34.760 align:middle line:84% But now since it's bounded and we've 00:30:34.760 --> 00:30:39.440 align:middle line:84% done this argument already, this is essentially the whole ball 00:30:39.440 --> 00:30:39.940 align:middle line:90% game. 00:30:39.940 --> 00:30:42.970 align:middle line:90% 00:30:42.970 --> 00:30:47.420 align:middle line:84% Since the subsequence v sub n is bounded, 00:30:47.420 --> 00:30:59.180 align:middle line:84% and A is a compact operator , we conclude that there exists 00:30:59.180 --> 00:31:06.150 align:middle line:84% a subsequence v sub n sub j such that-- 00:31:06.150 --> 00:31:08.640 align:middle line:84% so this has nothing to do with the previous argument now, 00:31:08.640 --> 00:31:12.060 align:middle line:84% but I just don't feel like using different letters-- 00:31:12.060 --> 00:31:20.230 align:middle line:84% such that a applied to v sub n sub j converges. 00:31:20.230 --> 00:31:22.280 align:middle line:84% So, remember, is A compact operator, 00:31:22.280 --> 00:31:27.370 align:middle line:84% which we stated it in terms of the closure 00:31:27.370 --> 00:31:30.640 align:middle line:84% of the image of the closed unit ball being compact. 00:31:30.640 --> 00:31:35.850 align:middle line:84% Equivalently by scaling the unit ball, 00:31:35.850 --> 00:31:39.060 align:middle line:84% it means that A takes any bounded sequence 00:31:39.060 --> 00:31:43.350 align:middle line:84% to a sequence that has a convergent subsequence. 00:31:43.350 --> 00:31:45.910 align:middle line:90% 00:31:45.910 --> 00:31:47.650 align:middle line:90% So we showed v sub n is bounded. 00:31:47.650 --> 00:31:49.900 align:middle line:84% And therefore since A is a compact operator, 00:31:49.900 --> 00:31:52.750 align:middle line:84% we can find a subsequence so that when A hits it, 00:31:52.750 --> 00:31:56.120 align:middle line:90% we have a convergence sequence. 00:31:56.120 --> 00:31:58.160 align:middle line:84% And by that same trick we used a minute ago, 00:31:58.160 --> 00:32:01.010 align:middle line:84% we conclude that v sub n sub j, which is-- 00:32:01.010 --> 00:32:03.320 align:middle line:84% and we can do this because lambda is nonzero. 00:32:03.320 --> 00:32:07.120 align:middle line:90% We can divide by it. 00:32:07.120 --> 00:32:15.560 align:middle line:84% So A v sub n sub j minus A minus lambda V sub n sub j-- 00:32:15.560 --> 00:32:20.000 align:middle line:84% now, again, this here is converging to something. 00:32:20.000 --> 00:32:22.100 align:middle line:90% This here is converging to f. 00:32:22.100 --> 00:32:26.870 align:middle line:84% So this linear combination of convergent sequences 00:32:26.870 --> 00:32:27.920 align:middle line:90% is convergent-- 00:32:27.920 --> 00:32:32.680 align:middle line:90% 00:32:32.680 --> 00:32:43.200 align:middle line:90% converges to and element v. 00:32:43.200 --> 00:32:49.170 align:middle line:84% And therefore I get that f which is the limit as n goes 00:32:49.170 --> 00:32:50.580 align:middle line:90% to infinity of the v sub n's. 00:32:50.580 --> 00:32:54.960 align:middle line:84% But convergence still holds if I look at a subsequence, 00:32:54.960 --> 00:32:59.370 align:middle line:84% A minus lambda v sub n sub j equals-- 00:32:59.370 --> 00:33:02.790 align:middle line:84% and since A-- so A is a bounded linear operator. 00:33:02.790 --> 00:33:05.280 align:middle line:84% Lambda times the identity is a bounded linear operator. 00:33:05.280 --> 00:33:09.810 align:middle line:84% This is equal to A minus lambda v. And therefore f is 00:33:09.810 --> 00:33:17.740 align:middle line:90% in the range of A minus lambda. 00:33:17.740 --> 00:33:23.850 align:middle line:84% OK, so Fredholm alternative tells you 00:33:23.850 --> 00:33:28.820 align:middle line:84% that the range of A minus lambda for a compact self-adjoint 00:33:28.820 --> 00:33:29.825 align:middle line:90% operator is closed. 00:33:29.825 --> 00:33:35.850 align:middle line:90% 00:33:35.850 --> 00:33:41.800 align:middle line:84% So where did we really used the fact that it was self adjoint? 00:33:41.800 --> 00:33:43.030 align:middle line:90% Nowhere in this argument. 00:33:43.030 --> 00:33:49.030 align:middle line:84% So this fact that the range of A minus lambda is close 00:33:49.030 --> 00:33:54.460 align:middle line:84% is still true if A is just a compact operator 00:33:54.460 --> 00:33:57.955 align:middle line:84% and lambda is just some nonzero complex number. 00:33:57.955 --> 00:34:00.600 align:middle line:90% 00:34:00.600 --> 00:34:03.750 align:middle line:84% But where we use that it's self adjoint is I 00:34:03.750 --> 00:34:05.400 align:middle line:84% guess in the rest of the conclusion, 00:34:05.400 --> 00:34:08.400 align:middle line:84% that the range of A minus lambda is therefore 00:34:08.400 --> 00:34:10.350 align:middle line:84% equal to the null space of A minus lambda. 00:34:10.350 --> 00:34:13.920 align:middle line:84% And therefore either A minus lambda is bijective, 00:34:13.920 --> 00:34:16.440 align:middle line:84% or the null space of A minus lambda 00:34:16.440 --> 00:34:19.590 align:middle line:84% is non-trivial and finite dimensional by what 00:34:19.590 --> 00:34:21.180 align:middle line:90% we did in the previous lecture. 00:34:21.180 --> 00:34:31.409 align:middle line:90% 00:34:31.409 --> 00:34:38.974 align:middle line:84% OK, now, again, this is a very powerful theorem. 00:34:38.974 --> 00:34:49.550 align:middle line:90% 00:34:49.550 --> 00:34:51.889 align:middle line:84% And, again, what this says is that if I 00:34:51.889 --> 00:34:54.620 align:middle line:84% look at the nonzero spectrum of a compact self-adjoint 00:34:54.620 --> 00:34:58.880 align:middle line:84% operator, then that consists entirely of eigenvalues of A. 00:34:58.880 --> 00:35:02.900 align:middle line:84% Now earlier we proved that plus or minus the norm of A 00:35:02.900 --> 00:35:08.090 align:middle line:84% has to be in the spectrum of a self-adjoint operator. 00:35:08.090 --> 00:35:10.250 align:middle line:84% And therefore what we can conclude 00:35:10.250 --> 00:35:14.330 align:middle line:84% is that if we have a non-trivial self-adjoint compact operator, 00:35:14.330 --> 00:35:18.860 align:middle line:84% then it has at least one eigenvalue. 00:35:18.860 --> 00:35:21.030 align:middle line:84% And we can characterize that eigenvalue. 00:35:21.030 --> 00:35:23.760 align:middle line:90% So this has a following theorem. 00:35:23.760 --> 00:35:40.640 align:middle line:84% Let A be a non-trivial compact self-adjoint operator, 00:35:40.640 --> 00:35:41.495 align:middle line:90% A equals A star. 00:35:41.495 --> 00:35:50.140 align:middle line:90% 00:35:50.140 --> 00:36:10.660 align:middle line:84% Then has a non-trivial eigenvalue lambda 1. 00:36:10.660 --> 00:36:14.290 align:middle line:84% And we can characterize lambda 1-- 00:36:14.290 --> 00:36:16.540 align:middle line:84% or at least the absolute value of lambda 1-- 00:36:16.540 --> 00:36:33.900 align:middle line:84% as the supremum over norm u equals 1 Au, u equals-- 00:36:33.900 --> 00:36:43.130 align:middle line:84% and this supremum is actually achieved 00:36:43.130 --> 00:36:55.210 align:middle line:84% where u1 is a normalized eigenvector 00:36:55.210 --> 00:36:56.760 align:middle line:90% corresponding to lambda 1. 00:36:56.760 --> 00:37:07.460 align:middle line:90% 00:37:07.460 --> 00:37:12.300 align:middle line:90% So why do we have-- 00:37:12.300 --> 00:37:16.890 align:middle line:84% or I should say, let me make sure I have everything here. 00:37:16.890 --> 00:37:18.885 align:middle line:90% OK, so why is this? 00:37:18.885 --> 00:37:24.430 align:middle line:90% 00:37:24.430 --> 00:37:30.940 align:middle line:84% So first off, we've shown that plus or minus the norm of A is, 00:37:30.940 --> 00:37:40.310 align:middle line:84% in fact, in the spectrum of A. for any self-adjoint bounded 00:37:40.310 --> 00:37:48.040 align:middle line:84% linear operator, not necessarily compact, that plus or minus-- 00:37:48.040 --> 00:37:51.010 align:middle line:84% not necessarily both of them, but plus or minus, 00:37:51.010 --> 00:38:00.760 align:middle line:84% one of these, at least one of plus or minus, since a 00:38:00.760 --> 00:38:15.360 align:middle line:84% is self adjoint, meaning A star equals A. 00:38:15.360 --> 00:38:22.460 align:middle line:84% Then lambda 1-- then I'm going to say plus or minus, 00:38:22.460 --> 00:38:29.980 align:middle line:84% meaning not both, actually at least one of these, 00:38:29.980 --> 00:38:34.850 align:middle line:84% is an eigenvalue of A by the Fredholm alternative. 00:38:34.850 --> 00:38:45.960 align:middle line:90% 00:38:45.960 --> 00:38:56.935 align:middle line:84% The Fredholm alternative and the fact that-- 00:38:56.935 --> 00:39:00.350 align:middle line:90% 00:39:00.350 --> 00:39:06.500 align:middle line:84% so lambda one is going to be either plus norm 00:39:06.500 --> 00:39:10.040 align:middle line:84% of A or minus norm of A, depending on 00:39:10.040 --> 00:39:13.700 align:middle line:84% whether which one is in the spectrum. 00:39:13.700 --> 00:39:18.590 align:middle line:84% Let's say plus if it's in the spectrum, minus if plus is not. 00:39:18.590 --> 00:39:24.380 align:middle line:84% And the fact that we can identify it as this quantity 00:39:24.380 --> 00:39:32.480 align:middle line:84% here is because we have that for self-adjoint operators, 00:39:32.480 --> 00:39:37.790 align:middle line:84% norm of A, which is the absolute value of one of those plus 00:39:37.790 --> 00:39:41.030 align:middle line:84% or minuses that are in the spectrum is equal to-- 00:39:41.030 --> 00:39:42.680 align:middle line:90% so this is where-- 00:39:42.680 --> 00:39:53.370 align:middle line:84% so this is equal to sup u equals 1, Au equals u. 00:39:53.370 --> 00:40:01.290 align:middle line:90% 00:40:01.290 --> 00:40:05.200 align:middle line:90% So that's the end of the proof. 00:40:05.200 --> 00:40:10.800 align:middle line:84% And now what we're going to do is we are going to keep going. 00:40:10.800 --> 00:40:13.920 align:middle line:84% So the end result will be that we 00:40:13.920 --> 00:40:19.700 align:middle line:84% can determine all of the eigenvalues 00:40:19.700 --> 00:40:24.950 align:middle line:84% via a certain maximum principle and build up 00:40:24.950 --> 00:40:28.010 align:middle line:84% a sequence of eigenvectors, normalized eigenvectors, which 00:40:28.010 --> 00:40:31.100 align:middle line:84% are pairwise orthogonal simply because they 00:40:31.100 --> 00:40:35.500 align:middle line:90% come from how they're built up. 00:40:35.500 --> 00:40:38.270 align:middle line:90% But we'll see. 00:40:38.270 --> 00:40:40.520 align:middle line:90% And what? 00:40:40.520 --> 00:40:43.670 align:middle line:84% And then we'll show that essentially that 00:40:43.670 --> 00:40:48.260 align:middle line:84% set of eigenvectors that we get, along with a set of-- 00:40:48.260 --> 00:40:52.190 align:middle line:84% or an orthonormal basis chosen for the null space of A, 00:40:52.190 --> 00:40:57.890 align:middle line:84% form an orthonormal basis for a separable Hilbert space H. 00:40:57.890 --> 00:40:59.820 align:middle line:90% So that's where we're going. 00:40:59.820 --> 00:41:05.550 align:middle line:84% But we can basically take this theorem and keep applying it. 00:41:05.550 --> 00:41:08.210 align:middle line:84% So we have the following maximum principle. 00:41:08.210 --> 00:41:11.300 align:middle line:90% 00:41:11.300 --> 00:41:13.940 align:middle line:84% If you like, this is the first step in a maximum principle. 00:41:13.940 --> 00:41:16.545 align:middle line:84% This says if you want to find the largest eigenvalue, lambda 00:41:16.545 --> 00:41:17.045 align:middle line:90% 1-- 00:41:17.045 --> 00:41:20.880 align:middle line:90% 00:41:20.880 --> 00:41:22.200 align:middle line:90% I should say-- 00:41:22.200 --> 00:41:29.170 align:middle line:84% I can even say largest eigenvalue, largest 00:41:29.170 --> 00:41:34.620 align:middle line:90% in the sense of absolute value. 00:41:34.620 --> 00:41:35.120 align:middle line:90% Why? 00:41:35.120 --> 00:41:37.370 align:middle line:84% Because remember the spectrum is contained 00:41:37.370 --> 00:41:43.250 align:middle line:84% in the interval minus norm A plus norm of A. 00:41:43.250 --> 00:41:48.290 align:middle line:84% So anything in the spectrum has to have absolute value less 00:41:48.290 --> 00:41:53.950 align:middle line:84% than or equal to the norm of A. And if it's eigenvalue, 00:41:53.950 --> 00:41:56.690 align:middle line:90% or if it's a-- 00:41:56.690 --> 00:42:02.940 align:middle line:84% anything other than 0 has to be an eigenvalue so we get this. 00:42:02.940 --> 00:42:05.740 align:middle line:84% So what's this maximum principle? 00:42:05.740 --> 00:42:06.990 align:middle line:90% And why was I saying all that? 00:42:06.990 --> 00:42:09.240 align:middle line:90% Oh, so this gives you a way to-- 00:42:09.240 --> 00:42:11.760 align:middle line:84% if you'd like try and find or at least 00:42:11.760 --> 00:42:18.000 align:middle line:84% approximate the first eigenvalue of a bounded linear operator, 00:42:18.000 --> 00:42:21.930 align:middle line:84% this is a maximization problem with a constraint. 00:42:21.930 --> 00:42:24.510 align:middle line:84% So you could use the method of Lagrange multipliers. 00:42:24.510 --> 00:42:27.090 align:middle line:90% 00:42:27.090 --> 00:42:30.920 align:middle line:84% That doesn't maybe make sense to you 00:42:30.920 --> 00:42:33.410 align:middle line:84% to be able to do on an infinite dimensional Hilbert space. 00:42:33.410 --> 00:42:38.270 align:middle line:84% But let's say you choose a big basis of your Hilbert space. 00:42:38.270 --> 00:42:42.530 align:middle line:84% And you just restrict to looking at that big but finite 00:42:42.530 --> 00:42:49.170 align:middle line:84% dimensional, or finite basis, or span of that finite basis, 00:42:49.170 --> 00:42:51.460 align:middle line:84% and try to solve the approximate problem, 00:42:51.460 --> 00:42:56.990 align:middle line:84% then you should get close to the eigenvalue 00:42:56.990 --> 00:42:59.390 align:middle line:84% and get an approximate eigenvector 00:42:59.390 --> 00:43:04.690 align:middle line:84% because as I've said, the eigenvector will be achieved-- 00:43:04.690 --> 00:43:07.910 align:middle line:84% or I should say that the eigenvector achieves this 00:43:07.910 --> 00:43:10.990 align:middle line:90% maximum here , or the supremum. 00:43:10.990 --> 00:43:14.780 align:middle line:84% So the maximum principle is the following. 00:43:14.780 --> 00:43:20.320 align:middle line:84% So let A-- again, we're only looking at compact self-adjoint 00:43:20.320 --> 00:43:30.790 align:middle line:90% operators, compact operator-- 00:43:30.790 --> 00:43:35.520 align:middle line:90% 00:43:35.520 --> 00:43:50.320 align:middle line:84% then the nonzero eigenvalues of A can be ordered. 00:43:50.320 --> 00:43:52.780 align:middle line:90% Now, this part we already know. 00:43:52.780 --> 00:43:57.340 align:middle line:84% They can be ordered lambda 1 less than or equal to lambda 2, 00:43:57.340 --> 00:44:02.410 align:middle line:84% less than or equal to lambda 3, counted with multiplicity 00:44:02.410 --> 00:44:08.010 align:middle line:84% meaning if lambda 1 has a two-dimensional eigenspace then 00:44:08.010 --> 00:44:08.910 align:middle line:90% lambda 2-- 00:44:08.910 --> 00:44:11.040 align:middle line:90% or the absolute value of-- 00:44:11.040 --> 00:44:13.050 align:middle line:90% or lambda 2 will be lambda 1. 00:44:13.050 --> 00:44:16.620 align:middle line:84% So we'll repeat it according to multiplicity. 00:44:16.620 --> 00:44:32.840 align:middle line:90% 00:44:32.840 --> 00:44:35.600 align:middle line:84% So we know that there's finitely many distinct eigenvalues. 00:44:35.600 --> 00:44:37.100 align:middle line:84% That was the part that I was saying. 00:44:37.100 --> 00:44:43.780 align:middle line:84% We know we can order them, but maybe it's not clear 00:44:43.780 --> 00:44:53.850 align:middle line:84% that you can order them with multiplicity 00:44:53.850 --> 00:45:15.210 align:middle line:84% with corresponding orthonormal eigenbasis functions uk. 00:45:15.210 --> 00:45:22.500 align:middle line:84% So u1 is a normalized eigenfunction for lambda 1. 00:45:22.500 --> 00:45:25.710 align:middle line:84% u2 will be a normalized function for lambda 2, which 00:45:25.710 --> 00:45:27.570 align:middle line:90% is orthogonal to u1. 00:45:27.570 --> 00:45:30.090 align:middle line:84% So these are pairwise orthonormal. 00:45:30.090 --> 00:45:32.750 align:middle line:90% 00:45:32.750 --> 00:45:38.340 align:middle line:84% And how do we obtain the eigenvalues 00:45:38.340 --> 00:45:40.140 align:middle line:84% in this order and these eigenfunctions 00:45:40.140 --> 00:45:48.860 align:middle line:84% via the following process so that lambda j 00:45:48.860 --> 00:45:58.770 align:middle line:84% is equal to the supremum over all unit vectors that 00:45:58.770 --> 00:46:03.090 align:middle line:84% are orthogonal to the first j minus 1? 00:46:03.090 --> 00:46:19.610 align:middle line:90% 00:46:19.610 --> 00:46:21.320 align:middle line:90% And this is equal to-- 00:46:21.320 --> 00:46:24.180 align:middle line:84% this will be achieved on the u sub j. 00:46:24.180 --> 00:46:29.950 align:middle line:90% 00:46:29.950 --> 00:46:35.480 align:middle line:84% So we have the first one, if you like. 00:46:35.480 --> 00:46:36.810 align:middle line:90% We built up lambda 1, u1. 00:46:36.810 --> 00:46:37.310 align:middle line:90% and. 00:46:37.310 --> 00:46:39.740 align:middle line:84% Now what this maximum principle theorem 00:46:39.740 --> 00:46:42.290 align:middle line:84% is saying is that we can repeat this, 00:46:42.290 --> 00:46:46.280 align:middle line:84% is that if we now look at this quantity here, 00:46:46.280 --> 00:46:50.210 align:middle line:84% the supremum over all norm 1 vectors 00:46:50.210 --> 00:46:55.250 align:middle line:84% that are orthogonal to u1, then we're 00:46:55.250 --> 00:47:00.920 align:middle line:84% going to pick up the next largest eigenvalue counted 00:47:00.920 --> 00:47:05.630 align:middle line:84% with multiplicity, meaning it'll be the absolute value of lambda 00:47:05.630 --> 00:47:07.700 align:middle line:84% 2, which will be less than lambda 1 00:47:07.700 --> 00:47:15.200 align:middle line:84% if lambda 1 only has a one dimensional eigenspace. 00:47:15.200 --> 00:47:18.470 align:middle line:84% Or it'll be lambda-- or the absolute value of lambda 2 00:47:18.470 --> 00:47:20.270 align:middle line:84% will be the absolute value of lambda 1 00:47:20.270 --> 00:47:23.330 align:middle line:84% again if lambda 1 has a, let's say, 00:47:23.330 --> 00:47:26.250 align:middle line:90% two-dimensional eigenspace. 00:47:26.250 --> 00:47:28.140 align:middle line:84% And if it had a three-dimensional eigenspace, 00:47:28.140 --> 00:47:32.970 align:middle line:84% we would get the absolute value of lambda 1 for-- 00:47:32.970 --> 00:47:36.160 align:middle line:84% it would equal this number and this number as well. 00:47:36.160 --> 00:47:42.750 align:middle line:84% So, again, these nonzero eigenvalues 00:47:42.750 --> 00:47:44.310 align:middle line:90% can be ordered in this way. 00:47:44.310 --> 00:47:49.000 align:middle line:90% 00:47:49.000 --> 00:47:50.170 align:middle line:90% Oh, and I left off one. 00:47:50.170 --> 00:47:54.720 align:middle line:90% 00:47:54.720 --> 00:47:56.235 align:middle line:90% And the lambda j is going to 0. 00:47:56.235 --> 00:48:08.100 align:middle line:90% 00:48:08.100 --> 00:48:14.200 align:middle line:84% OK, so we have the first one, the absolute value of lambda 1, 00:48:14.200 --> 00:48:15.980 align:middle line:90% and the first eigenvector u1. 00:48:15.980 --> 00:48:16.480 align:middle line:90% and. 00:48:16.480 --> 00:48:22.680 align:middle line:84% Now we're just going to basically repeat or apply 00:48:22.680 --> 00:48:34.380 align:middle line:84% the previous theorem to a modification of A. Now I should 00:48:34.380 --> 00:48:41.160 align:middle line:84% say this is not entirely true because there may only 00:48:41.160 --> 00:48:42.105 align:middle line:90% be finitely many. 00:48:42.105 --> 00:48:45.280 align:middle line:90% 00:48:45.280 --> 00:48:48.130 align:middle line:84% So really this should be a remark, 00:48:48.130 --> 00:49:14.010 align:middle line:84% if the decreasing sequence does not terminate, 00:49:14.010 --> 00:49:18.300 align:middle line:84% then the absolute value of the lambda j's goes to 0. 00:49:18.300 --> 00:49:31.270 align:middle line:90% 00:49:31.270 --> 00:49:42.790 align:middle line:84% Now, so what's the new bit of information here? 00:49:42.790 --> 00:49:45.810 align:middle line:90% We do know that we can-- 00:49:45.810 --> 00:49:50.280 align:middle line:84% for each, let's say, capital N, the number of eigenvalues 00:49:50.280 --> 00:49:51.840 align:middle line:90% outside of-- 00:49:51.840 --> 00:49:56.850 align:middle line:84% or with absolute value bigger than 1 over N has to be finite. 00:49:56.850 --> 00:49:59.550 align:middle line:84% So the fact that we can order them 00:49:59.550 --> 00:50:02.670 align:middle line:84% is not really that much new information. 00:50:02.670 --> 00:50:05.580 align:middle line:84% And the fact that they to go to 0 if this sequence is infinite, 00:50:05.580 --> 00:50:07.470 align:middle line:90% that's also not new information. 00:50:07.470 --> 00:50:10.110 align:middle line:84% I mean, we did prove that in a previous lecture 00:50:10.110 --> 00:50:10.890 align:middle line:90% that if you have-- 00:50:10.890 --> 00:50:15.450 align:middle line:90% 00:50:15.450 --> 00:50:17.940 align:middle line:84% so there we proved it for a sequence 00:50:17.940 --> 00:50:19.140 align:middle line:90% of distinct eigenvalues. 00:50:19.140 --> 00:50:21.420 align:middle line:84% But because we now know that each of these 00:50:21.420 --> 00:50:25.350 align:middle line:84% has a finite dimensional eigenspace, if you just 00:50:25.350 --> 00:50:27.840 align:middle line:84% look back at the same proof or that proof, 00:50:27.840 --> 00:50:29.400 align:middle line:84% you can make a small adjustment to be 00:50:29.400 --> 00:50:33.690 align:middle line:84% able to say that if you count the eigenvalues 00:50:33.690 --> 00:50:37.290 align:middle line:84% with multiplicity, then also the sequence has to go to 0, 00:50:37.290 --> 00:50:39.270 align:middle line:84% not just necessarily the sequence consisting 00:50:39.270 --> 00:50:41.680 align:middle line:90% of the distinct eigenvalues. 00:50:41.680 --> 00:50:48.120 align:middle line:84% So let me make that small remark. 00:50:48.120 --> 00:50:49.980 align:middle line:84% What the new piece is that we can 00:50:49.980 --> 00:50:52.710 align:middle line:84% compute the eigenvalues in this way 00:50:52.710 --> 00:50:57.300 align:middle line:84% and choose the eigenvectors in this way. 00:50:57.300 --> 00:50:57.800 align:middle line:90% That's a. 00:50:57.800 --> 00:51:00.680 align:middle line:84% New piece and that's what, in the end, 00:51:00.680 --> 00:51:06.110 align:middle line:84% we'll use to be able to show that H can be-- 00:51:06.110 --> 00:51:08.720 align:middle line:84% or that A can basically be diagonalized, 00:51:08.720 --> 00:51:12.380 align:middle line:84% or that you can find an orthonormal basis of a Hilbert 00:51:12.380 --> 00:51:15.170 align:middle line:84% space, separable Hilbert space consisting entirely 00:51:15.170 --> 00:51:19.880 align:middle line:90% of eigenvectors of A. 00:51:19.880 --> 00:51:35.990 align:middle line:84% So the construction proceeds inductively. 00:51:35.990 --> 00:51:39.877 align:middle line:84% So at one point I said, if you can d it for one 00:51:39.877 --> 00:51:41.210 align:middle line:90% then you can do it for the next. 00:51:41.210 --> 00:51:43.250 align:middle line:84% And then I won't be so formal every time 00:51:43.250 --> 00:51:46.640 align:middle line:90% I do an inductive construction. 00:51:46.640 --> 00:51:52.270 align:middle line:84% But in this case it pays to be careful. 00:51:52.270 --> 00:51:57.760 align:middle line:84% So we'll construct the sequence of lambda j's 00:51:57.760 --> 00:52:06.830 align:middle line:84% and uk's in this way via an inductive argument. 00:52:06.830 --> 00:52:13.570 align:middle line:84% So coming up with k equals 1, that was the previous theorem. 00:52:13.570 --> 00:52:21.450 align:middle line:84% So previous-- or I should say j equals 00:52:21.450 --> 00:52:26.670 align:middle line:90% 1, that's the previous theorem. 00:52:26.670 --> 00:52:31.050 align:middle line:84% We found the largest eigenvalue, or the eigenvalue with-- 00:52:31.050 --> 00:52:34.140 align:middle line:84% an eigenvalue with the largest absolute value 00:52:34.140 --> 00:52:36.720 align:middle line:90% via this theorem. 00:52:36.720 --> 00:52:40.990 align:middle line:84% And then we obtained an eigenvector this way, 00:52:40.990 --> 00:52:44.380 align:middle line:84% just basically as a consequence of the Fredholm alternative. 00:52:44.380 --> 00:52:46.780 align:middle line:84% And then that we know that plus or minus the norm of A 00:52:46.780 --> 00:52:48.620 align:middle line:90% has to be in the spectrum. 00:52:48.620 --> 00:52:52.750 align:middle line:84% | the fact that we can find the first eigenvalue of lambda 1, 00:52:52.750 --> 00:52:54.470 align:middle line:84% that follows from the previous theorem. 00:52:54.470 --> 00:53:04.130 align:middle line:84% So now we want to do the inductive step, i.e. 00:53:04.130 --> 00:53:19.640 align:middle line:84% we want to suppose we have found lambda 1 up to lambda-- 00:53:19.640 --> 00:53:22.740 align:middle line:90% let's say, what did I use here? 00:53:22.740 --> 00:53:41.420 align:middle line:84% Lambda n, so j equals n and with orthonormal eigenvectors 00:53:41.420 --> 00:53:51.370 align:middle line:84% u1 up to un, satisfying this maximum principal. 00:53:51.370 --> 00:53:56.410 align:middle line:84% So j starting from-- starting at 1 00:53:56.410 --> 00:54:01.270 align:middle line:84% up to n, we've constructed or found a lambda 2, lambda 3, up 00:54:01.270 --> 00:54:05.760 align:middle line:84% to lambda n, and the u went up to un, 00:54:05.760 --> 00:54:08.760 align:middle line:84% satisfying that maximum property. 00:54:08.760 --> 00:54:14.820 align:middle line:90% 00:54:14.820 --> 00:54:19.445 align:middle line:90% So then there's two cases. 00:54:19.445 --> 00:54:23.900 align:middle line:90% 00:54:23.900 --> 00:54:29.870 align:middle line:84% So in case 1 is the fact that A minus-- 00:54:29.870 --> 00:54:34.330 align:middle line:90% 00:54:34.330 --> 00:54:42.400 align:middle line:84% A is equal to the sum from k equals 1 to n of lambda 00:54:42.400 --> 00:54:46.675 align:middle line:90% k, u inner product uk, uk. 00:54:46.675 --> 00:54:50.130 align:middle line:90% 00:54:50.130 --> 00:55:04.270 align:middle line:84% And therefore what this shows is that we 00:55:04.270 --> 00:55:11.870 align:middle line:84% found all the eigenvalues and the process terminates. 00:55:11.870 --> 00:55:19.297 align:middle line:90% 00:55:19.297 --> 00:55:20.880 align:middle line:84% So this is kind of the degenerate case 00:55:20.880 --> 00:55:23.913 align:middle line:84% that A is a finite rank operator. 00:55:23.913 --> 00:55:25.830 align:middle line:84% So I could have started this whole theorem off 00:55:25.830 --> 00:55:29.160 align:middle line:84% with let's assume A is not a finite rank operator, 00:55:29.160 --> 00:55:31.410 align:middle line:84% and therefore we wouldn't have to deal with this case. 00:55:31.410 --> 00:55:34.260 align:middle line:84% But I just stated it for an arbitrary A. 00:55:34.260 --> 00:55:38.850 align:middle line:84% So there is the possibility that that sequence stopped. 00:55:38.850 --> 00:55:43.080 align:middle line:84% We found all of the eigenvalues with multiplicity 00:55:43.080 --> 00:55:45.160 align:middle line:90% in this process. 00:55:45.160 --> 00:55:48.180 align:middle line:84% And then the theorem-- or construction 00:55:48.180 --> 00:55:50.670 align:middle line:90% is done in that case. 00:55:50.670 --> 00:55:55.080 align:middle line:84% And the case 2 is that it is not a finite rank operator, 00:55:55.080 --> 00:55:57.380 align:middle line:84% so it's not equal to k equals 1 to n. 00:55:57.380 --> 00:56:02.790 align:middle line:90% 00:56:02.790 --> 00:56:08.640 align:middle line:84% Now we have to show how can we find lambda sub n plus 1 00:56:08.640 --> 00:56:11.220 align:middle line:84% and the eigenvector u sub n plus 1. 00:56:11.220 --> 00:56:14.060 align:middle line:90% 00:56:14.060 --> 00:56:22.160 align:middle line:84% Let a Sub n be the operator A minus k equals 1 to n lambda 00:56:22.160 --> 00:56:24.300 align:middle line:90% k inner product u. 00:56:24.300 --> 00:56:35.540 align:middle line:84% So should say Au minus u sub k, all right. 00:56:35.540 --> 00:56:41.410 align:middle line:90% 00:56:41.410 --> 00:56:41.910 align:middle line:90% Now. 00:56:41.910 --> 00:56:46.930 align:middle line:84% Note since we are in case 2, this operator is nonzero. 00:56:46.930 --> 00:56:49.680 align:middle line:84% So we're basically going to apply the previous theorem now 00:56:49.680 --> 00:56:51.090 align:middle line:90% to this operator. 00:56:51.090 --> 00:56:53.025 align:middle line:84% But let me first make a few remarks. 00:56:53.025 --> 00:56:57.310 align:middle line:90% 00:56:57.310 --> 00:57:00.925 align:middle line:90% Then A is A sub n is a-- 00:57:00.925 --> 00:57:05.417 align:middle line:90% 00:57:05.417 --> 00:57:06.750 align:middle line:90% this is something you can check. 00:57:06.750 --> 00:57:12.740 align:middle line:84% It's a self-adjoint compact operator. 00:57:12.740 --> 00:57:14.440 align:middle line:90% So why is it self adjoint? 00:57:14.440 --> 00:57:18.640 align:middle line:84% Basically because A is self adjoint and these lambda k's 00:57:18.640 --> 00:57:23.610 align:middle line:84% are real numbers because eigenvalues 00:57:23.610 --> 00:57:27.820 align:middle line:84% have to be real, because A is self adjoint 00:57:27.820 --> 00:57:29.480 align:middle line:90% and these are orthonormal. 00:57:29.480 --> 00:57:33.670 align:middle line:84% So that is why they're self adjoint. 00:57:33.670 --> 00:57:35.140 align:middle line:90% Why is it a compact operator? 00:57:35.140 --> 00:57:39.100 align:middle line:84% Well, it's the sum of a compact operator A and a finite rank 00:57:39.100 --> 00:57:40.220 align:middle line:90% operator here. 00:57:40.220 --> 00:57:44.090 align:middle line:90% So it's also a compact operator. 00:57:44.090 --> 00:57:46.550 align:middle line:84% And it's a nontrivial one because it's not 00:57:46.550 --> 00:57:49.960 align:middle line:90% identically equal to 0. 00:57:49.960 --> 00:57:52.300 align:middle line:84% So I shouldn't say not equal to 0, 00:57:52.300 --> 00:57:57.070 align:middle line:84% but which is not identically the zero operator. 00:57:57.070 --> 00:58:03.950 align:middle line:90% 00:58:03.950 --> 00:58:12.605 align:middle line:84% So here's a couple of facts that if u is in the span of u1 00:58:12.605 --> 00:58:21.780 align:middle line:84% up to un, then I get that A sub n applied to u 00:58:21.780 --> 00:58:24.630 align:middle line:90% is the zero vector. 00:58:24.630 --> 00:58:25.410 align:middle line:90% Why is that? 00:58:25.410 --> 00:58:29.070 align:middle line:84% So it suffices to check that this gives me-- 00:58:29.070 --> 00:58:32.430 align:middle line:84% this formula gives me 0 when u is one of the uk's. 00:58:32.430 --> 00:58:39.010 align:middle line:84% Now if u is one of the uk's, then this will be 1 only when-- 00:58:39.010 --> 00:58:41.050 align:middle line:84% or let's say it's one of the uj's. 00:58:41.050 --> 00:58:42.500 align:middle line:90% I need a different letter. 00:58:42.500 --> 00:58:47.410 align:middle line:84% So if u is one of these uj's, then this 00:58:47.410 --> 00:58:50.530 align:middle line:84% will give me 1 only when j equals k 00:58:50.530 --> 00:58:53.140 align:middle line:90% and I pick up lambda j times uj. 00:58:53.140 --> 00:58:56.440 align:middle line:84% And then I have A hitting u sub j. 00:58:56.440 --> 00:58:58.450 align:middle line:84% And that spits out lambda j times 00:58:58.450 --> 00:59:01.330 align:middle line:84% u sub j because u sub j is an eigenvector of A. 00:59:01.330 --> 00:59:03.280 align:middle line:84% So then I get the same thing here and there 00:59:03.280 --> 00:59:04.765 align:middle line:90% and subtracting them gives me 0. 00:59:04.765 --> 00:59:08.670 align:middle line:90% 00:59:08.670 --> 00:59:16.930 align:middle line:84% Now, if u is orthogonal to the span of the u1 up to un, 00:59:16.930 --> 00:59:23.760 align:middle line:90% then I get A n of u equals Au. 00:59:23.760 --> 00:59:24.930 align:middle line:90% So you just see this. 00:59:24.930 --> 00:59:29.120 align:middle line:84% If u is orthogonal to these, then this whole term is 0. 00:59:29.120 --> 00:59:31.250 align:middle line:84% Can't I just pick up An u equals Au. 00:59:31.250 --> 00:59:44.250 align:middle line:90% 00:59:44.250 --> 00:59:55.070 align:middle line:84% For all u in H, in the span of u1 up to un, 00:59:55.070 --> 01:00:00.980 align:middle line:84% if I compute A and u inner product v, 01:00:00.980 --> 01:00:07.190 align:middle line:84% this is equal to u and v since a n is self adjoint. 01:00:07.190 --> 01:00:09.500 align:middle line:84% And since v is in the span of the u1's 01:00:09.500 --> 01:00:13.160 align:middle line:84% up to un's, by the first property that's going to be 0. 01:00:13.160 --> 01:00:19.085 align:middle line:84% So this equals 0 and therefore, what have I shown? 01:00:19.085 --> 01:00:23.360 align:middle line:84% I've shown that A and u inner product v is 0 no matter 01:00:23.360 --> 01:00:26.810 align:middle line:84% what u is in H. And therefore, that 01:00:26.810 --> 01:00:30.770 align:middle line:84% means that v has to be in the orthogonal complement 01:00:30.770 --> 01:00:31.355 align:middle line:90% of the range. 01:00:31.355 --> 01:00:36.638 align:middle line:90% 01:00:36.638 --> 01:00:37.680 align:middle line:90% So I said that backwards. 01:00:37.680 --> 01:00:43.690 align:middle line:84% Anyways, here u is fixed. v is a thing that's changing. 01:00:43.690 --> 01:00:44.730 align:middle line:90% So what does this say? 01:00:44.730 --> 01:00:45.960 align:middle line:90% u as fixed. 01:00:45.960 --> 01:00:49.500 align:middle line:84% A and u inner product v when v is in the span has to be 0. 01:00:49.500 --> 01:00:52.020 align:middle line:84% This holds for all v in the span. 01:00:52.020 --> 01:00:55.260 align:middle line:84% And therefore A n u has to be in the orthogonal complement 01:00:55.260 --> 01:00:57.310 align:middle line:90% of the span of these guys. 01:00:57.310 --> 01:01:03.990 align:middle line:84% So this proves that the range of A n 01:01:03.990 --> 01:01:13.110 align:middle line:84% is a subset of your orthogonal complement of u1 up to un. 01:01:13.110 --> 01:01:21.510 align:middle line:90% 01:01:21.510 --> 01:01:24.420 align:middle line:84% And so from this previous property 01:01:24.420 --> 01:01:33.510 align:middle line:84% we get one last property, that if a n u equals 01:01:33.510 --> 01:01:37.050 align:middle line:84% lambda u is nonzero, meaning we have 01:01:37.050 --> 01:01:45.525 align:middle line:84% a nonzero eigenvalue of A sub n, then that implies that u-- 01:01:45.525 --> 01:01:50.240 align:middle line:90% 01:01:50.240 --> 01:01:55.140 align:middle line:84% so u is equal to 1 over lambda A sub n applied to u. 01:01:55.140 --> 01:01:58.500 align:middle line:84% In other words, u is equal to A sub n applied to u over lambda. 01:01:58.500 --> 01:02:05.180 align:middle line:84% So that implies u is in the range of A sub n, which again 01:02:05.180 --> 01:02:11.960 align:middle line:84% is contained in the orthogonal complement of u1 up to u sub n. 01:02:11.960 --> 01:02:13.910 align:middle line:84% And since it's in the orthogonal complement, 01:02:13.910 --> 01:02:16.270 align:middle line:90% that means A sub nu equals Au. 01:02:16.270 --> 01:02:29.151 align:middle line:90% 01:02:29.151 --> 01:02:33.000 align:middle line:84% So any nonzero eigenvalue of A sub n 01:02:33.000 --> 01:02:43.070 align:middle line:84% has to be a nonzero eigenvalue of A. 01:02:43.070 --> 01:02:50.250 align:middle line:84% Now we just apply the previous theorem 01:02:50.250 --> 01:03:00.140 align:middle line:84% to A sub n to the next eigenvalue lambda sub n plus 1 01:03:00.140 --> 01:03:02.450 align:middle line:90% and eigenvector u sub n plus 1. 01:03:02.450 --> 01:03:10.559 align:middle line:90% 01:03:10.559 --> 01:03:28.880 align:middle line:84% By the previous theorem, a sub n as a nonzero eigenvalue, 01:03:28.880 --> 01:03:42.260 align:middle line:84% which I will call lambda n plus 1, with unit eigenvector 01:03:42.260 --> 01:03:50.020 align:middle line:84% u sub n plus 1 so that lambda n plus 1 01:03:50.020 --> 01:03:57.460 align:middle line:84% is equal to the sup over all norm-- 01:03:57.460 --> 01:04:11.540 align:middle line:90% 01:04:11.540 --> 01:04:16.510 align:middle line:84% ah, OK, sup over all norm or unit length vectors 01:04:16.510 --> 01:04:21.010 align:middle line:84% of absolute value of A sub n applied to u, inner product u. 01:04:21.010 --> 01:04:30.350 align:middle line:84% Now, thos sup is the same sup over all 1 01:04:30.350 --> 01:04:32.760 align:middle line:90% so that A sub n u is nonzero. 01:04:32.760 --> 01:04:35.270 align:middle line:90% So that in particular includes-- 01:04:35.270 --> 01:04:51.700 align:middle line:90% 01:04:51.700 --> 01:04:52.200 align:middle line:90% OK. 01:04:52.200 --> 01:04:59.030 align:middle line:90% 01:04:59.030 --> 01:05:04.430 align:middle line:84% So if u is in the span u1 up to un, then A and u equals 0. 01:05:04.430 --> 01:05:08.810 align:middle line:84% So the supremum over all unit length is the same as u-- 01:05:08.810 --> 01:05:10.910 align:middle line:84% the supremum over everything in the orthogonal 01:05:10.910 --> 01:05:13.610 align:middle line:84% complement of these guys because these guys, 01:05:13.610 --> 01:05:16.730 align:middle line:84% when you stick them into A, and gives me a 0. 01:05:16.730 --> 01:05:18.770 align:middle line:90% So that's the same supremum. 01:05:18.770 --> 01:05:23.480 align:middle line:84% But now also when u is in the orthogonal complement of these 01:05:23.480 --> 01:05:41.490 align:middle line:84% u1's up to un, this is equal to sup of u on the orthogonal 01:05:41.490 --> 01:05:43.380 align:middle line:90% complement of the u1 up to un's. 01:05:43.380 --> 01:05:46.702 align:middle line:90% A n u, remember, equals Au. 01:05:46.702 --> 01:05:50.530 align:middle line:90% 01:05:50.530 --> 01:05:57.160 align:middle line:84% So that gives us the fact that I can find this 01:05:57.160 --> 01:05:59.695 align:middle line:90% from this supremum. 01:05:59.695 --> 01:06:03.150 align:middle line:90% 01:06:03.150 --> 01:06:10.560 align:middle line:84% And so why does this eigenvector has 01:06:10.560 --> 01:06:19.690 align:middle line:84% to be in the orthogonal complement of u1 up to un? 01:06:19.690 --> 01:06:20.190 align:middle line:90% it's. 01:06:20.190 --> 01:06:24.300 align:middle line:84% Because I can choose it that way because when A sub n hits 01:06:24.300 --> 01:06:27.210 align:middle line:90% anything in here, I get 0. 01:06:27.210 --> 01:06:34.220 align:middle line:90% 01:06:34.220 --> 01:06:37.700 align:middle line:84% So I should say this is also equal to A n 01:06:37.700 --> 01:06:39.440 align:middle line:90% apply to un plus 1. 01:06:39.440 --> 01:06:43.940 align:middle line:84% But that's the same as A applied to un plus 1-- 01:06:43.940 --> 01:06:44.840 align:middle line:90% un plus 1. 01:06:44.840 --> 01:06:51.690 align:middle line:90% 01:06:51.690 --> 01:06:56.220 align:middle line:84% And this is less than or equal to the sup over norm u equals 01:06:56.220 --> 01:07:06.530 align:middle line:84% 1 u in span u1 up to un minus 1, orthogonal complement 01:07:06.530 --> 01:07:13.520 align:middle line:84% Au applied to u, which equals the absolute value 01:07:13.520 --> 01:07:18.140 align:middle line:84% of the n-th eigenvalue counted with multiplicity. 01:07:18.140 --> 01:07:25.760 align:middle line:84% So we found the next eigenvalue in this sequence 01:07:25.760 --> 01:07:28.198 align:middle line:84% of eigenvalues countable multiplicity. 01:07:28.198 --> 01:07:41.380 align:middle line:90% 01:07:41.380 --> 01:07:50.140 align:middle line:84% So now, we can conclude the following spectral theorem 01:07:50.140 --> 01:07:52.750 align:middle line:84% for compact self-adjoint operators. 01:07:52.750 --> 01:07:59.360 align:middle line:90% 01:07:59.360 --> 01:08:23.859 align:middle line:84% Let A be a self-adjoint compact operator on a separable Hilbert 01:08:23.859 --> 01:08:31.560 align:middle line:90% space H. 01:08:31.560 --> 01:08:39.138 align:middle line:84% Let lambda 1 bigger than or equal to lambda 2 01:08:39.138 --> 01:08:47.839 align:middle line:84% be corresponding eigenvalues, be the-- 01:08:47.839 --> 01:08:57.340 align:middle line:90% 01:08:57.340 --> 01:09:00.819 align:middle line:84% eigenvalues-- or I should say nonzero eigenvalues-- 01:09:00.819 --> 01:09:06.420 align:middle line:90% 01:09:06.420 --> 01:09:17.410 align:middle line:84% of A counted again with multiplicity, so counted 01:09:17.410 --> 01:09:19.779 align:middle line:84% with multiplicity, as we've constructed 01:09:19.779 --> 01:09:26.649 align:middle line:84% in this theorem which I called the maximum principle, 01:09:26.649 --> 01:09:45.910 align:middle line:84% with corresponding orthonormal eigenvectors uk. 01:09:45.910 --> 01:09:56.050 align:middle line:84% So we have these eigenvectors coming from this process. 01:09:56.050 --> 01:10:12.310 align:middle line:84% And the conclusion is this subset of eigenvectors, 01:10:12.310 --> 01:10:22.040 align:middle line:84% or the orthonormal eigenvectors is an orthonormal basis 01:10:22.040 --> 01:10:27.240 align:middle line:90% for the closure-- 01:10:27.240 --> 01:10:31.390 align:middle line:84% or for the range of A. In fact, we 01:10:31.390 --> 01:10:40.900 align:middle line:84% can upgrade that the uk is, in fact, an orthonormal basis 01:10:40.900 --> 01:10:46.810 align:middle line:90% for the range of A closure. 01:10:46.810 --> 01:11:02.530 align:middle line:84% And there exists an orthonormal basis, call it f sub j, 01:11:02.530 --> 01:11:20.795 align:middle line:84% of the null space of A if it's nonzero, such that uk k fj. 01:11:20.795 --> 01:11:23.590 align:middle line:90% 01:11:23.590 --> 01:11:28.460 align:middle line:84% So, first off, the union of these two sets 01:11:28.460 --> 01:11:30.500 align:middle line:84% of orthonormal vectors is then going 01:11:30.500 --> 01:11:33.410 align:middle line:84% to be again a subset of orthonormal vectors 01:11:33.410 --> 01:11:39.740 align:middle line:84% because all the fj's would correspond to the eigenvalue 0. 01:11:39.740 --> 01:11:43.340 align:middle line:84% And these uk's would correspond to eigenvalues 01:11:43.340 --> 01:11:45.350 align:middle line:90% that are nonzero. 01:11:45.350 --> 01:11:49.490 align:middle line:84% And by what we proved last time, any eigenvector 01:11:49.490 --> 01:11:52.800 align:middle line:84% for two distinct eigenvalues would have to be orthonormal. 01:11:52.800 --> 01:11:57.200 align:middle line:84% So this subset is orthonormal from this subset. 01:11:57.200 --> 01:12:03.710 align:middle line:84% But, moreover, is orthonormal like I said, but also 01:12:03.710 --> 01:12:10.200 align:middle line:90% an orthonormal basis for H. 01:12:10.200 --> 01:12:15.150 align:middle line:84% So, in other words, I can find an orthonormal basis 01:12:15.150 --> 01:12:19.320 align:middle line:84% for H consisting entirely of eigenvectors 01:12:19.320 --> 01:12:27.270 align:middle line:84% of this self-adjoint compact operator, OK. 01:12:27.270 --> 01:12:30.150 align:middle line:84% So I will have one piece of this basis coming 01:12:30.150 --> 01:12:34.380 align:middle line:84% from the null space and the other piece corresponding 01:12:34.380 --> 01:12:37.560 align:middle line:90% to nonzero eigenvalues. 01:12:37.560 --> 01:12:40.410 align:middle line:90% 01:12:40.410 --> 01:12:42.905 align:middle line:90% So, really, 2 follows from 1. 01:12:42.905 --> 01:12:49.830 align:middle line:90% 01:12:49.830 --> 01:12:52.330 align:middle line:84% Not sure why I decided to state them separately, 01:12:52.330 --> 01:12:53.185 align:middle line:90% but here we are. 01:12:53.185 --> 01:12:58.240 align:middle line:90% 01:12:58.240 --> 01:13:01.600 align:middle line:84% So proof of 2 will show that uk is in orthonormal basis 01:13:01.600 --> 01:13:04.300 align:middle line:90% for the range of A. 01:13:04.300 --> 01:13:07.420 align:middle line:84% So first off note as we did in the previous proof 01:13:07.420 --> 01:13:18.460 align:middle line:84% that the process of obtaining the lambda 01:13:18.460 --> 01:13:33.450 align:middle line:84% 1 for the eigenvalues and eigenvectors, 01:13:33.450 --> 01:13:43.390 align:middle line:84% or orthonormal eigenvectors terminates if and only 01:13:43.390 --> 01:13:50.120 align:middle line:90% if A was finite rank. 01:13:50.120 --> 01:13:51.740 align:middle line:84% In other words, their exists an n 01:13:51.740 --> 01:14:07.405 align:middle line:84% so that Au is this finite rink operator u in a product uk, uk. 01:14:07.405 --> 01:14:18.720 align:middle line:90% 01:14:18.720 --> 01:14:20.790 align:middle line:84% In this case, if it's a finite rank operator, 01:14:20.790 --> 01:14:28.520 align:middle line:84% then the range is contained in uk, uk, 01:14:28.520 --> 01:14:37.670 align:middle line:84% which is what we wanted to prove for this case, 01:14:37.670 --> 01:14:40.960 align:middle line:84% that A is this finite rank operator. 01:14:40.960 --> 01:14:47.390 align:middle line:84% So suppose otherwise, in other words 01:14:47.390 --> 01:14:55.440 align:middle line:84% the process does not terminate, so that we 01:14:55.440 --> 01:15:01.520 align:middle line:84% have countably infinite many nonzero eigenvalues counted 01:15:01.520 --> 01:15:05.810 align:middle line:84% with multiplicity and corresponding orthonormal 01:15:05.810 --> 01:15:07.490 align:middle line:90% eigenvectors u sub k. 01:15:07.490 --> 01:15:10.540 align:middle line:90% So this is a more interesting-- 01:15:10.540 --> 01:15:24.320 align:middle line:90% 01:15:24.320 --> 01:15:26.560 align:middle line:84% so the eigenvalues are countably infinite. 01:15:26.560 --> 01:15:33.280 align:middle line:90% 01:15:33.280 --> 01:15:39.260 align:middle line:84% OK, so now we want to show-- so as in the remark 01:15:39.260 --> 01:15:44.720 align:middle line:84% that I made afterwards, you know that these lambda k's 01:15:44.720 --> 01:15:47.240 align:middle line:90% have to go to 0. 01:15:47.240 --> 01:15:51.840 align:middle line:84% Now, we want to prove that the uk's 01:15:51.840 --> 01:15:56.535 align:middle line:84% are an orthonormal basis for the range of A. 01:15:56.535 --> 01:15:57.540 align:middle line:90% What does that mean? 01:15:57.540 --> 01:15:59.250 align:middle line:84% By definition, that means that they're 01:15:59.250 --> 01:16:02.160 align:middle line:84% in maximal orthonormal subset of the range of. 01:16:02.160 --> 01:16:04.650 align:middle line:84% A So we have to show that if something's in the range 01:16:04.650 --> 01:16:08.940 align:middle line:84% and it's orthogonal to every one of these eigenvectors, 01:16:08.940 --> 01:16:10.440 align:middle line:90% then that thing has to be 0. 01:16:10.440 --> 01:16:14.880 align:middle line:90% 01:16:14.880 --> 01:16:26.338 align:middle line:84% So the claim if f is in the range of A, and for all k 01:16:26.338 --> 01:16:32.830 align:middle line:84% f inner product uk equals 0, then f is a 0 vector. 01:16:32.830 --> 01:16:44.047 align:middle line:90% 01:16:44.047 --> 01:16:45.630 align:middle line:84% So this is the claim we want to prove. 01:16:45.630 --> 01:16:47.460 align:middle line:84% So suppose we have something in the range. 01:16:47.460 --> 01:16:50.760 align:middle line:84% That means we can write it as A times u 01:16:50.760 --> 01:16:57.090 align:middle line:84% and f inner product uk equals 0 for all k. 01:16:57.090 --> 01:17:08.170 align:middle line:90% 01:17:08.170 --> 01:17:16.490 align:middle line:84% Then for all k, if I look at lambda ku inner product uk, 01:17:16.490 --> 01:17:19.880 align:middle line:90% this is equal to-- 01:17:19.880 --> 01:17:22.340 align:middle line:84% lambda k is a real number so I can bring it 01:17:22.340 --> 01:17:26.060 align:middle line:84% all the way in, get lambda k, uk. 01:17:26.060 --> 01:17:31.580 align:middle line:84% And this is equal to u A apply to uk. 01:17:31.580 --> 01:17:35.360 align:middle line:84% Now, a is self adjoint so I can move this a over here to u. 01:17:35.360 --> 01:17:38.330 align:middle line:90% 01:17:38.330 --> 01:17:43.520 align:middle line:84% And this is equal to f inner product uk equals 0 for all k. 01:17:43.520 --> 01:17:54.228 align:middle line:90% 01:17:54.228 --> 01:18:02.720 align:middle line:84% So by this maximum principle which 01:18:02.720 --> 01:18:06.640 align:middle line:84% we proved just a minute ago we conclude 01:18:06.640 --> 01:18:13.480 align:middle line:84% that the norm of f, which is equal to the norm of Au, which 01:18:13.480 --> 01:18:21.620 align:middle line:84% is equal to the norm of A minus sum from k equals 1 to n 01:18:21.620 --> 01:18:33.270 align:middle line:84% of lambda k u, uk, uk applied to u 01:18:33.270 --> 01:18:35.310 align:middle line:84% because every one of these numbers is 0, 01:18:35.310 --> 01:18:38.145 align:middle line:84% so I haven't subtracted off anything. 01:18:38.145 --> 01:18:47.060 align:middle line:90% 01:18:47.060 --> 01:18:55.040 align:middle line:84% So I can write this in terms of this A sub n applied to u, 01:18:55.040 --> 01:18:58.070 align:middle line:90% where A sub n was this thing-- 01:18:58.070 --> 01:19:03.530 align:middle line:84% or A sub n is this thing, which by the proof 01:19:03.530 --> 01:19:06.230 align:middle line:84% of the maximization-- or the maximum principle 01:19:06.230 --> 01:19:13.720 align:middle line:84% is less than or equal to lambda plus 1 n plus 1 of u 01:19:13.720 --> 01:19:20.580 align:middle line:84% because, again, remember this thing here 01:19:20.580 --> 01:19:23.670 align:middle line:84% is the supremum over all u's of unit length 01:19:23.670 --> 01:19:25.360 align:middle line:90% of A sub n applied to u. 01:19:25.360 --> 01:19:28.920 align:middle line:84% So lambda n plus 1 is less than-- 01:19:28.920 --> 01:19:32.220 align:middle line:84% or this quantity here divided by the norm of u 01:19:32.220 --> 01:19:34.740 align:middle line:84% so that you as a unit length is always less than 01:19:34.740 --> 01:19:36.690 align:middle line:90% or equal to this quantity here. 01:19:36.690 --> 01:19:38.910 align:middle line:90% But now lambda-- so this-- 01:19:38.910 --> 01:19:41.340 align:middle line:84% I had a fixed thing here, norm of f. 01:19:41.340 --> 01:19:43.560 align:middle line:84% And I've shown it's less than or equal to lambda n 01:19:43.560 --> 01:19:45.330 align:middle line:90% plus 1 times the norm of u. 01:19:45.330 --> 01:19:46.650 align:middle line:90% This is a fixed thing. 01:19:46.650 --> 01:19:48.570 align:middle line:84% And the lambda n's are converging to 0. 01:19:48.570 --> 01:19:53.140 align:middle line:90% 01:19:53.140 --> 01:19:56.950 align:middle line:84% And thus I started off with something non-negative, 01:19:56.950 --> 01:19:59.410 align:middle line:84% less than or equal to something converging to 0. 01:19:59.410 --> 01:20:04.690 align:middle line:84% And therefore that thing had to be 0. 01:20:04.690 --> 01:20:06.430 align:middle line:90% Therefore f is 0. 01:20:06.430 --> 01:20:11.350 align:middle line:84% So this proves, 1, that these eigenvectors 01:20:11.350 --> 01:20:24.305 align:middle line:84% are a maximal orthonormal subset of the range of A. 01:20:24.305 --> 01:20:35.810 align:middle line:84% And for 2 we simply note that by 1, we have 01:20:35.810 --> 01:20:41.250 align:middle line:84% that the range of A closure, this 01:20:41.250 --> 01:20:45.300 align:middle line:84% is since the eigenvectors are in orthonormal basis 01:20:45.300 --> 01:20:53.230 align:middle line:84% for the range of A, the closure is contained in-- 01:20:53.230 --> 01:21:01.770 align:middle line:90% 01:21:01.770 --> 01:21:05.580 align:middle line:84% closure of the spans of the uk's, here this 01:21:05.580 --> 01:21:13.440 align:middle line:84% is a finite span, which, remember, 01:21:13.440 --> 01:21:16.350 align:middle line:84% this is by an assignment exercise, 01:21:16.350 --> 01:21:20.850 align:middle line:84% this is equal to k ck, uk such that-- 01:21:20.850 --> 01:21:27.030 align:middle line:90% 01:21:27.030 --> 01:21:39.840 align:middle line:84% and therefore, this implies that is an orthonormal basis 01:21:39.840 --> 01:21:45.210 align:middle line:90% for the range of A closure. 01:21:45.210 --> 01:21:56.820 align:middle line:84% And this is equal to the range of A orthogonal complement, 01:21:56.820 --> 01:22:06.810 align:middle line:84% orthogonal complement, which is equal to the null space of-- 01:22:06.810 --> 01:22:09.850 align:middle line:84% or the orthogonal complement of the null space of A star. 01:22:09.850 --> 01:22:11.600 align:middle line:90% A star equals A, so we get that. 01:22:11.600 --> 01:22:17.440 align:middle line:84% So the uk's-- the eigenvectors form an orthonormal basis 01:22:17.440 --> 01:22:20.710 align:middle line:84% for the orthogonal complement of the null space of A. 01:22:20.710 --> 01:22:26.110 align:middle line:84% So once we've chosen orthonormal basis for null space of A, 01:22:26.110 --> 01:22:28.658 align:middle line:90% that's it. 01:22:28.658 --> 01:22:33.700 align:middle line:84% Since H is separable and the null space of A 01:22:33.700 --> 01:22:45.250 align:middle line:84% is a closed subspace of H, null space of A inseparable. 01:22:45.250 --> 01:22:48.580 align:middle line:84% And we've proven that every separable Hilbert 01:22:48.580 --> 01:22:52.780 align:middle line:90% space, or even just a-- 01:22:52.780 --> 01:22:56.050 align:middle line:84% so this is a close one so it's a Hilbert space anyways. 01:22:56.050 --> 01:22:57.490 align:middle line:84% But every separable Hilbert space 01:22:57.490 --> 01:23:06.000 align:middle line:90% has an orthonormal basis of-- 01:23:06.000 --> 01:23:10.940 align:middle line:90% 01:23:10.940 --> 01:23:17.010 align:middle line:84% and therefore, again, so since these two-- 01:23:17.010 --> 01:23:31.400 align:middle line:84% call this fj-- this is an orthonormal basis 01:23:31.400 --> 01:23:38.480 align:middle line:84% for the null space of A direct product null space 01:23:38.480 --> 01:23:41.990 align:middle line:90% of A orthogonal complement. 01:23:41.990 --> 01:23:44.630 align:middle line:90% That equals H. 01:23:44.630 --> 01:23:47.790 align:middle line:84% And just in the nick of time I'm finished. 01:23:47.790 --> 01:23:52.210 align:middle line:84% So next time where do we go from here? 01:23:52.210 --> 01:23:56.550 align:middle line:84% We'll see some of this applied in a concrete setting 01:23:56.550 --> 01:23:58.590 align:middle line:84% of differentiable equations and also discuss 01:23:58.590 --> 01:24:00.680 align:middle line:90% the functional calculus. 01:24:00.680 --> 01:24:06.000 align:middle line:90%