WEBVTT 00:00:00.000 --> 00:00:01.988 align:middle line:90% [SQUEAKING] 00:00:01.988 --> 00:00:04.224 align:middle line:90% [RUSTLING] 00:00:04.224 --> 00:00:18.400 align:middle line:90% [CLICKING] 00:00:18.400 --> 00:00:23.140 align:middle line:84% CASEY RODRIGUEZ: OK, so let's continue with our discussion 00:00:23.140 --> 00:00:24.910 align:middle line:90% of Hilbert spaces. 00:00:24.910 --> 00:00:28.810 align:middle line:84% And so last time, we finished with Riesz representation 00:00:28.810 --> 00:00:37.530 align:middle line:84% theorem, the only representation theory I remember, 00:00:37.530 --> 00:00:41.430 align:middle line:84% Riesz representation theorem, which 00:00:41.430 --> 00:00:58.710 align:middle line:84% states that if you have a Hilbert space, then 00:00:58.710 --> 00:01:01.920 align:middle line:90% for all f in the dual space-- 00:01:01.920 --> 00:01:07.650 align:middle line:84% which remember, this is a set of bounded linear maps 00:01:07.650 --> 00:01:10.485 align:middle line:90% from H to complex numbers-- 00:01:10.485 --> 00:01:13.560 align:middle line:90% 00:01:13.560 --> 00:01:23.280 align:middle line:84% there exists a unique v in H such that f of u 00:01:23.280 --> 00:01:32.010 align:middle line:84% is equal to u inner product v for all v, for all u and H. 00:01:32.010 --> 00:01:36.780 align:middle line:84% So every continuous linear functional on a Hilbert space 00:01:36.780 --> 00:01:41.115 align:middle line:84% can be realized as an inner product with a fixed vector. 00:01:41.115 --> 00:01:44.800 align:middle line:90% 00:01:44.800 --> 00:01:54.730 align:middle line:84% So using this, we can revisit the subject 00:01:54.730 --> 00:01:59.830 align:middle line:84% which you touched upon in the assignment of adjoints. 00:01:59.830 --> 00:02:02.485 align:middle line:84% So let me just state this as a theorem. 00:02:02.485 --> 00:02:09.270 align:middle line:90% 00:02:09.270 --> 00:02:19.930 align:middle line:84% Let H be a Hilbert space and A going from H to H 00:02:19.930 --> 00:02:21.520 align:middle line:90% be a bounded linear operator. 00:02:21.520 --> 00:02:39.390 align:middle line:90% 00:02:39.390 --> 00:02:43.930 align:middle line:84% The conclusion is then, there exists 00:02:43.930 --> 00:02:57.630 align:middle line:84% a unique bounded linear operator A star from H 00:02:57.630 --> 00:03:12.400 align:middle line:84% to H, which we call the adjoint of A such 00:03:12.400 --> 00:03:16.960 align:middle line:84% that A star satisfies the following property-- 00:03:16.960 --> 00:03:19.660 align:middle line:90% 00:03:19.660 --> 00:03:26.570 align:middle line:84% for all u, v in H, A u inner product 00:03:26.570 --> 00:03:35.640 align:middle line:84% v is equal to u inner product A star v. 00:03:35.640 --> 00:03:53.340 align:middle line:84% And moreover, the operator norm of the adjoint 00:03:53.340 --> 00:04:03.270 align:middle line:90% equals the operator norm of A. 00:04:03.270 --> 00:04:10.110 align:middle line:84% So again, the same way we proved uniqueness for the Riesz 00:04:10.110 --> 00:04:11.790 align:middle line:84% representation theorem, you can prove 00:04:11.790 --> 00:04:14.760 align:middle line:84% that such a linear operator satisfying 00:04:14.760 --> 00:04:17.620 align:middle line:84% this identity for all u and v has to be unique. 00:04:17.620 --> 00:04:23.660 align:middle line:84% So I'll just state here that uniqueness 00:04:23.660 --> 00:04:33.840 align:middle line:84% of such an operator A star follows 00:04:33.840 --> 00:04:44.850 align:middle line:84% from the identity it satisfies, just like the vector v, which 00:04:44.850 --> 00:04:48.360 align:middle line:84% spits out f of u when you take the inner product with u 00:04:48.360 --> 00:04:49.710 align:middle line:90% with it. 00:04:49.710 --> 00:04:52.680 align:middle line:84% Just like that vector is unique regardless 00:04:52.680 --> 00:04:54.630 align:middle line:90% of how you chose to define it-- 00:04:54.630 --> 00:04:59.250 align:middle line:84% I mean, how you created it based on that identity-- 00:04:59.250 --> 00:05:02.400 align:middle line:84% the same thing holds for the adjoint. 00:05:02.400 --> 00:05:05.543 align:middle line:90% 00:05:05.543 --> 00:05:07.460 align:middle line:84% So it's just the same argument is what I mean. 00:05:07.460 --> 00:05:12.900 align:middle line:90% 00:05:12.900 --> 00:05:17.200 align:middle line:84% So how do we define this operator? 00:05:17.200 --> 00:05:19.850 align:middle line:84% Well, we basically define it by this relation. 00:05:19.850 --> 00:05:31.130 align:middle line:84% So let v be in H. Define a map f sub v 00:05:31.130 --> 00:05:44.420 align:middle line:84% from H to C via f sub v of u is equal to A u inner product 00:05:44.420 --> 00:05:46.610 align:middle line:90% with v. 00:05:46.610 --> 00:05:50.255 align:middle line:84% Now, this is clearly a linear map. 00:05:50.255 --> 00:06:00.210 align:middle line:90% 00:06:00.210 --> 00:06:03.390 align:middle line:84% Maybe I shouldn't check this, maybe I should, 00:06:03.390 --> 00:06:07.500 align:middle line:90% but let's check it anyways. 00:06:07.500 --> 00:06:14.940 align:middle line:84% If I take f of v of lambda 1 times u1 plus lambda 2 u2, 00:06:14.940 --> 00:06:23.640 align:middle line:84% this is equal to A times lambda 1 u1 plus lambda 2 u2 v. 00:06:23.640 --> 00:06:27.850 align:middle line:84% And now, we use the fact that A is a linear operator 00:06:27.850 --> 00:06:33.780 align:middle line:84% so that it acts on the linear combination by linearity. 00:06:33.780 --> 00:06:39.330 align:middle line:84% So then I get lambda 1 A u1 plus lambda 2 00:06:39.330 --> 00:06:42.990 align:middle line:84% A applied to u2 inner product v. And because the inner product 00:06:42.990 --> 00:06:46.710 align:middle line:90% is linear in the first entry-- 00:06:46.710 --> 00:06:50.700 align:middle line:84% it's conjugate linear in the second entry, 00:06:50.700 --> 00:06:52.320 align:middle line:84% but it's linear in the first entry-- 00:06:52.320 --> 00:07:05.230 align:middle line:84% this is equal to lambda 1 times A u v plus lambda 2 A u2 v, 00:07:05.230 --> 00:07:14.930 align:middle line:84% and which is lambda 1 f v of u1 plus lambda 2 f v of u2. 00:07:14.930 --> 00:07:16.735 align:middle line:90% So this thing is a linear map. 00:07:16.735 --> 00:07:19.570 align:middle line:90% 00:07:19.570 --> 00:07:22.990 align:middle line:84% I claim it's also continuous, i.e., 00:07:22.990 --> 00:07:26.290 align:middle line:90% it's an element of the dual. 00:07:26.290 --> 00:07:27.115 align:middle line:90% And we just check. 00:07:27.115 --> 00:07:30.730 align:middle line:90% 00:07:30.730 --> 00:07:34.900 align:middle line:84% If the norm of u equals 1, then if I 00:07:34.900 --> 00:07:43.330 align:middle line:84% take the absolute value of f sub v applied to u, 00:07:43.330 --> 00:07:50.260 align:middle line:84% this is, by definition, A u v, which is less than or equal to 00:07:50.260 --> 00:07:53.680 align:middle line:84% by the Cauchy-Schwarz inequality, 00:07:53.680 --> 00:08:02.710 align:middle line:84% the norm of A u times the norm of v. And the norm of A u 00:08:02.710 --> 00:08:06.310 align:middle line:84% is less than or equal to the norm of A, 00:08:06.310 --> 00:08:10.000 align:middle line:84% since is a bounded linear operator, times the norm of v. 00:08:10.000 --> 00:08:17.710 align:middle line:84% So this holds for all u and norm u equals 1. 00:08:17.710 --> 00:08:22.240 align:middle line:84% And therefore, norm of this operator 00:08:22.240 --> 00:08:29.190 align:middle line:84% is less than or equal to norm of A times norm of v. 00:08:29.190 --> 00:08:32.819 align:middle line:84% So that proves that it's an element. 00:08:32.819 --> 00:08:37.380 align:middle line:90% 00:08:37.380 --> 00:08:43.740 align:middle line:84% So thus, f sub v is an element of the dual. 00:08:43.740 --> 00:08:46.830 align:middle line:84% And now we use the Riesz representation theorem. 00:08:46.830 --> 00:08:55.250 align:middle line:84% So by the Riesz representation theorem, 00:08:55.250 --> 00:09:09.870 align:middle line:84% there exists a unique element, which we denote A star v, 00:09:09.870 --> 00:09:23.220 align:middle line:84% in H such that for all u in H, f v of u equals u inner product 00:09:23.220 --> 00:09:33.590 align:middle line:84% with A star v. I.e., for all u in H A u v-- 00:09:33.590 --> 00:09:35.960 align:middle line:84% because that's the definition of this guy on the left-- 00:09:35.960 --> 00:09:47.280 align:middle line:90% is equal to u A star v. 00:09:47.280 --> 00:09:53.550 align:middle line:84% So this just defines, for each v, an element A star v in H. 00:09:53.550 --> 00:10:00.660 align:middle line:84% So now let's prove that this map v to A star v 00:10:00.660 --> 00:10:03.270 align:middle line:90% is a bounded linear operator. 00:10:03.270 --> 00:10:05.540 align:middle line:90% That then defines the adjoint. 00:10:05.540 --> 00:10:14.510 align:middle line:90% 00:10:14.510 --> 00:10:25.870 align:middle line:84% So first, we claim v, which maps to A star v-- 00:10:25.870 --> 00:10:28.810 align:middle line:84% the element that satisfies this identity-- 00:10:28.810 --> 00:10:31.920 align:middle line:90% 00:10:31.920 --> 00:10:34.820 align:middle line:90% is linear. 00:10:34.820 --> 00:10:46.860 align:middle line:84% So let lambda 1, let v1 v2 be in H, lambda 1, lambda 2 00:10:46.860 --> 00:10:47.820 align:middle line:90% be complex numbers. 00:10:47.820 --> 00:10:54.490 align:middle line:90% 00:10:54.490 --> 00:11:05.710 align:middle line:84% Then for all u in H, we have what? 00:11:05.710 --> 00:11:12.790 align:middle line:84% If I take-- let's see, how did I spell this out-- 00:11:12.790 --> 00:11:15.910 align:middle line:90% 00:11:15.910 --> 00:11:23.500 align:middle line:84% u inner product with the element A star lambda 1 v1 plus lambda 00:11:23.500 --> 00:11:26.260 align:middle line:90% 2 v2-- 00:11:26.260 --> 00:11:30.430 align:middle line:84% so remember, this is the element or this is the element such 00:11:30.430 --> 00:11:35.260 align:middle line:84% that this inner product is defined by-- 00:11:35.260 --> 00:11:39.640 align:middle line:84% this is equal to A u plus inner product 00:11:39.640 --> 00:11:44.650 align:middle line:84% with lambda 1 v1 plus lambda 2 v2. 00:11:44.650 --> 00:11:45.985 align:middle line:90% And this is equal to-- 00:11:45.985 --> 00:11:50.200 align:middle line:90% 00:11:50.200 --> 00:11:53.770 align:middle line:84% so first off, the inner product is conjugate linear 00:11:53.770 --> 00:11:54.610 align:middle line:90% in the second entry. 00:11:54.610 --> 00:11:58.330 align:middle line:84% So this is equal to lambda 1 complex conjugate 00:11:58.330 --> 00:12:06.940 align:middle line:84% A u v1 plus lambda 2 complex conjugate A u v2. 00:12:06.940 --> 00:12:11.890 align:middle line:90% Now, how A star-- 00:12:11.890 --> 00:12:16.030 align:middle line:84% how this symbol, if you like-- is defined 00:12:16.030 --> 00:12:20.320 align:middle line:84% is that it's equal to this quantity 00:12:20.320 --> 00:12:28.880 align:middle line:84% here is equal to lambda 1 times A u A star v1 00:12:28.880 --> 00:12:36.480 align:middle line:90% plus lambda 2 bar u A star v2. 00:12:36.480 --> 00:12:40.200 align:middle line:84% And pulling the complex numbers back in, 00:12:40.200 --> 00:12:49.200 align:middle line:84% this is equal to u lambda 1 A star v1 00:12:49.200 --> 00:12:54.570 align:middle line:90% plus lambda 2 A star v2. 00:12:54.570 --> 00:12:59.590 align:middle line:90% 00:12:59.590 --> 00:13:05.650 align:middle line:84% Now, this holds for all u in H, i.e., 00:13:05.650 --> 00:13:12.070 align:middle line:84% the inner product of u with this minus this is 0 for all u in H. 00:13:12.070 --> 00:13:14.110 align:middle line:84% And the only thing orthogonal to everything in H 00:13:14.110 --> 00:13:15.370 align:middle line:90% is the zero vector. 00:13:15.370 --> 00:13:23.650 align:middle line:84% So we conclude that this element is equal to-- 00:13:23.650 --> 00:13:26.890 align:middle line:84% so this element corresponding to this linear combination 00:13:26.890 --> 00:13:31.525 align:middle line:84% is a linear combination of these elements. 00:13:31.525 --> 00:13:40.750 align:middle line:90% 00:13:40.750 --> 00:13:49.870 align:middle line:84% Thus, the map v that's mapped to A star v is linear. 00:13:49.870 --> 00:13:53.320 align:middle line:84% And so we denote this map now simply by A star. 00:13:53.320 --> 00:13:57.940 align:middle line:84% It acts on a vector v by operating by A star v, 00:13:57.940 --> 00:13:59.240 align:middle line:90% by multiplying by A star. 00:13:59.240 --> 00:14:15.705 align:middle line:84% So now, A star is a linear operator. 00:14:15.705 --> 00:14:18.330 align:middle line:84% When I write A star here, I mean the map that takes v to A star 00:14:18.330 --> 00:14:24.390 align:middle line:84% v. So now I want to check that this linear operator is 00:14:24.390 --> 00:14:26.220 align:middle line:90% a bounded linear operator. 00:14:26.220 --> 00:14:28.020 align:middle line:84% I need to check that it's continuous. 00:14:28.020 --> 00:14:31.980 align:middle line:84% And in the process, we'll end up showing that the norm of A star 00:14:31.980 --> 00:14:37.950 align:middle line:90% equals the norm of A. 00:14:37.950 --> 00:14:46.580 align:middle line:90% Now, suppose norm of v equals 1. 00:14:46.580 --> 00:14:48.680 align:middle line:84% So I want to bound A star applied 00:14:48.680 --> 00:15:02.270 align:middle line:84% to v. So if A star v equals 0, then clearly, A star 00:15:02.270 --> 00:15:08.840 align:middle line:84% v is less than or equal to the norm of A. This, in the end, 00:15:08.840 --> 00:15:10.100 align:middle line:90% is what I will show. 00:15:10.100 --> 00:15:12.320 align:middle line:84% But I'm kind of dealing with the stupid case, which 00:15:12.320 --> 00:15:16.340 align:middle line:84% is when this thing on the left-hand side is 0. 00:15:16.340 --> 00:15:17.930 align:middle line:90% So suppose A star v is non-zero. 00:15:17.930 --> 00:15:25.230 align:middle line:90% 00:15:25.230 --> 00:15:32.370 align:middle line:84% Then, if I take the norm of A star v squared, 00:15:32.370 --> 00:15:42.060 align:middle line:84% this is equal to A star v inner product A star 00:15:42.060 --> 00:15:51.810 align:middle line:84% v. Now, by the definition of the inner product, 00:15:51.810 --> 00:16:01.380 align:middle line:84% this is equal to A times A star v, v, taking this A star 00:16:01.380 --> 00:16:02.580 align:middle line:90% and moving it over here. 00:16:02.580 --> 00:16:05.380 align:middle line:90% 00:16:05.380 --> 00:16:10.390 align:middle line:84% And by Cauchy-Schwarz-- well, first off, all of these numbers 00:16:10.390 --> 00:16:12.640 align:middle line:84% are real so it makes sense to actually write less than 00:16:12.640 --> 00:16:13.140 align:middle line:90% or equal to. 00:16:13.140 --> 00:16:16.540 align:middle line:84% But by Cauchy-Schwarz, this is equal to, or less than 00:16:16.540 --> 00:16:22.780 align:middle line:84% or equal to, A times A star v norm times norm 00:16:22.780 --> 00:16:24.550 align:middle line:90% of v, which equals 1. 00:16:24.550 --> 00:16:29.190 align:middle line:90% 00:16:29.190 --> 00:16:32.300 align:middle line:90% Now, this equals 1. 00:16:32.300 --> 00:16:35.600 align:middle line:84% And the norm of A applied to A star v, 00:16:35.600 --> 00:16:38.870 align:middle line:84% that's bounded by the norm of A times 00:16:38.870 --> 00:16:49.600 align:middle line:84% the norm of the input, which is A star v. And therefore-- 00:16:49.600 --> 00:16:52.990 align:middle line:84% so we started off with A star v squared less than 00:16:52.990 --> 00:16:56.800 align:middle line:84% or equal to the norm of A times the norm of A star v. This 00:16:56.800 --> 00:16:57.560 align:middle line:90% is non-zero. 00:16:57.560 --> 00:17:01.720 align:middle line:84% So I can divide by it and conclude 00:17:01.720 --> 00:17:11.950 align:middle line:84% that A star v norm of that is less than or equal to norm 00:17:11.950 --> 00:17:16.119 align:middle line:84% of A. Now remember, the norm of an operator 00:17:16.119 --> 00:17:19.390 align:middle line:84% or a linear operator is the sup of this 00:17:19.390 --> 00:17:22.839 align:middle line:90% for all v which have linked 1. 00:17:22.839 --> 00:17:28.540 align:middle line:84% And since this is bounded above by the norm of A, 00:17:28.540 --> 00:17:33.310 align:middle line:84% that implies that norm of A star is less than 00:17:33.310 --> 00:17:38.680 align:middle line:90% or equal to the norm of A. 00:17:38.680 --> 00:17:44.020 align:middle line:84% So we've shown that basically all of this theorem, 00:17:44.020 --> 00:17:47.890 align:middle line:84% with the exception of equality of the norms. 00:17:47.890 --> 00:17:50.830 align:middle line:84% So every bounded linear operator has 00:17:50.830 --> 00:17:55.660 align:middle line:84% an adjoint which is a bounded linear operator and satisfies-- 00:17:55.660 --> 00:17:57.610 align:middle line:84% and it's a unique linear operator 00:17:57.610 --> 00:18:02.770 align:middle line:84% satisfying that identity up there for A star. 00:18:02.770 --> 00:18:08.070 align:middle line:90% 00:18:08.070 --> 00:18:20.020 align:middle line:84% Now, note that for all u, v, and H, 00:18:20.020 --> 00:18:24.790 align:middle line:84% if I look at A star u inner product v, 00:18:24.790 --> 00:18:31.120 align:middle line:84% this is equal to the complex conjugate of v inner product 00:18:31.120 --> 00:18:40.780 align:middle line:84% A star u, which is equal to, by the definition of A star, 00:18:40.780 --> 00:18:44.080 align:middle line:90% moves over to A v u. 00:18:44.080 --> 00:18:46.570 align:middle line:84% And again, applying the complex conjugate 00:18:46.570 --> 00:18:49.310 align:middle line:84% switches the order of the entries. 00:18:49.310 --> 00:18:52.600 align:middle line:90% So this is equal to u A v. 00:18:52.600 --> 00:18:54.550 align:middle line:90% So what have we shown? 00:18:54.550 --> 00:18:59.350 align:middle line:84% We've shown that the adjoint of the adjoint-- 00:18:59.350 --> 00:19:07.540 align:middle line:84% that is equal to A. So we've shown that A star u 00:19:07.540 --> 00:19:14.800 align:middle line:84% v is equal to u times A v. And recall 00:19:14.800 --> 00:19:21.670 align:middle line:84% this is supposed to be equal to u A star A star, so the adjoint 00:19:21.670 --> 00:19:23.950 align:middle line:90% of the adjoint. 00:19:23.950 --> 00:19:25.990 align:middle line:84% And we've shown that this quantity is 00:19:25.990 --> 00:19:30.670 align:middle line:84% equal to this quantity for all u and v. 00:19:30.670 --> 00:19:37.090 align:middle line:84% And therefore, the adjoint of the adjoint 00:19:37.090 --> 00:19:38.650 align:middle line:90% is equal to the operator again. 00:19:38.650 --> 00:19:43.420 align:middle line:90% 00:19:43.420 --> 00:19:50.870 align:middle line:84% Thus, by what we've done previously-- 00:19:50.870 --> 00:19:52.490 align:middle line:84% the previous argument, if you like-- 00:19:52.490 --> 00:19:58.910 align:middle line:84% applied now where A is replaced by A star, the norm of A, 00:19:58.910 --> 00:20:04.100 align:middle line:84% which is equal to the norm of A-- 00:20:04.100 --> 00:20:07.520 align:middle line:84% the adjoint of the adjoint because this operator 00:20:07.520 --> 00:20:09.230 align:middle line:90% is equal to A-- 00:20:09.230 --> 00:20:11.900 align:middle line:84% by what we've done, we've shown that the norm of the adjoint 00:20:11.900 --> 00:20:14.000 align:middle line:84% is always less than or equal to the norm of the thing you're 00:20:14.000 --> 00:20:15.050 align:middle line:90% taking the adjoint of. 00:20:15.050 --> 00:20:26.010 align:middle line:84% So this is less than or equal to norm of A. And so we have this. 00:20:26.010 --> 00:20:34.970 align:middle line:84% And we have this, which implies norm of the adjoint 00:20:34.970 --> 00:20:40.650 align:middle line:84% is equal to the norm of the operator. 00:20:40.650 --> 00:20:46.810 align:middle line:90% 00:20:46.810 --> 00:20:53.750 align:middle line:84% So I mean, what is this creature in practice? 00:20:53.750 --> 00:21:00.170 align:middle line:84% So let's take maybe a second simplest example. 00:21:00.170 --> 00:21:06.980 align:middle line:84% The simplest example would be if you're on C n and R n-- well, 00:21:06.980 --> 00:21:09.840 align:middle line:90% so let's talk about that. 00:21:09.840 --> 00:21:11.240 align:middle line:90% Suppose now we have a matrix. 00:21:11.240 --> 00:21:20.480 align:middle line:90% So u is equal to u1, u n. 00:21:20.480 --> 00:21:22.670 align:middle line:90% And this is in C n. 00:21:22.670 --> 00:21:31.010 align:middle line:84% And A applied to u, if I want the i-th coordinate of that, 00:21:31.010 --> 00:21:37.800 align:middle line:84% is given by sum of j equals 1 to n 00:21:37.800 --> 00:21:42.720 align:middle line:84% of u j, where these are just complex 00:21:42.720 --> 00:21:47.325 align:middle line:84% numbers, some fixed complex numbers. 00:21:47.325 --> 00:21:50.010 align:middle line:90% 00:21:50.010 --> 00:21:53.186 align:middle line:90% So A is just a matrix. 00:21:53.186 --> 00:21:55.850 align:middle line:84% A linear transformation on C n is always 00:21:55.850 --> 00:21:58.590 align:middle line:90% represented by a matrix. 00:21:58.590 --> 00:22:02.300 align:middle line:84% And here, I'm writing u in terms of the standard basis 00:22:02.300 --> 00:22:04.250 align:middle line:90% vectors 1, 0, and so on. 00:22:04.250 --> 00:22:09.040 align:middle line:90% 00:22:09.040 --> 00:22:11.890 align:middle line:84% Then, to determine what the adjoint is, 00:22:11.890 --> 00:22:15.580 align:middle line:84% we figure out what's the operator that 00:22:15.580 --> 00:22:19.690 align:middle line:84% satisfies the identity the adjoint has to satisfy. 00:22:19.690 --> 00:22:27.590 align:middle line:84% So if we take A u and take its inner product, 00:22:27.590 --> 00:22:30.970 align:middle line:84% we want to write this-- so let me write this in a box-- 00:22:30.970 --> 00:22:36.250 align:middle line:84% we want to be able to write this as u with some operator B 00:22:36.250 --> 00:22:38.950 align:middle line:84% applied to v. And then this would be-- 00:22:38.950 --> 00:22:41.230 align:middle line:84% and therefore, that would be the adjoint. 00:22:41.230 --> 00:22:44.200 align:middle line:84% So if we write A u inner product v, 00:22:44.200 --> 00:22:58.230 align:middle line:84% this is equal to sum i equals 1 to n of v i complex conjugate. 00:22:58.230 --> 00:23:01.260 align:middle line:84% And this is, by definition, equal to-- 00:23:01.260 --> 00:23:03.300 align:middle line:84% I'm just going to write sum over i 00:23:03.300 --> 00:23:06.540 align:middle line:84% and j, i and j are going from 1 to n, 00:23:06.540 --> 00:23:18.890 align:middle line:84% of A i, j, u i v j complex conjugate. 00:23:18.890 --> 00:23:26.990 align:middle line:84% And if I now switch this over and sum first 00:23:26.990 --> 00:23:30.560 align:middle line:84% with respect to i and then with respect to j, 00:23:30.560 --> 00:23:35.570 align:middle line:84% I can write this as u j sum i equals 1 to n 00:23:35.570 --> 00:23:46.178 align:middle line:84% of A i j applied to v i complex conjugate, 00:23:46.178 --> 00:23:47.345 align:middle line:90% the complex conjugate there. 00:23:47.345 --> 00:23:53.180 align:middle line:90% 00:23:53.180 --> 00:23:56.035 align:middle line:90% And so this tells me that-- 00:23:56.035 --> 00:24:00.700 align:middle line:90% 00:24:00.700 --> 00:24:04.870 align:middle line:84% so this is equal to sum j equals 1 00:24:04.870 --> 00:24:19.890 align:middle line:84% to n u j and times what I'll call the adjoint applied to v j 00:24:19.890 --> 00:24:20.460 align:middle line:90% where-- 00:24:20.460 --> 00:24:22.290 align:middle line:90% what is the adjoint? 00:24:22.290 --> 00:24:28.080 align:middle line:84% Or how does the adjoint operate on an element v? 00:24:28.080 --> 00:24:35.440 align:middle line:90% 00:24:35.440 --> 00:24:39.145 align:middle line:90% This is equal to sum from-- 00:24:39.145 --> 00:24:41.680 align:middle line:90% 00:24:41.680 --> 00:24:44.290 align:middle line:84% let's say I want the i-th member now-- 00:24:44.290 --> 00:24:52.720 align:middle line:84% this is sum from j equals 1 to n A j i complex conjugate applied 00:24:52.720 --> 00:24:59.060 align:middle line:90% to v j. 00:24:59.060 --> 00:25:03.400 align:middle line:90% 00:25:03.400 --> 00:25:09.160 align:middle line:84% So for matrices-- again, this should be review-- 00:25:09.160 --> 00:25:12.980 align:middle line:90% 00:25:12.980 --> 00:25:21.570 align:middle line:84% if A is represented by the matrix, 00:25:21.570 --> 00:25:27.560 align:middle line:84% then the adjoint, which is also representable by a matrix, 00:25:27.560 --> 00:25:30.300 align:middle line:90% is representable by the matrix. 00:25:30.300 --> 00:25:34.520 align:middle line:84% So I should say the matrix for the adjoint 00:25:34.520 --> 00:25:41.060 align:middle line:84% A i j is equal to A j i complex conjugate. 00:25:41.060 --> 00:25:50.450 align:middle line:90% 00:25:50.450 --> 00:25:52.400 align:middle line:90% So we did this for C n. 00:25:52.400 --> 00:25:57.937 align:middle line:84% There's no reason we can't do it for, let's say, little l2 now 00:25:57.937 --> 00:25:59.520 align:middle line:84% and make it a little more interesting. 00:25:59.520 --> 00:26:12.250 align:middle line:84% So suppose now I have now infinitely many numbers 00:26:12.250 --> 00:26:26.200 align:middle line:84% A i j, so a double sequence in C n, 00:26:26.200 --> 00:26:35.110 align:middle line:84% such that sum i j, sum A i j squared, which 00:26:35.110 --> 00:26:45.340 align:middle line:84% is linked as n goes to infinity, of sum of i equals 1 to n, 00:26:45.340 --> 00:26:54.055 align:middle line:90% j equals 1 to n, is finite. 00:26:54.055 --> 00:27:04.830 align:middle line:90% 00:27:04.830 --> 00:27:18.350 align:middle line:84% Now, let's define A now from little l 2 to little l 2 via A 00:27:18.350 --> 00:27:22.290 align:middle line:84% applied to a sequence, which I put an underline underneath, 00:27:22.290 --> 00:27:25.550 align:middle line:90% or let's make it u. 00:27:25.550 --> 00:27:31.790 align:middle line:84% OK, we've been using A's for elements in little l p 00:27:31.790 --> 00:27:41.810 align:middle line:84% as sum j equals 1 to infinity A i j A j. 00:27:41.810 --> 00:27:49.670 align:middle line:84% And here, this is for A equals A little l 2. 00:27:49.670 --> 00:27:51.350 align:middle line:84% Now, by the Cauchy-Schwarz inequality, 00:27:51.350 --> 00:27:54.860 align:middle line:84% you can check that if this condition is satisfied, then 00:27:54.860 --> 00:27:57.860 align:middle line:84% this is a bounded linear operator. 00:27:57.860 --> 00:28:00.620 align:middle line:84% The absolute value of this thing will 00:28:00.620 --> 00:28:05.060 align:middle line:84% be less than or equal to the sum j equals 1 to infinity of A i j 00:28:05.060 --> 00:28:11.900 align:middle line:84% sum of j squared, square root times the l 2 norm of A. 00:28:11.900 --> 00:28:15.210 align:middle line:84% And therefore, I can sum that using-- 00:28:15.210 --> 00:28:21.150 align:middle line:84% that's l 2 summable using the fact that this is finite. 00:28:21.150 --> 00:28:24.600 align:middle line:84% Don't worry about the fact that I'm 00:28:24.600 --> 00:28:28.590 align:middle line:84% summing this double sum in a certain way 00:28:28.590 --> 00:28:32.100 align:middle line:84% by j going from 1 to n, i going from 1 to n. 00:28:32.100 --> 00:28:36.703 align:middle line:84% In fact, if it's absolutely summable in this, which 00:28:36.703 --> 00:28:38.620 align:middle line:84% it is-- it's the sum of non-negative numbers-- 00:28:38.620 --> 00:28:42.940 align:middle line:84% this sum doesn't depend on how I'm summing it in this way. 00:28:42.940 --> 00:28:45.790 align:middle line:90% 00:28:45.790 --> 00:28:50.650 align:middle line:84% So then this is a bounded linear operator. 00:28:50.650 --> 00:28:57.130 align:middle line:90% 00:28:57.130 --> 00:28:58.150 align:middle line:90% And what's its adjoint? 00:28:58.150 --> 00:29:01.300 align:middle line:84% Its adjoint is going to be of the same flavor 00:29:01.300 --> 00:29:07.090 align:middle line:84% as what we got for the finite dimensional case. 00:29:07.090 --> 00:29:15.130 align:middle line:84% And for all A B and little l 2, if I take the inner product 00:29:15.130 --> 00:29:21.280 align:middle line:84% of A applied to u, A, B, and little l 2, 00:29:21.280 --> 00:29:25.540 align:middle line:90% this is equal to sum over i-- 00:29:25.540 --> 00:29:28.090 align:middle line:84% I mean, the same proof applies, essentially-- 00:29:28.090 --> 00:29:35.060 align:middle line:84% sum i j A i B i complex conjugate, 00:29:35.060 --> 00:29:46.420 align:middle line:84% which is equal to sum j A j i A i j B i complex conjugate, 00:29:46.420 --> 00:29:59.830 align:middle line:84% complex conjugate, which is equal to inner product with A 00:29:59.830 --> 00:30:10.805 align:middle line:84% star B, where A star B is defined via sum-- 00:30:10.805 --> 00:30:15.650 align:middle line:90% 00:30:15.650 --> 00:30:18.140 align:middle line:84% so I should have said this is the i-th entry 00:30:18.140 --> 00:30:24.140 align:middle line:90% in this new sequence in l 2. 00:30:24.140 --> 00:30:30.110 align:middle line:84% And so here, the i-th entry in this new sequence in l 2 00:30:30.110 --> 00:30:35.540 align:middle line:84% is defined to be j equals 1 to infinity A j i 00:30:35.540 --> 00:30:39.230 align:middle line:90% complex conjugate B j. 00:30:39.230 --> 00:30:44.040 align:middle line:90% 00:30:44.040 --> 00:30:47.730 align:middle line:84% So just like in the finite dimensional case, 00:30:47.730 --> 00:30:52.620 align:middle line:84% for this case, if I switch the variables of this A-- 00:30:52.620 --> 00:30:59.970 align:middle line:84% if you'd like a double infinite matrix, if you 00:30:59.970 --> 00:31:01.810 align:middle line:90% want to think of it that way-- 00:31:01.810 --> 00:31:05.280 align:middle line:84% it's the same as in the finite dimensional case. 00:31:05.280 --> 00:31:07.810 align:middle line:90% Let's do one last example. 00:31:07.810 --> 00:31:11.010 align:middle line:84% So I'm not going to go through again the computations, which 00:31:11.010 --> 00:31:14.130 align:middle line:90% are very similar. 00:31:14.130 --> 00:31:16.650 align:middle line:84% Here, we're having kind of finite sums. 00:31:16.650 --> 00:31:18.563 align:middle line:84% If you can do something with sums, 00:31:18.563 --> 00:31:19.980 align:middle line:84% you should consider whether or not 00:31:19.980 --> 00:31:22.180 align:middle line:90% you can do it for integrals. 00:31:22.180 --> 00:31:28.080 align:middle line:84% So let's suppose K is a continuous function 00:31:28.080 --> 00:31:29.940 align:middle line:90% on 0, 1 cross 0, 1. 00:31:29.940 --> 00:31:33.630 align:middle line:90% 00:31:33.630 --> 00:31:48.000 align:middle line:84% And we define a map A which we can show goes from L 2 00:31:48.000 --> 00:31:53.970 align:middle line:84% to L 2 via A f of x is equal to the integral from 0 00:31:53.970 --> 00:32:00.660 align:middle line:90% to 1 of K of x, y, f of y dy. 00:32:00.660 --> 00:32:04.722 align:middle line:90% 00:32:04.722 --> 00:32:12.480 align:middle line:84% In fact, this is for each f in L 2, A f-- 00:32:12.480 --> 00:32:14.910 align:middle line:84% this thing is, in fact, a continuous function. 00:32:14.910 --> 00:32:16.570 align:middle line:90% It's more than just in L2. 00:32:16.570 --> 00:32:18.070 align:middle line:84% It's in fact, a continuous function. 00:32:18.070 --> 00:32:22.850 align:middle line:84% But continuous functions on 0, 1 are elements of L2 on 0, 1. 00:32:22.850 --> 00:32:25.570 align:middle line:90% 00:32:25.570 --> 00:32:28.270 align:middle line:84% Then you can check, just as we've 00:32:28.270 --> 00:32:36.100 align:middle line:84% done in these two other examples, that the adjoint of f 00:32:36.100 --> 00:32:38.320 align:middle line:90% is equal to-- 00:32:38.320 --> 00:32:40.600 align:middle line:90% or let's make this different-- 00:32:40.600 --> 00:32:44.440 align:middle line:84% say adjoint applied to g is equal to the integral 00:32:44.440 --> 00:32:54.370 align:middle line:84% from 0 to 1 K y, x complex conjugate g of y dy. 00:32:54.370 --> 00:32:56.770 align:middle line:84% So again, it's like flipping the indices 00:32:56.770 --> 00:32:58.330 align:middle line:84% and taking the complex conjugate. 00:32:58.330 --> 00:33:08.460 align:middle line:90% 00:33:08.460 --> 00:33:11.620 align:middle line:84% I said at the end of last class that you can tell something 00:33:11.620 --> 00:33:20.330 align:middle line:84% about the solvability of equations 00:33:20.330 --> 00:33:23.930 align:middle line:90% based on how the adjoint-- 00:33:23.930 --> 00:33:25.355 align:middle line:90% on properties of the adjoint. 00:33:25.355 --> 00:33:34.290 align:middle line:90% 00:33:34.290 --> 00:33:41.470 align:middle line:84% We'll see another way this is manifested later. 00:33:41.470 --> 00:33:46.030 align:middle line:90% That's kind of the simplest way. 00:33:46.030 --> 00:33:56.625 align:middle line:84% And so suppose H is a Hilbert space. 00:33:56.625 --> 00:34:05.470 align:middle line:90% 00:34:05.470 --> 00:34:10.239 align:middle line:84% And A from H to H is a bounded linear operator. 00:34:10.239 --> 00:34:18.489 align:middle line:90% 00:34:18.489 --> 00:34:28.750 align:middle line:84% Then-- I used Ran in the assignments, 00:34:28.750 --> 00:34:31.150 align:middle line:90% so I'll use it this way here-- 00:34:31.150 --> 00:34:34.250 align:middle line:84% the range of A, orthogonal complement-- 00:34:34.250 --> 00:34:41.550 align:middle line:84% so the range you can show is a subspace of H. In fact, I 00:34:41.550 --> 00:34:44.440 align:middle line:84% think that was in one of the earlier assignments. 00:34:44.440 --> 00:34:47.380 align:middle line:84% If you take the orthogonal complement of that, 00:34:47.380 --> 00:35:02.450 align:middle line:84% that is equal to the null space of the adjoint. 00:35:02.450 --> 00:35:06.780 align:middle line:90% 00:35:06.780 --> 00:35:16.150 align:middle line:84% So here, let me recall that range of an operator B-- 00:35:16.150 --> 00:35:17.350 align:middle line:90% this is exactly what it is. 00:35:17.350 --> 00:35:22.960 align:middle line:84% This is a set of all vectors B u where u is in H 00:35:22.960 --> 00:35:27.810 align:middle line:90% and null space of B-- 00:35:27.810 --> 00:35:32.730 align:middle line:84% I might have called it the kernel, 00:35:32.730 --> 00:35:37.290 align:middle line:84% might have labeled it as the kernel of B, 00:35:37.290 --> 00:35:39.400 align:middle line:90% but both mean the same set. 00:35:39.400 --> 00:35:45.540 align:middle line:84% This is a set of all u in H such that B u equals 0. 00:35:45.540 --> 00:35:51.600 align:middle line:90% 00:35:51.600 --> 00:35:56.400 align:middle line:84% So in particular, if we know that we have an operator A such 00:35:56.400 --> 00:36:03.810 align:middle line:84% that the range is a closed subspace, then being 00:36:03.810 --> 00:36:06.420 align:middle line:84% able to solve the equation A u equals 00:36:06.420 --> 00:36:12.390 align:middle line:84% v is equivalent to showing that the null space of the adjoint 00:36:12.390 --> 00:36:13.965 align:middle line:90% is simply 0. 00:36:13.965 --> 00:36:17.940 align:middle line:90% 00:36:17.940 --> 00:36:19.380 align:middle line:90% So let me make that as a remark. 00:36:19.380 --> 00:36:26.460 align:middle line:90% 00:36:26.460 --> 00:36:35.790 align:middle line:84% Suppose that the range of A is closed. 00:36:35.790 --> 00:36:47.320 align:middle line:84% Then A from H to H is surjective if and only 00:36:47.320 --> 00:36:58.330 align:middle line:84% if the adjoint is injective, meaning 00:36:58.330 --> 00:37:02.680 align:middle line:84% the only thing that gets set to 0 is the zero vector. 00:37:02.680 --> 00:37:07.360 align:middle line:84% Because if the null space is equal to the 0-- so first off, 00:37:07.360 --> 00:37:11.680 align:middle line:84% it's easy to see if the range of A is equal to H, 00:37:11.680 --> 00:37:14.980 align:middle line:84% then the orthogonal complement of H is the zero vector. 00:37:14.980 --> 00:37:19.870 align:middle line:84% So null space of the adjoint is just a set 00:37:19.870 --> 00:37:21.550 align:middle line:90% containing the zero vector. 00:37:21.550 --> 00:37:27.360 align:middle line:84% Now, if the null phase of the adjoint 00:37:27.360 --> 00:37:31.170 align:middle line:84% is just a zero vector, then taking 00:37:31.170 --> 00:37:33.540 align:middle line:84% the orthogonal complements of both sides, 00:37:33.540 --> 00:37:35.970 align:middle line:84% and using the fact that the range is closed 00:37:35.970 --> 00:37:37.650 align:middle line:84% so that the orthogonal complement 00:37:37.650 --> 00:37:40.470 align:middle line:84% of the orthogonal complement gives me back the set, 00:37:40.470 --> 00:37:42.480 align:middle line:84% I conclude that the range of A equals 00:37:42.480 --> 00:37:45.570 align:middle line:84% the orthogonal complement of the zero vector, which 00:37:45.570 --> 00:37:47.505 align:middle line:90% is H, the entire space. 00:37:47.505 --> 00:37:50.050 align:middle line:90% 00:37:50.050 --> 00:37:54.670 align:middle line:84% So we know that we have an operator which 00:37:54.670 --> 00:37:55.780 align:middle line:90% has closed range. 00:37:55.780 --> 00:37:58.870 align:middle line:84% Then being surjective is equivalent to the adjoint 00:37:58.870 --> 00:37:59.850 align:middle line:90% being injective. 00:37:59.850 --> 00:38:13.650 align:middle line:90% 00:38:13.650 --> 00:38:15.885 align:middle line:84% So the proof of this, though, is pretty easy. 00:38:15.885 --> 00:38:20.680 align:middle line:90% 00:38:20.680 --> 00:38:30.210 align:middle line:84% So B is in the null space of A star 00:38:30.210 --> 00:38:42.570 align:middle line:84% if and only if u A star B equals 0 for all u in H. 00:38:42.570 --> 00:38:47.100 align:middle line:84% A star v equals 0 if and only if this holds for all u in H. 00:38:47.100 --> 00:38:53.340 align:middle line:84% And this is equivalent to, by the property of the adjoint, 00:38:53.340 --> 00:38:57.240 align:middle line:84% u inner product of A star v is equal to A times u 00:38:57.240 --> 00:39:05.910 align:middle line:84% inner product v, so A u, v equals 0 for all u in H. 00:39:05.910 --> 00:39:08.810 align:middle line:90% So v is a fixed thing. 00:39:08.810 --> 00:39:12.680 align:middle line:84% So this says that v is orthogonal to all 00:39:12.680 --> 00:39:15.230 align:middle line:84% of the elements of H of the form A times u. 00:39:15.230 --> 00:39:18.950 align:middle line:84% But that's just the range of A. So 00:39:18.950 --> 00:39:27.740 align:middle line:84% this is equivalent to B is in the orthogonal complement 00:39:27.740 --> 00:40:01.770 align:middle line:90% of the range of A. 00:40:01.770 --> 00:40:05.160 align:middle line:84% So we're soon going to get into the realm of more 00:40:05.160 --> 00:40:08.220 align:middle line:84% refined things we can say about solving 00:40:08.220 --> 00:40:11.850 align:middle line:84% certain equations involving operators, 00:40:11.850 --> 00:40:16.840 align:middle line:84% meaning when can you solve A u equals v and so on. 00:40:16.840 --> 00:40:28.650 align:middle line:84% But now, the best theorem about that you should know or maybe 00:40:28.650 --> 00:40:31.560 align:middle line:84% heard at some point from linear algebra is the rank-nullity 00:40:31.560 --> 00:40:38.040 align:middle line:84% theorem, that says the dimension of the range plus the dimension 00:40:38.040 --> 00:40:40.800 align:middle line:84% of the null space equals the dimension of the-- 00:40:40.800 --> 00:40:44.370 align:middle line:84% let's say you're going from spaces with the same dimension, 00:40:44.370 --> 00:40:49.860 align:middle line:84% then the dimension of the range plus the dimension 00:40:49.860 --> 00:40:52.318 align:middle line:90% of the kernel-- 00:40:52.318 --> 00:40:53.860 align:middle line:84% I may be actually getting this wrong; 00:40:53.860 --> 00:40:56.793 align:middle line:84% it's the end of a day; it's the end of a long day-- 00:40:56.793 --> 00:40:58.460 align:middle line:84% equals the dimension of the whole space. 00:40:58.460 --> 00:41:09.980 align:middle line:84% So what this says is that in essence, 00:41:09.980 --> 00:41:13.410 align:middle line:84% in order to be able to solve a given equation, 00:41:13.410 --> 00:41:18.230 align:middle line:84% your input has to satisfy finitely many linear relations. 00:41:18.230 --> 00:41:20.660 align:middle line:84% And the solution of that equation 00:41:20.660 --> 00:41:25.010 align:middle line:90% is unique up to finally many-- 00:41:25.010 --> 00:41:26.615 align:middle line:84% or a finite dimensional subspace. 00:41:26.615 --> 00:41:31.770 align:middle line:90% 00:41:31.770 --> 00:41:35.690 align:middle line:84% Now this is the best thing you can say. 00:41:35.690 --> 00:41:40.730 align:middle line:84% Input has to satisfy finitely many linear relations 00:41:40.730 --> 00:41:43.020 align:middle line:90% and unique. 00:41:43.020 --> 00:41:46.700 align:middle line:84% So that existence is predicated upon your data 00:41:46.700 --> 00:41:50.390 align:middle line:84% satisfying finitely many linear conditions, 00:41:50.390 --> 00:41:58.700 align:middle line:84% and you have uniqueness up to a finite dimensional subspace, 00:41:58.700 --> 00:42:00.360 align:middle line:90% meaning the null space. 00:42:00.360 --> 00:42:03.260 align:middle line:84% Now, this is from finite dimensional linear algebra. 00:42:03.260 --> 00:42:06.590 align:middle line:84% It would be great if we could do the same thing for infinite 00:42:06.590 --> 00:42:08.990 align:middle line:84% dimensions, not just because it would be fun, 00:42:08.990 --> 00:42:12.950 align:middle line:84% but in the end because we need to do these things, 00:42:12.950 --> 00:42:19.167 align:middle line:84% or I should say need to, but not in the sense that an airplane 00:42:19.167 --> 00:42:20.750 align:middle line:84% wing is going to fall off if we don't. 00:42:20.750 --> 00:42:29.100 align:middle line:84% But in order to learn more about certain problems that arise, 00:42:29.100 --> 00:42:31.440 align:middle line:84% it would be great to be able to do these things. 00:42:31.440 --> 00:42:35.220 align:middle line:84% And we'll be able to do them for certain operators, which 00:42:35.220 --> 00:42:38.170 align:middle line:90% are in some way-- 00:42:38.170 --> 00:42:42.010 align:middle line:84% these aren't the only operators, but for certain operators 00:42:42.010 --> 00:42:45.595 align:middle line:84% that are close to being matrices in a certain sense. 00:42:45.595 --> 00:42:46.970 align:middle line:84% So we'll get to that in a minute. 00:42:46.970 --> 00:42:58.560 align:middle line:84% Now, these operators that we will eventually 00:42:58.560 --> 00:43:02.490 align:middle line:84% study the solvability properties of 00:43:02.490 --> 00:43:10.980 align:middle line:84% have a very special nature or property on how 00:43:10.980 --> 00:43:14.420 align:middle line:90% it acts on bounded sequences. 00:43:14.420 --> 00:43:17.390 align:middle line:84% Now, something that you take for granted in R n 00:43:17.390 --> 00:43:19.490 align:middle line:84% is that if I have a linear operator 00:43:19.490 --> 00:43:23.120 align:middle line:84% and I have a bounded sequence of vectors, then 00:43:23.120 --> 00:43:25.670 align:middle line:84% A maps that to a bounded sequence of vectors 00:43:25.670 --> 00:43:28.350 align:middle line:84% because a matrix is a bounded linear operator. 00:43:28.350 --> 00:43:30.260 align:middle line:84% It takes bounded sets of bounded sets. 00:43:30.260 --> 00:43:37.430 align:middle line:84% And therefore, by the Heine-Borel theorem, 00:43:37.430 --> 00:43:42.130 align:middle line:84% there will be a subsequence of the images of these vectors, 00:43:42.130 --> 00:43:43.150 align:middle line:90% which converges. 00:43:43.150 --> 00:43:46.100 align:middle line:90% 00:43:46.100 --> 00:43:51.470 align:middle line:84% So there's some compactness hidden in what 00:43:51.470 --> 00:43:53.840 align:middle line:90% you're doing on R n and C n. 00:43:53.840 --> 00:43:56.900 align:middle line:90% And why am I rambling right now? 00:43:56.900 --> 00:43:59.180 align:middle line:84% I'm getting to the final point that we 00:43:59.180 --> 00:44:05.390 align:middle line:84% need to study a little bit about compactness and Hilbert spaces 00:44:05.390 --> 00:44:08.720 align:middle line:84% to eventually get to the place where we can say more 00:44:08.720 --> 00:44:10.880 align:middle line:84% about being able to solve equations involving 00:44:10.880 --> 00:44:13.400 align:middle line:84% bounded linear operators, now in infinite dimensions. 00:44:13.400 --> 00:44:16.410 align:middle line:90% 00:44:16.410 --> 00:44:19.590 align:middle line:84% So with that rambling build-up, hopefully, I 00:44:19.590 --> 00:44:21.630 align:middle line:84% convinced you that we should study a little bit 00:44:21.630 --> 00:44:30.320 align:middle line:84% about what it means for sets to be compact in a Hilbert space. 00:44:30.320 --> 00:44:41.200 align:middle line:84% So let's look at compactness in Hilbert space. 00:44:41.200 --> 00:44:46.080 align:middle line:84% So let me recall for you what it means 00:44:46.080 --> 00:44:51.306 align:middle line:84% for a set to be compact in a metric space. 00:44:51.306 --> 00:45:05.770 align:middle line:84% If X is a metric space, We say a subset K in X 00:45:05.770 --> 00:45:22.940 align:middle line:84% is compact if every sequence of elements in K 00:45:22.940 --> 00:45:40.120 align:middle line:84% has a subsequence converging to an element in K. 00:45:40.120 --> 00:45:48.480 align:middle line:84% So a set is compact if every sequence of elements in the set 00:45:48.480 --> 00:45:50.070 align:middle line:84% has a subsequence which converges 00:45:50.070 --> 00:45:51.390 align:middle line:90% to an element in the set. 00:45:51.390 --> 00:45:55.020 align:middle line:84% So the simplest on Earth types of compact sets 00:45:55.020 --> 00:45:55.920 align:middle line:90% are finite sets. 00:45:55.920 --> 00:45:58.390 align:middle line:90% 00:45:58.390 --> 00:46:00.390 align:middle line:84% This just follows from the pigeonhole principle, 00:46:00.390 --> 00:46:01.140 align:middle line:90% basically. 00:46:01.140 --> 00:46:03.930 align:middle line:90% 00:46:03.930 --> 00:46:07.320 align:middle line:84% So the simplest example are finite sets 00:46:07.320 --> 00:46:11.415 align:middle line:84% of finite subsets of a metric space, of any metric space. 00:46:11.415 --> 00:46:25.420 align:middle line:90% 00:46:25.420 --> 00:46:31.880 align:middle line:84% Now, we have this very cool theorem from Intro Analysis, 00:46:31.880 --> 00:46:39.090 align:middle line:84% which goes by the name of Heine-Borel, 00:46:39.090 --> 00:46:44.630 align:middle line:90% that says a subset K in R-- 00:46:44.630 --> 00:46:47.720 align:middle line:84% you can replace R by R n or C n, but this 00:46:47.720 --> 00:46:49.760 align:middle line:84% follows from the one-dimensional case, 00:46:49.760 --> 00:47:02.270 align:middle line:84% essentially-- a subset K of R is compact if and only if K is 00:47:02.270 --> 00:47:07.265 align:middle line:90% closed subset of R and bounded. 00:47:07.265 --> 00:47:09.990 align:middle line:90% 00:47:09.990 --> 00:47:12.170 align:middle line:84% So for example, the closed and bounded intervals 00:47:12.170 --> 00:47:13.700 align:middle line:84% a, b-- those are all compact sets. 00:47:13.700 --> 00:47:19.620 align:middle line:90% 00:47:19.620 --> 00:47:26.720 align:middle line:84% a, b, the set consisting of 1 over n, 00:47:26.720 --> 00:47:31.340 align:middle line:84% n a natural number, union the element 0. 00:47:31.340 --> 00:47:33.020 align:middle line:90% That's also a compact set. 00:47:33.020 --> 00:47:34.235 align:middle line:90% It's closed and it's bounded. 00:47:34.235 --> 00:47:38.980 align:middle line:90% 00:47:38.980 --> 00:47:43.320 align:middle line:84% Now, this is a theorem about elements 00:47:43.320 --> 00:47:51.570 align:middle line:84% of compact subsets of R. One can build off of the proof of this, 00:47:51.570 --> 00:47:56.160 align:middle line:84% really, to get the result for R n and C n. 00:47:56.160 --> 00:47:59.370 align:middle line:84% The metric properties of C n are the same as R 2n, 00:47:59.370 --> 00:48:03.600 align:middle line:84% so let's say just R n-- that a subset of R n is compact if 00:48:03.600 --> 00:48:06.640 align:middle line:84% and only if it's closed and bounded. 00:48:06.640 --> 00:48:13.350 align:middle line:84% Now, does that hold for arbitrary metric spaces? 00:48:13.350 --> 00:48:14.640 align:middle line:90% No. 00:48:14.640 --> 00:48:16.750 align:middle line:90% Does it hold for Banach spaces? 00:48:16.750 --> 00:48:17.250 align:middle line:90% No. 00:48:17.250 --> 00:48:19.583 align:middle line:84% You did an example on one of the assignments that proved 00:48:19.583 --> 00:48:22.560 align:middle line:90% that little l p is not compact. 00:48:22.560 --> 00:48:24.840 align:middle line:84% What about if we specialize the Hilbert spaces? 00:48:24.840 --> 00:48:25.860 align:middle line:90% The answer is still no. 00:48:25.860 --> 00:48:30.360 align:middle line:84% So let's make this a non example, since we 00:48:30.360 --> 00:48:32.080 align:middle line:90% did examples a minute ago. 00:48:32.080 --> 00:48:37.230 align:middle line:84% So all of these, as subsets of R, these are all compact. 00:48:37.230 --> 00:49:08.010 align:middle line:84% So let's suppose H is a infinite-dimensional Hilbert 00:49:08.010 --> 00:49:10.290 align:middle line:90% space. 00:49:10.290 --> 00:49:20.520 align:middle line:84% Then the closed ball B equals the set of u and H-- 00:49:20.520 --> 00:49:21.285 align:middle line:90% let's make it F-- 00:49:21.285 --> 00:49:25.110 align:middle line:90% 00:49:25.110 --> 00:49:28.650 align:middle line:84% such that length of u is less than or equal to 1. 00:49:28.650 --> 00:49:31.450 align:middle line:84% This is a closed and bounded set. 00:49:31.450 --> 00:49:34.140 align:middle line:90% You can check that. 00:49:34.140 --> 00:49:40.530 align:middle line:90% This is not compact. 00:49:40.530 --> 00:49:43.730 align:middle line:90% Let's get back to compactness. 00:49:43.730 --> 00:49:46.580 align:middle line:84% So let's suppose H is a infinite-dimensional-- 00:49:46.580 --> 00:49:53.680 align:middle line:84% let's even make it separable, as all good Hilbert spaces are. 00:49:53.680 --> 00:49:59.380 align:middle line:84% Then the closed unit ball is not compact. 00:49:59.380 --> 00:50:00.430 align:middle line:90% So why is this? 00:50:00.430 --> 00:50:04.430 align:middle line:90% 00:50:04.430 --> 00:50:11.410 align:middle line:84% Let e n equals 1 to infinity b n-- 00:50:11.410 --> 00:50:18.110 align:middle line:84% or in fact, it doesn't have to be separable. 00:50:18.110 --> 00:50:21.950 align:middle line:84% In fact, if you just take any infinite-dimensional Hilbert 00:50:21.950 --> 00:50:23.480 align:middle line:90% space, you can find by the-- 00:50:23.480 --> 00:50:26.730 align:middle line:90% 00:50:26.730 --> 00:50:29.910 align:middle line:84% we don't need to make it separable. 00:50:29.910 --> 00:50:34.020 align:middle line:84% You can always find an orthonormal, countably infinite 00:50:34.020 --> 00:50:37.593 align:middle line:90% orthonormal, subset of H. 00:50:37.593 --> 00:50:39.510 align:middle line:84% Now, in the separable case, we could choose it 00:50:39.510 --> 00:50:40.948 align:middle line:84% so that it's a orthonormal basis. 00:50:40.948 --> 00:50:42.240 align:middle line:90% But that really doesn't matter. 00:50:42.240 --> 00:50:44.970 align:middle line:84% Just the fact that H is an infinite-dimensional Hilbert 00:50:44.970 --> 00:51:00.410 align:middle line:84% space, we can find a countable subset. 00:51:00.410 --> 00:51:02.590 align:middle line:84% So we can find a countably infinite subset 00:51:02.590 --> 00:51:04.690 align:middle line:90% of orthonormal vectors. 00:51:04.690 --> 00:51:05.350 align:middle line:90% Why is this? 00:51:05.350 --> 00:51:08.980 align:middle line:84% Because we can find a countable set of linearly independent 00:51:08.980 --> 00:51:11.300 align:middle line:90% vectors-- 00:51:11.300 --> 00:51:14.350 align:middle line:84% an infinite-dimensional Hilbert space 00:51:14.350 --> 00:51:18.100 align:middle line:84% or infinite-dimensional vector space. 00:51:18.100 --> 00:51:21.040 align:middle line:84% And then we can make them orthonormal by applying 00:51:21.040 --> 00:51:25.300 align:middle line:84% the Gram-Schmidt process to that collection, 00:51:25.300 --> 00:51:29.830 align:middle line:84% and come up with countably infinite orthonormal subset. 00:51:29.830 --> 00:51:34.270 align:middle line:84% Then, for all n not equal to k, we 00:51:34.270 --> 00:51:40.120 align:middle line:84% get that e n norm minus e k squared-- 00:51:40.120 --> 00:51:45.130 align:middle line:84% this is equal to, again, using the parallelogram law-- 00:51:45.130 --> 00:51:49.457 align:middle line:90% 00:51:49.457 --> 00:51:51.540 align:middle line:84% well, I don't need to apply the parallelogram law. 00:51:51.540 --> 00:51:53.370 align:middle line:90% I can just apply what this is. 00:51:53.370 --> 00:52:00.230 align:middle line:84% This is equal to norm n squared plus norm e k squared plus 2 00:52:00.230 --> 00:52:04.140 align:middle line:90% real part e n e k. 00:52:04.140 --> 00:52:07.110 align:middle line:84% And since they're orthogonal, this is 0. 00:52:07.110 --> 00:52:10.440 align:middle line:84% And since they're normalized to 1, this is equal to 2. 00:52:10.440 --> 00:52:17.220 align:middle line:84% So for any two distinct elements in the sequence, 00:52:17.220 --> 00:52:22.620 align:middle line:84% the distance between the two of them is equal to 2. 00:52:22.620 --> 00:52:26.880 align:middle line:84% So there's no way the sequence of these orthonormal vectors 00:52:26.880 --> 00:52:29.550 align:middle line:84% can have a convergent subsequence. 00:52:29.550 --> 00:52:31.290 align:middle line:84% Because a convergent subsequence would 00:52:31.290 --> 00:52:35.610 align:middle line:84% mean that if I go along the subsequence, 00:52:35.610 --> 00:52:37.320 align:middle line:84% or this subsequence has to be Cauchy 00:52:37.320 --> 00:52:41.640 align:middle line:84% so that the difference between vectors has to get small. 00:52:41.640 --> 00:52:42.692 align:middle line:90% But it's never small. 00:52:42.692 --> 00:52:44.400 align:middle line:84% It's always at least bigger than or equal 00:52:44.400 --> 00:52:49.095 align:middle line:84% to 2, at least the length squared or distance squared. 00:52:49.095 --> 00:52:52.130 align:middle line:90% 00:52:52.130 --> 00:52:59.120 align:middle line:84% So this is a closed and bounded subset which is not compact. 00:52:59.120 --> 00:53:04.470 align:middle line:90% 00:53:04.470 --> 00:53:07.680 align:middle line:84% There exist sequences which have no-- 00:53:07.680 --> 00:53:11.580 align:middle line:84% or this sequence has no convergent subsequence. 00:53:11.580 --> 00:53:15.570 align:middle line:84% So the question is, what's the additional condition 00:53:15.570 --> 00:53:26.280 align:middle line:84% I need to conclude that a subset of a metric space is compact? 00:53:26.280 --> 00:53:34.720 align:middle line:84% From Real Analysis 18.100, every compact set has to be closed 00:53:34.720 --> 00:53:36.370 align:middle line:90% and bounded. 00:53:36.370 --> 00:53:40.220 align:middle line:84% One implication is that if something is compact, 00:53:40.220 --> 00:53:42.280 align:middle line:90% it's closed and it's bounded. 00:53:42.280 --> 00:53:46.090 align:middle line:84% What additional condition on the set-- so being 00:53:46.090 --> 00:53:48.370 align:middle line:84% closed and bounded is a necessity of the set. 00:53:48.370 --> 00:53:52.240 align:middle line:84% What additional condition guarantees that it's compact? 00:53:52.240 --> 00:53:53.860 align:middle line:84% Maybe not if and only if, like we 00:53:53.860 --> 00:53:58.660 align:middle line:84% have for the Heine-Borel, but at least something simple 00:53:58.660 --> 00:54:06.580 align:middle line:84% to verify so that the subset of the Hilbert space is compact. 00:54:06.580 --> 00:54:12.350 align:middle line:84% Now, if you didn't see this in 18.100, that's fine. 00:54:12.350 --> 00:54:19.750 align:middle line:84% But this question can or could have been already asked 00:54:19.750 --> 00:54:25.870 align:middle line:84% in a different context in 18.100. 00:54:25.870 --> 00:54:30.370 align:middle line:84% And at least when I taught it, 100B, it 00:54:30.370 --> 00:54:36.160 align:middle line:84% was that you can ask the same question about subsets 00:54:36.160 --> 00:54:40.060 align:middle line:90% of continuous functions. 00:54:40.060 --> 00:54:43.900 align:middle line:84% So definition of a compact set is with respect 00:54:43.900 --> 00:54:45.400 align:middle line:90% to a metric space. 00:54:45.400 --> 00:54:47.150 align:middle line:84% Doesn't have to be an inner product space. 00:54:47.150 --> 00:54:51.400 align:middle line:84% So the space of continuous functions on, say, 0, 1-- 00:54:51.400 --> 00:54:54.760 align:middle line:84% that's a perfectly good metric space with the infinity norm. 00:54:54.760 --> 00:55:00.130 align:middle line:84% You could ask, what are the conditions on a set to ensure 00:55:00.130 --> 00:55:04.000 align:middle line:90% that the set is compact? 00:55:04.000 --> 00:55:07.210 align:middle line:84% And the three conditions are closed, bounded-- 00:55:07.210 --> 00:55:10.960 align:middle line:84% because these are necessary if the set is to be compact-- 00:55:10.960 --> 00:55:14.150 align:middle line:84% and what's called "equicontinuous." 00:55:14.150 --> 00:55:21.500 align:middle line:84% So for subsets of the space of continuous functions, 00:55:21.500 --> 00:55:23.990 align:middle line:84% if the subset is closed, bounded, 00:55:23.990 --> 00:55:25.970 align:middle line:84% and what's called equicontinuous, 00:55:25.970 --> 00:55:28.310 align:middle line:90% then the subset is compact. 00:55:28.310 --> 00:55:30.380 align:middle line:84% This is the famous Arzela-Ascoli theorem. 00:55:30.380 --> 00:55:33.290 align:middle line:90% 00:55:33.290 --> 00:55:39.560 align:middle line:84% And that extra condition, equicontinuity, 00:55:39.560 --> 00:55:48.250 align:middle line:84% is in some sense what allows you to reduce 00:55:48.250 --> 00:55:51.820 align:middle line:84% your problem of showing your space is 00:55:51.820 --> 00:55:55.810 align:middle line:84% compact to the finite-dimensional case. 00:55:55.810 --> 00:55:59.770 align:middle line:84% It takes care of the infinite dimensions 00:55:59.770 --> 00:56:06.890 align:middle line:84% or it takes care of the infinite part, in a sense. 00:56:06.890 --> 00:56:09.310 align:middle line:84% That's not very specific, but it's 00:56:09.310 --> 00:56:14.980 align:middle line:84% needed to control all but finitely many things, 00:56:14.980 --> 00:56:16.160 align:middle line:90% in some sense. 00:56:16.160 --> 00:56:18.760 align:middle line:84% Equicontinuity in Arzela-Ascoli is 00:56:18.760 --> 00:56:22.150 align:middle line:84% what helps you control all but finitely many things. 00:56:22.150 --> 00:56:33.590 align:middle line:84% Now, there is a similar theorem that you can prove, now 00:56:33.590 --> 00:56:34.445 align:middle line:90% in Hilbert spaces. 00:56:34.445 --> 00:56:43.092 align:middle line:90% 00:56:43.092 --> 00:56:44.300 align:middle line:90% Let's just state the theorem. 00:56:44.300 --> 00:56:46.700 align:middle line:84% Part of the theorem will be a definition, unfortunately, 00:56:46.700 --> 00:56:54.445 align:middle line:90% but let H be a Hilbert space. 00:56:54.445 --> 00:57:03.190 align:middle line:90% 00:57:03.190 --> 00:57:07.618 align:middle line:84% Well, we're not going to prove the theorem 00:57:07.618 --> 00:57:08.660 align:middle line:90% I want to prove just yet. 00:57:08.660 --> 00:57:09.827 align:middle line:90% I just skipped ahead by one. 00:57:09.827 --> 00:57:16.600 align:middle line:90% 00:57:16.600 --> 00:57:19.010 align:middle line:84% So we'll motivate a certain condition. 00:57:19.010 --> 00:57:24.850 align:middle line:84% So I got ahead of myself just a minute by just one theorem. 00:57:24.850 --> 00:57:27.850 align:middle line:84% But let me introduce the definition. 00:57:27.850 --> 00:57:30.550 align:middle line:90% 00:57:30.550 --> 00:57:43.995 align:middle line:90% So let H be a Hilbert space. 00:57:43.995 --> 00:57:50.040 align:middle line:90% 00:57:50.040 --> 00:58:10.180 align:middle line:84% A subset K of H sets has equismall tails with respect 00:58:10.180 --> 00:58:25.150 align:middle line:84% to a countable orthonormal subset. 00:58:25.150 --> 00:58:34.990 align:middle line:84% So the new bit of terminology is "equismall tails--" e n if-- 00:58:34.990 --> 00:58:38.970 align:middle line:90% 00:58:38.970 --> 00:58:39.980 align:middle line:90% let's make this K-- 00:58:39.980 --> 00:58:52.280 align:middle line:90% 00:58:52.280 --> 00:59:02.236 align:middle line:84% in some sense-- so by Bessel's inequality, 00:59:02.236 --> 00:59:06.380 align:middle line:84% the sum of the, if you like, Fourier coefficients or the sum 00:59:06.380 --> 00:59:07.940 align:middle line:84% of the squares of the inner product 00:59:07.940 --> 00:59:09.800 align:middle line:84% e K with a fixed vector-- this is always 00:59:09.800 --> 00:59:12.770 align:middle line:84% bounded by the norm of the vector squared. 00:59:12.770 --> 00:59:16.790 align:middle line:84% And therefore, it always converges. 00:59:16.790 --> 00:59:21.860 align:middle line:84% So equismall tails means that somehow, that series converges, 00:59:21.860 --> 00:59:24.800 align:middle line:84% or the tail end of that series converges, 00:59:24.800 --> 00:59:30.020 align:middle line:84% uniformly with respect to this orthonormal subset. 00:59:30.020 --> 00:59:35.040 align:middle line:84% So in other words, for all epsilon positive, 00:59:35.040 --> 00:59:41.300 align:middle line:84% there exists a natural number N such that 00:59:41.300 --> 00:59:53.950 align:middle line:84% for all v in the set K, if I look at the tail, 00:59:53.950 --> 00:59:56.250 align:middle line:84% this is less than epsilon squared. 00:59:56.250 --> 00:59:59.200 align:middle line:90% 00:59:59.200 --> 01:00:01.870 align:middle line:84% So somehow, given epsilon, I can always 01:00:01.870 --> 01:00:06.490 align:middle line:84% choose an N which is independent of the element in K 01:00:06.490 --> 01:00:11.890 align:middle line:84% so that the tail end of, if you like, the Fourier series 01:00:11.890 --> 01:00:14.290 align:middle line:90% is small. 01:00:14.290 --> 01:00:16.570 align:middle line:84% And this N can be chosen independent 01:00:16.570 --> 01:00:20.530 align:middle line:90% of the element in K. 01:00:20.530 --> 01:00:34.910 align:middle line:84% So I leave it to you to verify if K is simply a finite set, 01:00:34.910 --> 01:00:48.920 align:middle line:84% this implies K has equismall tails with respect 01:00:48.920 --> 01:01:00.735 align:middle line:90% to any orthonormal subset. 01:01:00.735 --> 01:01:11.150 align:middle line:90% 01:01:11.150 --> 01:01:14.720 align:middle line:84% So what's the motivation for this definition 01:01:14.720 --> 01:01:21.860 align:middle line:84% is that the next simplest type of set which is compact. 01:01:21.860 --> 01:01:28.510 align:middle line:84% So like I said, any finite subset of a metric space 01:01:28.510 --> 01:01:29.440 align:middle line:90% is compact. 01:01:29.440 --> 01:01:34.150 align:middle line:84% Now, I'll give an example of another, 01:01:34.150 --> 01:01:36.610 align:middle line:90% more interesting compact set. 01:01:36.610 --> 01:01:39.760 align:middle line:84% But that compact set also satisfies this property, 01:01:39.760 --> 01:01:41.770 align:middle line:84% that it has equismall tails with respect 01:01:41.770 --> 01:01:46.510 align:middle line:84% to any countable orthonormal subset. 01:01:46.510 --> 01:01:51.220 align:middle line:84% And so then, we should hope that if a set is closed, 01:01:51.220 --> 01:01:54.460 align:middle line:84% bounded, and has the property that, with respect 01:01:54.460 --> 01:01:58.990 align:middle line:84% to an orthonormal basis it has equismall tails, then that set 01:01:58.990 --> 01:01:59.560 align:middle line:90% is compact. 01:01:59.560 --> 01:02:00.820 align:middle line:84% And that's eventually what will prove, 01:02:00.820 --> 01:02:02.500 align:middle line:84% but I don't think we'll get to it today. 01:02:02.500 --> 01:02:08.590 align:middle line:84% But so one more bit of motivation 01:02:08.590 --> 01:02:12.610 align:middle line:84% on why this might be the right extra condition 01:02:12.610 --> 01:02:17.350 align:middle line:84% to add to closed and bounded to produce compact sets 01:02:17.350 --> 01:02:18.890 align:middle line:90% is provided by the following. 01:02:18.890 --> 01:02:28.490 align:middle line:84% So again, let H be a Hilbert space 01:02:28.490 --> 01:02:36.530 align:middle line:90% and be in a convergent sequence. 01:02:36.530 --> 01:02:44.590 align:middle line:90% 01:02:44.590 --> 01:02:49.720 align:middle line:84% Now let e K be some arbitrary orthonormal, 01:02:49.720 --> 01:02:51.070 align:middle line:90% countable orthonormal subset. 01:02:51.070 --> 01:03:00.020 align:middle line:90% 01:03:00.020 --> 01:03:02.120 align:middle line:90% So then two things hold-- 01:03:02.120 --> 01:03:11.540 align:middle line:84% one is that the set K equals a set of elements v N 01:03:11.540 --> 01:03:14.750 align:middle line:90% union the limit. 01:03:14.750 --> 01:03:15.530 align:middle line:90% This is compact. 01:03:15.530 --> 01:03:21.940 align:middle line:90% 01:03:21.940 --> 01:03:25.770 align:middle line:84% The proof is given in [? Mel ?] [? Rose's ?] notes, 01:03:25.770 --> 01:03:28.270 align:middle line:84% and I'm going to leave it to you to look up or work your way 01:03:28.270 --> 01:03:28.770 align:middle line:90% through. 01:03:28.770 --> 01:03:32.380 align:middle line:84% It's not that difficult to work your way through. 01:03:32.380 --> 01:03:48.240 align:middle line:84% And K has equismall tails with respect to e K. 01:03:48.240 --> 01:03:54.880 align:middle line:84% So this subset consisting of a convergent sequence 01:03:54.880 --> 01:03:56.950 align:middle line:84% along with this limit is compact. 01:03:56.950 --> 01:03:59.170 align:middle line:84% And it has equismall details with respect 01:03:59.170 --> 01:04:01.225 align:middle line:84% to any countable orthonormal subset. 01:04:01.225 --> 01:04:06.820 align:middle line:90% 01:04:06.820 --> 01:04:12.100 align:middle line:84% So like I said, I'm going to leave one for you. 01:04:12.100 --> 01:04:14.560 align:middle line:90% 2 we will prove. 01:04:14.560 --> 01:04:15.580 align:middle line:90% How much time do I got? 01:04:15.580 --> 01:04:20.440 align:middle line:84% I don't have much time left, but we 01:04:20.440 --> 01:04:23.770 align:middle line:90% should be able to finish this. 01:04:23.770 --> 01:04:27.170 align:middle line:84% We have to verify this definition. 01:04:27.170 --> 01:04:28.900 align:middle line:90% So let epsilon be positive. 01:04:28.900 --> 01:04:32.260 align:middle line:90% 01:04:32.260 --> 01:04:35.370 align:middle line:90% So why should we-- 01:04:35.370 --> 01:04:38.490 align:middle line:84% what's the thinking on why we expect this set to have 01:04:38.490 --> 01:04:40.860 align:middle line:90% equismall tails? 01:04:40.860 --> 01:04:43.260 align:middle line:84% So I have to check that I can find a large enough integer 01:04:43.260 --> 01:04:47.590 align:middle line:84% that all the squares of these coefficients is small. 01:04:47.590 --> 01:04:50.190 align:middle line:84% Now, the point is that the v N's are 01:04:50.190 --> 01:04:56.100 align:middle line:84% essentially very close to being v, at least for N very large. 01:04:56.100 --> 01:04:59.220 align:middle line:84% And for a fixed v, I can always choose a capital N 01:04:59.220 --> 01:05:01.780 align:middle line:90% so that this holds. 01:05:01.780 --> 01:05:04.770 align:middle line:84% And so that N will take care of infinitely many, 01:05:04.770 --> 01:05:08.130 align:middle line:84% and then I just need to choose N big enough for v, 01:05:08.130 --> 01:05:12.900 align:middle line:84% and also the finitely many that are not very close-- 01:05:12.900 --> 01:05:14.830 align:middle line:84% finitely many elements of this sequence 01:05:14.830 --> 01:05:18.510 align:middle line:84% which are not close to v. That's essentially the argument. 01:05:18.510 --> 01:05:34.210 align:middle line:84% So since v N converges to v, there exists a natural number 01:05:34.210 --> 01:05:42.670 align:middle line:84% M such that all N bigger than or equal to M, I have v N minus v, 01:05:42.670 --> 01:05:45.760 align:middle line:84% and norm is less than epsilon over 2. 01:05:45.760 --> 01:05:52.610 align:middle line:90% 01:05:52.610 --> 01:06:02.650 align:middle line:84% Now, choose natural number N so large 01:06:02.650 --> 01:06:17.240 align:middle line:84% so that sum k bigger than N of v v k-- 01:06:17.240 --> 01:06:19.850 align:middle line:84% this is small, so here, v is fixed. 01:06:19.850 --> 01:06:22.820 align:middle line:84% And we always have that this whole sum converges 01:06:22.820 --> 01:06:24.350 align:middle line:90% by Bessel's inequality. 01:06:24.350 --> 01:06:26.120 align:middle line:84% And therefore, we can always choose an N 01:06:26.120 --> 01:06:27.740 align:middle line:90% so that the tail is small. 01:06:27.740 --> 01:06:30.530 align:middle line:84% And we can do the same for the first M 01:06:30.530 --> 01:06:50.610 align:middle line:84% minus 1 elements of the sequence, so plus max of 1, 01:06:50.610 --> 01:06:55.470 align:middle line:84% so that this is all less than epsilon squared over 4. 01:06:55.470 --> 01:06:59.250 align:middle line:84% So for each v N, I can find a capital-- 01:06:59.250 --> 01:07:03.690 align:middle line:84% so for v1, I can find an N1 so that this is less than epsilon 01:07:03.690 --> 01:07:07.350 align:middle line:90% squared over 8, say. 01:07:07.350 --> 01:07:11.070 align:middle line:84% And then for N equals 2, N equals 3, 01:07:11.070 --> 01:07:16.350 align:middle line:84% all the way up to N minus 1, I can find a capital N sub n. 01:07:16.350 --> 01:07:18.240 align:middle line:84% I then take capital N to be the maximum 01:07:18.240 --> 01:07:24.720 align:middle line:84% of those finitely many capital N sub n's along with the N 01:07:24.720 --> 01:07:26.610 align:middle line:90% that I need for this. 01:07:26.610 --> 01:07:29.970 align:middle line:84% So that's the argument on why I can find such a capital N. 01:07:29.970 --> 01:07:32.470 align:middle line:84% I just have finitely many things I need to make small, 01:07:32.470 --> 01:07:33.945 align:middle line:90% which I individually can. 01:07:33.945 --> 01:07:46.160 align:middle line:90% 01:07:46.160 --> 01:07:48.000 align:middle line:84% Now, I claim this capital N works. 01:07:48.000 --> 01:07:52.850 align:middle line:84% So I have to show that if I take any element in K, then 01:07:52.850 --> 01:07:54.320 align:middle line:90% it has a small tail. 01:07:54.320 --> 01:07:58.070 align:middle line:90% 01:07:58.070 --> 01:08:09.180 align:middle line:84% Then I have that, by how I've chosen N, 01:08:09.180 --> 01:08:15.330 align:middle line:84% sum over k bigger than N of v e k squared-- 01:08:15.330 --> 01:08:18.750 align:middle line:84% this is less than epsilon squared over 4, 01:08:18.750 --> 01:08:21.189 align:middle line:84% which is less than epsilon squared. 01:08:21.189 --> 01:08:28.979 align:middle line:84% And for all 1 less than or equal to n is less than 01:08:28.979 --> 01:08:30.780 align:middle line:84% or equal to n minus 1, the same bound. 01:08:30.780 --> 01:08:41.720 align:middle line:90% 01:08:41.720 --> 01:08:45.590 align:middle line:84% So I just need to check now that this is small 01:08:45.590 --> 01:08:50.950 align:middle line:84% when n is bigger than or equal to M. 01:08:50.950 --> 01:09:02.130 align:middle line:84% Now, if n is bigger than or equal to M, by Bessel's 01:09:02.130 --> 01:09:11.250 align:middle line:84% inequality, we have that some K bigger than our N v n 01:09:11.250 --> 01:09:17.069 align:middle line:84% e K squared, if I can take the 1/2 power 01:09:17.069 --> 01:09:19.630 align:middle line:84% and show that's less than epsilon. 01:09:19.630 --> 01:09:35.529 align:middle line:84% So this is equal to K bigger than N of v N minus v plus v. 01:09:35.529 --> 01:09:43.340 align:middle line:90% So I can-- e K plus-- 01:09:43.340 --> 01:09:45.080 align:middle line:90% so let's write it this way. 01:09:45.080 --> 01:09:59.360 align:middle line:90% 01:09:59.360 --> 01:10:04.940 align:middle line:84% Now, this is the little l2 norm of a sum of two sequences in K, 01:10:04.940 --> 01:10:08.960 align:middle line:84% not N, but in K. So by the triangle inequality 01:10:08.960 --> 01:10:12.380 align:middle line:84% for sequences in little l2, this is less than or equal 01:10:12.380 --> 01:10:27.990 align:middle line:84% to sum K bigger than N of v N minus v e K 1/2 plus sum from K 01:10:27.990 --> 01:10:38.630 align:middle line:84% bigger than N of v e K to the 1/2 power. 01:10:38.630 --> 01:10:42.890 align:middle line:84% Now, this quantity I already have control over. 01:10:42.890 --> 01:10:46.610 align:middle line:90% That's less than epsilon over 2. 01:10:46.610 --> 01:10:48.950 align:middle line:90% I'm taking the 1/2 power here. 01:10:48.950 --> 01:10:51.300 align:middle line:84% And I can bound this by Bessel's inequality. 01:10:51.300 --> 01:10:56.990 align:middle line:84% This is less than or equal to norm of v N minus v plus-- 01:10:56.990 --> 01:10:59.600 align:middle line:84% and then using the bound for this-- 01:10:59.600 --> 01:11:01.100 align:middle line:90% epsilon over 2. 01:11:01.100 --> 01:11:05.210 align:middle line:84% And now remember, how is capital M chosen? 01:11:05.210 --> 01:11:08.905 align:middle line:84% Capital M is chosen to ensure that this is less 01:11:08.905 --> 01:11:10.280 align:middle line:84% than epsilon over 2, which we can 01:11:10.280 --> 01:11:12.440 align:middle line:84% do since the v N's are converging to v. 01:11:12.440 --> 01:11:17.930 align:middle line:84% So this is less than epsilon over 2 plus epsilon over 2 01:11:17.930 --> 01:11:23.600 align:middle line:84% equals epsilon, proving that this set consisting 01:11:23.600 --> 01:11:28.610 align:middle line:84% of the elements of a convergent sequence along with the limit, 01:11:28.610 --> 01:11:32.190 align:middle line:84% has equismall tails with respect to an arbitrary orthonormal 01:11:32.190 --> 01:11:32.690 align:middle line:90% subset. 01:11:32.690 --> 01:11:34.940 align:middle line:84% So next time, we will prove that if we 01:11:34.940 --> 01:11:40.970 align:middle line:84% have a subset of a separable Hilbert space which is closed, 01:11:40.970 --> 01:11:43.940 align:middle line:84% bounded, and has equismall details with respect 01:11:43.940 --> 01:11:47.930 align:middle line:84% to A orthonormal basis, which exists 01:11:47.930 --> 01:11:54.540 align:middle line:84% because it's a separable Hilbert space, then the set is compact. 01:11:54.540 --> 01:11:56.850 align:middle line:84% And then, we'll rephrase that in a way that 01:11:56.850 --> 01:12:02.157 align:middle line:84% doesn't involve Hilbert spaces, and go from there, 01:12:02.157 --> 01:12:03.990 align:middle line:84% and start looking at some of these operators 01:12:03.990 --> 01:12:08.210 align:middle line:84% I said, which are close to matrices. 01:12:08.210 --> 01:12:10.480 align:middle line:90% All right, we'll stop there. 01:12:10.480 --> 01:12:17.000 align:middle line:90%