WEBVTT 00:00:00.000 --> 00:00:18.700 align:middle line:90% 00:00:18.700 --> 00:00:32.490 align:middle line:84% PROFESSOR: So last time we introduced pre-Hilbert spaces 00:00:32.490 --> 00:00:36.210 align:middle line:84% as vector spaces that come equipped with a Hermitian 00:00:36.210 --> 00:00:37.380 align:middle line:90% inner product. 00:00:37.380 --> 00:00:39.870 align:middle line:84% Our Hermitian inner product is linear. 00:00:39.870 --> 00:00:42.330 align:middle line:84% So it's a pairing between two elements 00:00:42.330 --> 00:00:44.340 align:middle line:84% that gives you a complex number that's 00:00:44.340 --> 00:00:48.160 align:middle line:90% linear in the first entry. 00:00:48.160 --> 00:00:50.310 align:middle line:84% And if you switch the entries, that's 00:00:50.310 --> 00:00:54.630 align:middle line:84% equal to the pairing of the original two entries, 00:00:54.630 --> 00:00:56.590 align:middle line:84% taking the complex conjugate of that. 00:00:56.590 --> 00:00:59.640 align:middle line:84% And it's positive definite, meaning 00:00:59.640 --> 00:01:03.840 align:middle line:84% if I take the inner product of a vector with itself, 00:01:03.840 --> 00:01:10.580 align:middle line:84% it's non-negative and 0 if and only if the element is zero. 00:01:10.580 --> 00:01:13.200 align:middle line:84% And we proved the Cauchy-Schwarz inequality. 00:01:13.200 --> 00:01:20.100 align:middle line:84% So let me actually just recall that we defined for if h 00:01:20.100 --> 00:01:28.190 align:middle line:90% is a pre-Hilbert space. 00:01:28.190 --> 00:01:33.718 align:middle line:84% We defined what we called, or using this norm notation, 00:01:33.718 --> 00:01:35.260 align:middle line:84% although I haven't proved it's a norm 00:01:35.260 --> 00:01:44.520 align:middle line:84% yet to be the inner product of v with itself 00:01:44.520 --> 00:01:46.560 align:middle line:90% raised to the 1/2 power. 00:01:46.560 --> 00:01:48.810 align:middle line:84% And this is a non-negative real number. 00:01:48.810 --> 00:01:51.570 align:middle line:90% 00:01:51.570 --> 00:01:52.890 align:middle line:90% This is a non-negative number. 00:01:52.890 --> 00:01:58.760 align:middle line:84% So taking the 1/2 power is meaningful. 00:01:58.760 --> 00:02:01.180 align:middle line:84% And we proved at the end of last time 00:02:01.180 --> 00:02:13.630 align:middle line:84% the Cauchy-Schwarz inequality that for all u, v in H if I 00:02:13.630 --> 00:02:17.180 align:middle line:84% take the inner product of u and v, take its absolute value, 00:02:17.180 --> 00:02:24.120 align:middle line:84% this is less than or equal to norm of u times the norm of v. 00:02:24.120 --> 00:02:29.130 align:middle line:84% OK, so now let's use this to actually prove or to prove 00:02:29.130 --> 00:02:34.960 align:middle line:84% that this thing that I have been denoting with this norm 00:02:34.960 --> 00:02:40.510 align:middle line:84% notation is, in fact, a norm on a pre-Hilbert space. 00:02:40.510 --> 00:02:53.640 align:middle line:84% So theorem H is a pre-Hilbert space, 00:02:53.640 --> 00:03:07.567 align:middle line:84% then this thing here defined in this way is norm on h, OK. 00:03:07.567 --> 00:03:10.490 align:middle line:90% 00:03:10.490 --> 00:03:12.650 align:middle line:84% So remember, we have to prove two-- 00:03:12.650 --> 00:03:15.290 align:middle line:84% or we have to prove three things for something to be a norm. 00:03:15.290 --> 00:03:17.210 align:middle line:84% We have to prove it's positive definite. 00:03:17.210 --> 00:03:20.720 align:middle line:84% And then we also have to prove homogeneity and the triangle 00:03:20.720 --> 00:03:21.690 align:middle line:90% inequality. 00:03:21.690 --> 00:03:34.700 align:middle line:84% So note that this quantity here equals 0 if 00:03:34.700 --> 00:03:39.110 align:middle line:84% and only if the inner product of v with itself 00:03:39.110 --> 00:03:45.050 align:middle line:84% is zero, which by the positive definite quantity 00:03:45.050 --> 00:03:48.890 align:middle line:84% of a Hermitian inner product implies that v equals 0. 00:03:48.890 --> 00:03:53.120 align:middle line:84% So that proves that this function here on h 00:03:53.120 --> 00:03:57.410 align:middle line:90% is in fact positive definite. 00:03:57.410 --> 00:04:00.300 align:middle line:84% Now if lambda is a complex number 00:04:00.300 --> 00:04:02.780 align:middle line:90% and v is in H, if I take-- 00:04:02.780 --> 00:04:05.770 align:middle line:90% 00:04:05.770 --> 00:04:09.880 align:middle line:84% so we kind of saw this at the proof of the Cauchy-Schwarz 00:04:09.880 --> 00:04:10.520 align:middle line:90% inequality. 00:04:10.520 --> 00:04:15.160 align:middle line:84% But if I take lambda times v and inner product lambda times v, 00:04:15.160 --> 00:04:19.300 align:middle line:84% this is equal to lambda times lambda bar. 00:04:19.300 --> 00:04:24.550 align:middle line:84% A scalar pulls out of the first entry unfazed. 00:04:24.550 --> 00:04:26.140 align:middle line:84% If a scalar is in the second entry, 00:04:26.140 --> 00:04:28.030 align:middle line:84% it comes out with a complex conjugate. 00:04:28.030 --> 00:04:32.710 align:middle line:90% 00:04:32.710 --> 00:04:34.760 align:middle line:84% And then of course for a complex number 00:04:34.760 --> 00:04:37.780 align:middle line:84% of times this complex conjugate, I get the norm of that 00:04:37.780 --> 00:04:40.310 align:middle line:90% or the length of that-- 00:04:40.310 --> 00:04:44.040 align:middle line:84% the absolute value of that complex number squared. 00:04:44.040 --> 00:04:46.970 align:middle line:84% So taking the square root of both sides 00:04:46.970 --> 00:04:50.300 align:middle line:84% of this, which is equal to this, I 00:04:50.300 --> 00:05:03.350 align:middle line:84% get that this quantity here is equal to the absolute value 00:05:03.350 --> 00:05:11.090 align:middle line:84% of lambda times this norm of v, which proves homogeneity 00:05:11.090 --> 00:05:13.670 align:middle line:90% of this function on h. 00:05:13.670 --> 00:05:17.210 align:middle line:84% OK, so all that remains to prove that this is an actual norm-- 00:05:17.210 --> 00:05:22.490 align:middle line:84% so I don't have to keep being or at least trying to be careful 00:05:22.490 --> 00:05:24.035 align:middle line:90% with what words I'm using-- 00:05:24.035 --> 00:05:27.640 align:middle line:90% 00:05:27.640 --> 00:05:29.050 align:middle line:90% will prove that this is a norm. 00:05:29.050 --> 00:05:31.825 align:middle line:90% 00:05:31.825 --> 00:05:36.300 align:middle line:84% OK, so now we need to prove the triangle inequality. 00:05:36.300 --> 00:05:48.230 align:middle line:84% You let uv be in H, then I compute u plus v squared. 00:05:48.230 --> 00:05:52.970 align:middle line:84% This is equal to u plus v plus v, which 00:05:52.970 --> 00:05:57.220 align:middle line:84% is equal to just how we computed it 00:05:57.220 --> 00:06:01.988 align:middle line:84% for when we had a t here and a t here, 00:06:01.988 --> 00:06:04.280 align:middle line:84% when we did the proof of the Cauchy-Schwarz inequality. 00:06:04.280 --> 00:06:07.210 align:middle line:84% And we'll use this identity quite often 00:06:07.210 --> 00:06:10.240 align:middle line:84% that the norm of u plus v squared 00:06:10.240 --> 00:06:14.410 align:middle line:84% is norm u squared plus norm b squared 00:06:14.410 --> 00:06:20.440 align:middle line:84% plus 2 times the real part of u and v, 00:06:20.440 --> 00:06:25.036 align:middle line:84% the real part of the inner product of u and v. 00:06:25.036 --> 00:06:31.200 align:middle line:84% OK, now this is less than or equal to norm squared 00:06:31.200 --> 00:06:35.550 align:middle line:84% plus v squared plus the absolute value of the real part. 00:06:35.550 --> 00:06:40.730 align:middle line:90% 00:06:40.730 --> 00:06:45.890 align:middle line:84% Now the absolute value of the real part of a complex number 00:06:45.890 --> 00:06:47.750 align:middle line:84% is less than or equal to the absolute value 00:06:47.750 --> 00:06:49.260 align:middle line:90% of that complex number. 00:06:49.260 --> 00:06:52.010 align:middle line:84% So this is less than or equal to the absolute value 00:06:52.010 --> 00:06:54.260 align:middle line:90% of the inner product of u and v. 00:06:54.260 --> 00:07:03.840 align:middle line:84% And by Cauchy-Schwarz that is less than or equal to still 00:07:03.840 --> 00:07:13.180 align:middle line:84% what I have before plus 2 times the inner product of u and v. 00:07:13.180 --> 00:07:20.650 align:middle line:84% And I guess let me go to the next board. 00:07:20.650 --> 00:07:22.075 align:middle line:84% All the proof is essentially done. 00:07:22.075 --> 00:07:24.670 align:middle line:90% 00:07:24.670 --> 00:07:30.280 align:middle line:84% What I had before is equal to norm of u 00:07:30.280 --> 00:07:36.320 align:middle line:90% plus normal v squared, i.e. 00:07:36.320 --> 00:07:42.660 align:middle line:84% So I had started off with the norm of u plus v squared, 00:07:42.660 --> 00:07:45.150 align:middle line:84% and I had proved that's less than or equal to this thing 00:07:45.150 --> 00:07:47.260 align:middle line:90% here squared. 00:07:47.260 --> 00:07:49.900 align:middle line:84% So taking the square root of both sides, 00:07:49.900 --> 00:07:56.370 align:middle line:84% I get the norm of u plus v is less than or equal to the norm 00:07:56.370 --> 00:07:57.360 align:middle line:90% u plus the norm of v. 00:07:57.360 --> 00:08:01.950 align:middle line:84% OK, so this thing that I defined before is, in fact, the norm. 00:08:01.950 --> 00:08:11.795 align:middle line:84% I'm not just using that notation to denote an impostor. 00:08:11.795 --> 00:08:20.292 align:middle line:90% 00:08:20.292 --> 00:08:24.670 align:middle line:84% OK, now using the Cauchy-Schwarz inequality, 00:08:24.670 --> 00:08:29.220 align:middle line:84% we can also prove that taking the inner product is-- 00:08:29.220 --> 00:08:32.830 align:middle line:90% 00:08:32.830 --> 00:08:38.289 align:middle line:84% it's a function on h cross h that's continuous, OK. 00:08:38.289 --> 00:08:45.380 align:middle line:84% So let me label this as continuity 00:08:45.380 --> 00:08:49.230 align:middle line:90% of the inner product. 00:08:49.230 --> 00:08:51.170 align:middle line:84% So let me state it as the following. 00:08:51.170 --> 00:08:57.770 align:middle line:84% If we're in a Hilbert space and un converges to u and vm 00:08:57.770 --> 00:09:22.560 align:middle line:84% converges to v in a pre-Hilbert space with norm as defined 00:09:22.560 --> 00:09:24.840 align:middle line:90% as before-- 00:09:24.840 --> 00:09:27.350 align:middle line:84% so now we have a norm on a pre-Hilbert space, 00:09:27.350 --> 00:09:30.825 align:middle line:84% so we can define convergence since it's a norm space. 00:09:30.825 --> 00:09:36.020 align:middle line:90% 00:09:36.020 --> 00:09:47.130 align:middle line:84% Pre-Hilbert space H with norm this, then un, inner product vn 00:09:47.130 --> 00:09:52.635 align:middle line:90% converges to u inner product v. 00:09:52.635 --> 00:09:53.135 align:middle line:90% OK. 00:09:53.135 --> 00:09:56.530 align:middle line:90% 00:09:56.530 --> 00:09:59.810 align:middle line:90% All right, so why is this? 00:09:59.810 --> 00:10:02.470 align:middle line:90% 00:10:02.470 --> 00:10:06.640 align:middle line:84% So last time in the previous proof, I mean, 00:10:06.640 --> 00:10:09.520 align:middle line:84% we really didn't use the full strength of the Cauchy-Schwarz 00:10:09.520 --> 00:10:10.390 align:middle line:90% inequality. 00:10:10.390 --> 00:10:13.660 align:middle line:84% We could have just gotten by with the real part, 00:10:13.660 --> 00:10:18.210 align:middle line:84% having just proven that the real part of the inner product 00:10:18.210 --> 00:10:21.240 align:middle line:84% is an absolute value less than or equal to the inner product 00:10:21.240 --> 00:10:25.560 align:middle line:84% of-- or the product of the norms of u and v. Here 00:10:25.560 --> 00:10:28.690 align:middle line:84% we'll actually use the full Cauchy-Schwarz inequality. 00:10:28.690 --> 00:10:31.170 align:middle line:90% So the proof is quite simple. 00:10:31.170 --> 00:10:33.850 align:middle line:90% 00:10:33.850 --> 00:10:37.935 align:middle line:84% If un converges to u and vn converges to v-- 00:10:37.935 --> 00:10:40.560 align:middle line:90% 00:10:40.560 --> 00:10:45.480 align:middle line:84% i.e, let me just spell this out for you-- 00:10:45.480 --> 00:10:48.660 align:middle line:90% un minus u converges to 0. 00:10:48.660 --> 00:10:54.060 align:middle line:84% And norm of vn minus v converges to 0. 00:10:54.060 --> 00:10:57.330 align:middle line:84% As n converges to-- as n goes to infinity, 00:10:57.330 --> 00:11:08.840 align:middle line:84% then we'll use the squeeze theorem 00:11:08.840 --> 00:11:11.360 align:middle line:84% to show that this quantity converges 00:11:11.360 --> 00:11:13.850 align:middle line:90% to u in a product with uv. 00:11:13.850 --> 00:11:17.997 align:middle line:84% So I have to show that this quantity here-- hold on. 00:11:17.997 --> 00:11:18.830 align:middle line:90% Let me not put that. 00:11:18.830 --> 00:11:22.130 align:middle line:90% We're not in 18100. 00:11:22.130 --> 00:11:25.017 align:middle line:84% Typically in 18100 when I first started 00:11:25.017 --> 00:11:26.600 align:middle line:84% teaching the squeeze theorem, I always 00:11:26.600 --> 00:11:28.260 align:middle line:90% included the lower bound. 00:11:28.260 --> 00:11:31.340 align:middle line:84% But it should be clear that this is always 00:11:31.340 --> 00:11:33.380 align:middle line:84% bigger than or equal to zero since it's 00:11:33.380 --> 00:11:35.855 align:middle line:84% absolute value of a complex number. 00:11:35.855 --> 00:11:38.730 align:middle line:90% 00:11:38.730 --> 00:11:47.990 align:middle line:84% OK, so this is equal to un minus u, vn plus-- 00:11:47.990 --> 00:11:50.780 align:middle line:90% 00:11:50.780 --> 00:11:51.440 align:middle line:90% let's see. 00:11:51.440 --> 00:11:54.490 align:middle line:90% 00:11:54.490 --> 00:12:04.290 align:middle line:84% U, inner product, vn minus v. By the triangle inequality now, 00:12:04.290 --> 00:12:08.100 align:middle line:84% just for the modulus or absolute value 00:12:08.100 --> 00:12:10.590 align:middle line:84% of complex numbers, that's less than 00:12:10.590 --> 00:12:19.285 align:middle line:84% or equal to un minus u, vn, plus-- 00:12:19.285 --> 00:12:22.680 align:middle line:90% 00:12:22.680 --> 00:12:32.640 align:middle line:84% that's the value of u, vn minus v. 00:12:32.640 --> 00:12:34.050 align:middle line:84% And this is less than or equal to 00:12:34.050 --> 00:12:41.180 align:middle line:84% by the Cauchy-Schwarz inequality norm of un minus 00:12:41.180 --> 00:12:46.610 align:middle line:84% u times the norm of vn plus the norm of u times 00:12:46.610 --> 00:12:55.510 align:middle line:84% the norm of vn minus v. And now the vns are. 00:12:55.510 --> 00:12:59.170 align:middle line:84% So the vns are converging to v. And therefore, 00:12:59.170 --> 00:13:02.410 align:middle line:84% the norms of the vns are converging to the norm of v. 00:13:02.410 --> 00:13:13.270 align:middle line:84% So recall that if the simple fact that vn convergence to v, 00:13:13.270 --> 00:13:20.270 align:middle line:84% then norm of vn converges to norm of v, OK. 00:13:20.270 --> 00:13:23.770 align:middle line:90% 00:13:23.770 --> 00:13:26.250 align:middle line:90% All right. 00:13:26.250 --> 00:13:28.500 align:middle line:90% And, again, so let me add-- 00:13:28.500 --> 00:13:34.880 align:middle line:84% since one can prove as you do in real analysis 00:13:34.880 --> 00:13:37.820 align:middle line:84% the following reverse triangle inequality since if I 00:13:37.820 --> 00:13:43.520 align:middle line:90% have two vectors v and h. 00:13:43.520 --> 00:13:45.980 align:middle line:84% So this just works in any Banach space, not necessarily 00:13:45.980 --> 00:13:47.630 align:middle line:90% a pre-Hilbert space. 00:13:47.630 --> 00:13:49.790 align:middle line:84% The absolute value of the difference in norms 00:13:49.790 --> 00:13:55.650 align:middle line:84% is less than or equal to the norm of the difference, OK. 00:13:55.650 --> 00:14:00.600 align:middle line:84% So this is a convergent sequence of real numbers. 00:14:00.600 --> 00:14:01.650 align:middle line:90% So it must be bounded. 00:14:01.650 --> 00:14:08.660 align:middle line:84% So I can say that's less than or equal to times sup of n. 00:14:08.660 --> 00:14:18.980 align:middle line:84% Or let's use a different letter, k, times plus norm 00:14:18.980 --> 00:14:23.030 align:middle line:90% of vn minus v times norm of u. 00:14:23.030 --> 00:14:33.360 align:middle line:84% Now, this is something converging to 0 times 00:14:33.360 --> 00:14:34.210 align:middle line:90% a fixed number. 00:14:34.210 --> 00:14:37.560 align:middle line:84% This is something converging to 0 times a fixed number. 00:14:37.560 --> 00:14:41.650 align:middle line:84% And therefore, this goes to 0 as n goes to infinity. 00:14:41.650 --> 00:14:45.430 align:middle line:84% And therefore, the thing which we started with, 00:14:45.430 --> 00:14:48.630 align:middle line:84% so this quantity here, an absolute value, 00:14:48.630 --> 00:14:52.140 align:middle line:84% is less than or equal to something converging to zero. 00:14:52.140 --> 00:14:54.390 align:middle line:90% And it's also non-negative. 00:14:54.390 --> 00:14:59.860 align:middle line:90% So therefore I get that. 00:14:59.860 --> 00:15:09.810 align:middle line:90% 00:15:09.810 --> 00:15:13.700 align:middle line:84% So this is by the squeeze theorem, i.e. 00:15:13.700 --> 00:15:27.340 align:middle line:90% 00:15:27.340 --> 00:15:40.430 align:middle line:84% OK, so on pre-Hilbert space, we can define 00:15:40.430 --> 00:15:44.600 align:middle line:90% a norm using the inner product. 00:15:44.600 --> 00:15:47.750 align:middle line:84% And this inner product is continuous with respect 00:15:47.750 --> 00:15:49.258 align:middle line:90% to this norm, OK. 00:15:49.258 --> 00:15:51.300 align:middle line:84% Because once we have a norm, we have convergence. 00:15:51.300 --> 00:15:53.447 align:middle line:90% So we can talk about-- 00:15:53.447 --> 00:15:55.530 align:middle line:84% once we have a norm, we have a notion of distance, 00:15:55.530 --> 00:16:02.210 align:middle line:84% so we can talk about convergence of sequences and things 00:16:02.210 --> 00:16:03.450 align:middle line:90% like that. 00:16:03.450 --> 00:16:07.350 align:middle line:84% So now we have pre-Hilbert spaces. 00:16:07.350 --> 00:16:15.770 align:middle line:90% What is a Hilbert space? 00:16:15.770 --> 00:16:17.270 align:middle line:84% This is simply a pre-Hilbert space 00:16:17.270 --> 00:16:20.210 align:middle line:84% which is complete with respect to this norm. 00:16:20.210 --> 00:16:43.760 align:middle line:84% So a Hilbert space, H, is a pre-Hilbert space, 00:16:43.760 --> 00:17:06.980 align:middle line:84% which is complete with respect to the norm, 00:17:06.980 --> 00:17:11.670 align:middle line:84% with respect to this norm, which again remember was defined as-- 00:17:11.670 --> 00:17:13.748 align:middle line:90% I took a vector. 00:17:13.748 --> 00:17:16.040 align:middle line:84% The norm of a vector here is equal to the inner product 00:17:16.040 --> 00:17:19.810 align:middle line:84% of the vector with itself raised to the 1/2 power. 00:17:19.810 --> 00:17:22.180 align:middle line:84% All right, so this is the new terminology. 00:17:22.180 --> 00:17:28.980 align:middle line:84% And as we'll see, or at least we'll explicitly spell out 00:17:28.980 --> 00:17:33.990 align:middle line:84% in the form of a theorem-- but we will see probably by the end 00:17:33.990 --> 00:17:41.570 align:middle line:84% of this lecture that really there's-- for a reasonable 00:17:41.570 --> 00:17:46.140 align:middle line:84% Hilbert space, you're either one of two things. 00:17:46.140 --> 00:17:50.540 align:middle line:84% So let me first give the examples, basic examples 00:17:50.540 --> 00:17:53.240 align:middle line:90% of Hilbert space. 00:17:53.240 --> 00:18:00.560 align:middle line:84% So for example Cn which is a set of N-tuples. 00:18:00.560 --> 00:18:01.700 align:middle line:90% How do I denote this? 00:18:01.700 --> 00:18:07.090 align:middle line:90% 00:18:07.090 --> 00:18:17.010 align:middle line:84% Complex numbers where the inner product of two vectors, 00:18:17.010 --> 00:18:24.920 align:middle line:84% z and w, is equal to sum j equals 1 to the n, zj, wj. 00:18:24.920 --> 00:18:27.740 align:middle line:90% 00:18:27.740 --> 00:18:28.240 align:middle line:90% OK. 00:18:28.240 --> 00:18:32.540 align:middle line:90% 00:18:32.540 --> 00:18:35.270 align:middle line:84% So this is an example of a Hilbert 00:18:35.270 --> 00:18:38.120 align:middle line:84% space, finite dimensional one, meaning 00:18:38.120 --> 00:18:41.880 align:middle line:84% it's a vector space of finite dimension. 00:18:41.880 --> 00:18:47.970 align:middle line:84% The other example we have is this space, little l, 00:18:47.970 --> 00:19:02.200 align:middle line:84% to which this was the space of sequences such that each 00:19:02.200 --> 00:19:04.240 align:middle line:90% of these is a complex number. 00:19:04.240 --> 00:19:16.570 align:middle line:84% And this series is convergent, is finite. 00:19:16.570 --> 00:19:19.570 align:middle line:84% Sum of the absolute value of the ak squares, this 00:19:19.570 --> 00:19:23.920 align:middle line:84% is a Hilbert space as well with what's 00:19:23.920 --> 00:19:30.670 align:middle line:84% the inner product of two elements of little l2. 00:19:30.670 --> 00:19:43.090 align:middle line:84% This is the sum from a plus 1 to infinity of ak, bk. 00:19:43.090 --> 00:19:43.590 align:middle line:90% OK. 00:19:43.590 --> 00:19:47.350 align:middle line:90% 00:19:47.350 --> 00:19:48.280 align:middle line:90% All right. 00:19:48.280 --> 00:19:52.870 align:middle line:90% 00:19:52.870 --> 00:19:55.020 align:middle line:90% So this is pretty clear. 00:19:55.020 --> 00:20:05.630 align:middle line:84% But note that the norm I get from this inner product, 00:20:05.630 --> 00:20:10.060 align:middle line:90% this is simply-- 00:20:10.060 --> 00:20:13.200 align:middle line:90% 00:20:13.200 --> 00:20:19.635 align:middle line:90% which was little l2 norm, right? 00:20:19.635 --> 00:20:22.960 align:middle line:90% 00:20:22.960 --> 00:20:23.950 align:middle line:90% OK. 00:20:23.950 --> 00:20:36.870 align:middle line:84% So these are two basic examples of a Hilbert space. 00:20:36.870 --> 00:20:43.560 align:middle line:84% We will in fact show that every separable Hilbert space 00:20:43.560 --> 00:20:46.860 align:middle line:90% can be in a inner product. 00:20:46.860 --> 00:20:49.560 align:middle line:84% And therefore length preserving way-- 00:20:49.560 --> 00:20:52.270 align:middle line:90% 00:20:52.270 --> 00:20:57.120 align:middle line:84% so that's usually called an isomorphism form-- 00:20:57.120 --> 00:21:04.950 align:middle line:84% can be mapped isometrically to either Cn or little l2. 00:21:04.950 --> 00:21:06.450 align:middle line:90% So these two kind kof-- 00:21:06.450 --> 00:21:12.820 align:middle line:84% if one wants to go about categorizing 00:21:12.820 --> 00:21:15.070 align:middle line:84% all of the possible Hilbert spaces 00:21:15.070 --> 00:21:17.680 align:middle line:84% as far as separable Hilbert spaces, which 00:21:17.680 --> 00:21:25.670 align:middle line:84% are the only reasonable ones, of course, 00:21:25.670 --> 00:21:28.870 align:middle line:84% one can come up with wild examples of Hilbert spaces 00:21:28.870 --> 00:21:29.590 align:middle line:90% which are not. 00:21:29.590 --> 00:21:31.660 align:middle line:90% Then these are the only two-- 00:21:31.660 --> 00:21:33.310 align:middle line:84% I shouldn't say two because this one 00:21:33.310 --> 00:21:34.540 align:middle line:90% is indexed by its dimension. 00:21:34.540 --> 00:21:36.970 align:middle line:84% But these are the only two types that you come up with, 00:21:36.970 --> 00:21:42.512 align:middle line:84% either a finite dimensional Cn, or it's isometric to little l2, 00:21:42.512 --> 00:21:43.012 align:middle line:90% OK. 00:21:43.012 --> 00:21:48.250 align:middle line:90% 00:21:48.250 --> 00:21:51.775 align:middle line:84% So but I will still write down another example for-- 00:21:51.775 --> 00:22:00.160 align:middle line:90% 00:22:00.160 --> 00:22:02.920 align:middle line:84% since we went to all that work, if E 00:22:02.920 --> 00:22:13.650 align:middle line:84% is a measurable subset of r and l2 of e, the big L2 of e, 00:22:13.650 --> 00:22:19.925 align:middle line:84% which, remember, this was the space of measurable functions 00:22:19.925 --> 00:22:25.755 align:middle line:84% f from e to c such that f squared e is finite. 00:22:25.755 --> 00:22:36.720 align:middle line:90% 00:22:36.720 --> 00:22:37.808 align:middle line:90% This is a Hilbert space. 00:22:37.808 --> 00:22:38.850 align:middle line:90% What's the inner product? 00:22:38.850 --> 00:22:45.270 align:middle line:90% 00:22:45.270 --> 00:22:50.070 align:middle line:84% The inner product of two elements in L2 00:22:50.070 --> 00:22:53.940 align:middle line:84% defines v, just the analog of little l2 where 00:22:53.940 --> 00:22:58.610 align:middle line:84% we replace the sum with an integral, f times g, 00:22:58.610 --> 00:23:03.660 align:middle line:84% OK, the Lesbesgue integral of f times the complex conjugate 00:23:03.660 --> 00:23:04.180 align:middle line:90% of g. 00:23:04.180 --> 00:23:29.890 align:middle line:90% 00:23:29.890 --> 00:23:42.600 align:middle line:84% OK, now, I wrote down capital L2 I wrote down little l2. 00:23:42.600 --> 00:23:45.960 align:middle line:84% But what about the other little lp and big LP spaces 00:23:45.960 --> 00:23:49.050 align:middle line:90% or any of those Hilbert spaces? 00:23:49.050 --> 00:23:56.470 align:middle line:84% So, of course, if I define the inner product in this way, 00:23:56.470 --> 00:24:01.150 align:middle line:84% as I did before, then that only induces the little l2 00:24:01.150 --> 00:24:03.970 align:middle line:90% and the big L2 norm. 00:24:03.970 --> 00:24:09.280 align:middle line:84% But is there perhaps some other kind of magic inner product 00:24:09.280 --> 00:24:14.440 align:middle line:84% out there that I can put on little lp or big LP 00:24:14.440 --> 00:24:20.410 align:middle line:84% so that I would get out the little lp or big LP 00:24:20.410 --> 00:24:22.420 align:middle line:84% norm, when I define the norm according 00:24:22.420 --> 00:24:25.850 align:middle line:90% to how I've been doing it. 00:24:25.850 --> 00:24:29.590 align:middle line:90% So the question on hand is-- 00:24:29.590 --> 00:24:39.340 align:middle line:90% 00:24:39.340 --> 00:24:51.235 align:middle line:84% the other little lp or big LP spaces, also Hilbert spaces. 00:24:51.235 --> 00:24:54.430 align:middle line:90% 00:24:54.430 --> 00:24:56.930 align:middle line:84% All right, so again it's clear that if I 00:24:56.930 --> 00:24:58.970 align:middle line:84% were to define the inner product in the way 00:24:58.970 --> 00:25:01.220 align:middle line:84% I did in the two examples, then that only 00:25:01.220 --> 00:25:03.860 align:middle line:90% is going to give the l2 norm. 00:25:03.860 --> 00:25:06.830 align:middle line:84% I'm asking now is there some magical inner product 00:25:06.830 --> 00:25:12.470 align:middle line:84% I can define on these spaces that spits out the little lp 00:25:12.470 --> 00:25:13.880 align:middle line:90% or big LP norm? 00:25:13.880 --> 00:25:16.490 align:middle line:90% 00:25:16.490 --> 00:25:18.330 align:middle line:90% So the answer is no. 00:25:18.330 --> 00:25:25.460 align:middle line:84% And there's in fact, a way to determine whether or not 00:25:25.460 --> 00:25:28.400 align:middle line:84% a space is a Hilbert space because if you think about it, 00:25:28.400 --> 00:25:34.600 align:middle line:84% the way we've come about introducing what 00:25:34.600 --> 00:25:37.060 align:middle line:84% a pre-Hilbert space is and a Hilbert space is, 00:25:37.060 --> 00:25:39.257 align:middle line:90% we first had a norm in our hand. 00:25:39.257 --> 00:25:40.840 align:middle line:84% And then we'd define an inner product. 00:25:40.840 --> 00:25:42.460 align:middle line:84% Well, if that's the way you're building your space, 00:25:42.460 --> 00:25:45.010 align:middle line:84% then you know automatically that it's a Hilbert space 00:25:45.010 --> 00:25:47.290 align:middle line:84% because you had an inner product first 00:25:47.290 --> 00:25:49.330 align:middle line:84% and then you define the norm second. 00:25:49.330 --> 00:25:54.760 align:middle line:84% Let's suppose the data you're given is some norm on a space. 00:25:54.760 --> 00:25:58.570 align:middle line:84% When can you determine if that norm 00:25:58.570 --> 00:26:01.390 align:middle line:90% comes from an inner product? 00:26:01.390 --> 00:26:06.190 align:middle line:84% That's the question that is underlying this question I 00:26:06.190 --> 00:26:07.090 align:middle line:90% wrote on the board. 00:26:07.090 --> 00:26:10.700 align:middle line:90% If I have a norm space-- 00:26:10.700 --> 00:26:13.190 align:middle line:84% all right, so my initial data is the norm. 00:26:13.190 --> 00:26:16.430 align:middle line:84% When can I tell that that norm comes from an inner product? 00:26:16.430 --> 00:26:25.550 align:middle line:84% And so you'll prove this basically 00:26:25.550 --> 00:26:27.170 align:middle line:90% by direct calculation. 00:26:27.170 --> 00:26:30.300 align:middle line:90% 00:26:30.300 --> 00:26:39.080 align:middle line:84% And this is the following parallelogram law, 00:26:39.080 --> 00:26:40.940 align:middle line:90% which is the following. 00:26:40.940 --> 00:26:59.350 align:middle line:84% If H is a pre-Hilbert space, then for all uv in H, 00:26:59.350 --> 00:27:06.700 align:middle line:84% if I take the norm of u plus v squared, and I 00:27:06.700 --> 00:27:10.660 align:middle line:84% add the norm of u minus v squared, 00:27:10.660 --> 00:27:15.760 align:middle line:84% this is equal to twice the norm of u squared 00:27:15.760 --> 00:27:18.880 align:middle line:90% plus the norm of v squared. 00:27:18.880 --> 00:27:25.230 align:middle line:84% So this is a condition that is stated 00:27:25.230 --> 00:27:26.895 align:middle line:90% purely in terms of the norm. 00:27:26.895 --> 00:27:33.210 align:middle line:90% 00:27:33.210 --> 00:27:56.680 align:middle line:84% Moreover, if h is a norm space satisfying star, 00:27:56.680 --> 00:28:00.830 align:middle line:90% then h is a pre-Hilbert space. 00:28:00.830 --> 00:28:04.359 align:middle line:90% 00:28:04.359 --> 00:28:09.600 align:middle line:84% OK, in other words, although it seems 00:28:09.600 --> 00:28:13.890 align:middle line:84% we defined Hilbert spaces or pre-Hilbert spaces 00:28:13.890 --> 00:28:16.540 align:middle line:84% initially in terms of an inner product, in fact, 00:28:16.540 --> 00:28:19.950 align:middle line:84% you can say a norm space is a pre-Hilbert space if 00:28:19.950 --> 00:28:23.860 align:middle line:84% and only if it satisfies this parallelogram law. 00:28:23.860 --> 00:28:27.420 align:middle line:84% So if you have on your hands, just the norm, 00:28:27.420 --> 00:28:30.990 align:middle line:84% then as long as that norm satisfies this identity, 00:28:30.990 --> 00:28:34.120 align:middle line:84% that norm can be derived from an inner product. 00:28:34.120 --> 00:28:42.230 align:middle line:84% So using this theorem, you can check that the answer is only 00:28:42.230 --> 00:28:47.210 align:middle line:90% for people to-- 00:28:47.210 --> 00:28:51.830 align:middle line:84% so in other words, if you can come with u and v 00:28:51.830 --> 00:28:54.920 align:middle line:84% so that this inequality is not satisfied 00:28:54.920 --> 00:28:56.980 align:middle line:90% when p is not equal to 2. 00:28:56.980 --> 00:28:57.480 align:middle line:90% OK. 00:28:57.480 --> 00:29:07.180 align:middle line:90% 00:29:07.180 --> 00:29:14.350 align:middle line:84% OK, so now we have the notion of a Hilbert space 00:29:14.350 --> 00:29:19.570 align:middle line:84% where this norm given in terms of an inner product, this space 00:29:19.570 --> 00:29:21.860 align:middle line:90% is complete. 00:29:21.860 --> 00:29:27.000 align:middle line:84% Now, we have an inner product, so we 00:29:27.000 --> 00:29:35.310 align:middle line:84% can start talking about vectors being 00:29:35.310 --> 00:29:45.030 align:middle line:84% orthogonal to other vectors or orthonormal sets, which 00:29:45.030 --> 00:29:48.660 align:middle line:84% when I first started lecturing about this stuff, 00:29:48.660 --> 00:29:52.350 align:middle line:84% I was already using the terminology, this thing being 00:29:52.350 --> 00:29:54.600 align:middle line:84% orthogonal to this thing or something like that. 00:29:54.600 --> 00:30:01.381 align:middle line:90% 00:30:01.381 --> 00:30:03.370 align:middle line:84% But of course, if you don't remember 00:30:03.370 --> 00:30:08.570 align:middle line:84% what those words mean from linear algebra, 00:30:08.570 --> 00:30:10.630 align:middle line:90% I'll quickly remind you. 00:30:10.630 --> 00:30:13.975 align:middle line:84% So suppose we're in a pre-Hilbert space. 00:30:13.975 --> 00:30:26.760 align:middle line:90% 00:30:26.760 --> 00:30:29.040 align:middle line:84% We say that two elements, u and v, 00:30:29.040 --> 00:30:41.210 align:middle line:84% are orthogonal, if their inner product is zero. 00:30:41.210 --> 00:30:43.940 align:middle line:90% 00:30:43.940 --> 00:30:49.800 align:middle line:84% If instead of saying orthogonal I want to write this in words, 00:30:49.800 --> 00:30:53.820 align:middle line:90% I'll write u per v, OK. 00:30:53.820 --> 00:31:06.490 align:middle line:90% 00:31:06.490 --> 00:31:09.610 align:middle line:90% So throughout H is going to be-- 00:31:09.610 --> 00:31:13.840 align:middle line:84% well, a pre-Hilbert space if I don't actually write it. 00:31:13.840 --> 00:31:33.740 align:middle line:84% If h is a pre-Hilbert space, a subset, 00:31:33.740 --> 00:31:38.430 align:middle line:84% which I'll denote by a lambda, lambda, and capital lambda, 00:31:38.430 --> 00:31:43.260 align:middle line:84% so just some subset indexed by some indexing set capital 00:31:43.260 --> 00:31:43.760 align:middle line:90% lambda. 00:31:43.760 --> 00:31:46.430 align:middle line:90% 00:31:46.430 --> 00:31:58.830 align:middle line:84% We say this set is orthonormal if for all lambda 00:31:58.830 --> 00:32:01.740 align:middle line:84% each one of these vectors in the subset has unit length. 00:32:01.740 --> 00:32:05.720 align:middle line:90% 00:32:05.720 --> 00:32:17.240 align:middle line:84% And I need two different indexed elements, 00:32:17.240 --> 00:32:24.650 align:middle line:84% implies that they are orthogonal. 00:32:24.650 --> 00:32:25.150 align:middle line:90% OK. 00:32:25.150 --> 00:32:28.950 align:middle line:90% 00:32:28.950 --> 00:32:33.350 align:middle line:84% So maybe this notation scares you 00:32:33.350 --> 00:32:35.420 align:middle line:84% because what's this indexing set? 00:32:35.420 --> 00:32:37.290 align:middle line:84% Typically we just use the natural number. 00:32:37.290 --> 00:32:40.220 align:middle line:84% So let me just make that remark although, I'll 00:32:40.220 --> 00:32:43.820 align:middle line:84% make a few remarks in general about orthonormal sets that 00:32:43.820 --> 00:32:47.000 align:middle line:84% are not necessarily indexed by the natural numbers 00:32:47.000 --> 00:33:11.870 align:middle line:84% will mainly be interested in a finite set or a countably 00:33:11.870 --> 00:33:12.560 align:middle line:90% infinite set. 00:33:12.560 --> 00:33:16.652 align:middle line:90% 00:33:16.652 --> 00:33:23.960 align:middle line:84% OK, so although an orthonormal subset of Hs 00:33:23.960 --> 00:33:27.380 align:middle line:84% could be a very crazy type of subset, 00:33:27.380 --> 00:33:29.120 align:middle line:84% mainly we're going to be interested only 00:33:29.120 --> 00:33:34.730 align:middle line:84% in finite or countable, countably infinite 00:33:34.730 --> 00:33:35.645 align:middle line:90% orthonormal sets. 00:33:35.645 --> 00:33:43.530 align:middle line:90% 00:33:43.530 --> 00:33:47.750 align:middle line:90% So what are some examples? 00:33:47.750 --> 00:33:50.795 align:middle line:90% 00:33:50.795 --> 00:34:00.170 align:middle line:90% OK, so simplest examples if-- 00:34:00.170 --> 00:34:15.250 align:middle line:84% so this is an example of a set of orthonormal-- 00:34:15.250 --> 00:34:17.860 align:middle line:90% or an orthonormal subset of c2. 00:34:17.860 --> 00:34:23.464 align:middle line:90% Or this is a 1. 00:34:23.464 --> 00:34:27.100 align:middle line:90% 00:34:27.100 --> 00:34:32.790 align:middle line:84% This is also an example of an orthonormal subset of c3, 00:34:32.790 --> 00:34:38.650 align:middle line:84% let's say, using notation from before. 00:34:38.650 --> 00:34:44.830 align:middle line:84% If I denote by e sub n, this is the sequence consisting 00:34:44.830 --> 00:34:48.250 align:middle line:84% of zeros up until I hit the end spot and then 00:34:48.250 --> 00:34:59.300 align:middle line:84% 0 afterwards and entry, which is an element of little l2, 00:34:59.300 --> 00:35:12.640 align:middle line:84% then this is an orthonormal subset of l2, OK. 00:35:12.640 --> 00:35:16.660 align:middle line:90% 00:35:16.660 --> 00:35:23.450 align:middle line:90% One other example is-- 00:35:23.450 --> 00:35:28.130 align:middle line:84% Let's look at the functions 1 over square root of 2pi times 00:35:28.130 --> 00:35:32.100 align:middle line:90% e to the i and x. 00:35:32.100 --> 00:35:37.605 align:middle line:84% And let's think of these as elements of l2 minus pi to pi. 00:35:37.605 --> 00:35:42.450 align:middle line:90% 00:35:42.450 --> 00:35:56.995 align:middle line:84% Then this is an orthonormal subset of-- 00:35:56.995 --> 00:36:01.000 align:middle line:90% 00:36:01.000 --> 00:36:05.930 align:middle line:84% I might write O-N instead of writing out orthonormal, 00:36:05.930 --> 00:36:10.640 align:middle line:84% but this is an orthonormal subset of L2. 00:36:10.640 --> 00:36:17.940 align:middle line:84% If I take the inner product of two of these guys, I get-- 00:36:17.940 --> 00:36:21.960 align:middle line:90% 00:36:21.960 --> 00:36:24.510 align:middle line:84% let's say I just look at two different ones. 00:36:24.510 --> 00:36:28.140 align:middle line:90% 00:36:28.140 --> 00:36:30.540 align:middle line:84% let's say m does not equal n, so then I 00:36:30.540 --> 00:36:39.070 align:middle line:84% take the inner product of imx with inner product of inx, 00:36:39.070 --> 00:36:41.360 align:middle line:90% complex conjugate. 00:36:41.360 --> 00:36:44.920 align:middle line:84% So that's the inner product on big L2, 00:36:44.920 --> 00:36:52.850 align:middle line:84% and this is equal to 1 over 2 pi times minus pi to pi. 00:36:52.850 --> 00:36:57.580 align:middle line:84% Now if m equals n, I just get the length of e to the i, 00:36:57.580 --> 00:36:59.860 align:middle line:90% nmx squared. 00:36:59.860 --> 00:37:02.380 align:middle line:84% The length of e to the i, mx is 1. 00:37:02.380 --> 00:37:04.750 align:middle line:84% And therefore, I would just get 1 here, 00:37:04.750 --> 00:37:08.230 align:middle line:90% dx over 2 pi, which gives me 1. 00:37:08.230 --> 00:37:09.710 align:middle line:90% But now they're not equal. 00:37:09.710 --> 00:37:17.630 align:middle line:84% So I get e to the i, m minus nx, dx. 00:37:17.630 --> 00:37:28.190 align:middle line:84% And now the integral of this creature-- so here e to the i, 00:37:28.190 --> 00:37:35.940 align:middle line:84% let's say I had to come up with some e to the i 00:37:35.940 --> 00:37:40.620 align:middle line:84% y when y is a real number, this is by definition cosine of y 00:37:40.620 --> 00:37:42.270 align:middle line:90% plus i sine y. 00:37:42.270 --> 00:37:45.830 align:middle line:90% 00:37:45.830 --> 00:37:49.400 align:middle line:84% Now you can check that fundamental theorem of calculus 00:37:49.400 --> 00:37:54.590 align:middle line:84% still holds for e to the i, m minus n 00:37:54.590 --> 00:38:01.790 align:middle line:84% over x And this equals e to the im minus nx over i times 00:38:01.790 --> 00:38:03.650 align:middle line:90% m minus n. 00:38:03.650 --> 00:38:09.720 align:middle line:84% And e to the i, m minus n times x is 2 pi periodic. 00:38:09.720 --> 00:38:14.210 align:middle line:84% So when I evaluate it at minus pi, 00:38:14.210 --> 00:38:16.860 align:middle line:84% I get the same value, if I evaluated at pi. 00:38:16.860 --> 00:38:19.842 align:middle line:90% So this equals 0. 00:38:19.842 --> 00:38:22.280 align:middle line:90% OK. 00:38:22.280 --> 00:38:26.805 align:middle line:84% So that's an orthonormal subset of big L2. 00:38:26.805 --> 00:38:36.900 align:middle line:90% 00:38:36.900 --> 00:38:40.770 align:middle line:90% So this collection of vectors-- 00:38:40.770 --> 00:38:43.170 align:middle line:84% and I'm calling them vectors, but this collection 00:38:43.170 --> 00:38:45.810 align:middle line:84% of elements in big L2 is still countable, 00:38:45.810 --> 00:38:48.638 align:middle line:84% even though I'm indexing it by the integers rather 00:38:48.638 --> 00:38:49.680 align:middle line:90% than the natural numbers. 00:38:49.680 --> 00:38:54.840 align:middle line:90% 00:38:54.840 --> 00:38:56.460 align:middle line:90% OK, so we have the following. 00:38:56.460 --> 00:39:02.360 align:middle line:90% 00:39:02.360 --> 00:39:05.180 align:middle line:84% Now most of what I'm going to say with 00:39:05.180 --> 00:39:09.215 align:middle line:84% regards to countable subsets of-- 00:39:09.215 --> 00:39:13.280 align:middle line:90% 00:39:13.280 --> 00:39:17.990 align:middle line:84% so countable orthonormal sets still carries through 00:39:17.990 --> 00:39:21.470 align:middle line:84% to possibly uncountable orthonormal sets 00:39:21.470 --> 00:39:27.620 align:middle line:84% where now a infinite sum over an uncountable number of elements 00:39:27.620 --> 00:39:29.780 align:middle line:84% has to be defined in a precise way. 00:39:29.780 --> 00:39:34.310 align:middle line:84% But I will really just stick mostly to the countable case, 00:39:34.310 --> 00:39:37.160 align:middle line:84% and if you're interested, you can always look that stuff up. 00:39:37.160 --> 00:39:39.670 align:middle line:90% 00:39:39.670 --> 00:39:40.665 align:middle line:90% So I should say-- 00:39:40.665 --> 00:39:44.430 align:middle line:90% 00:39:44.430 --> 00:39:50.340 align:middle line:84% what I haven't said is that whether or not the lp spaces, 00:39:50.340 --> 00:39:53.980 align:middle line:84% let's say over an interval ab or over r, 00:39:53.980 --> 00:39:57.945 align:middle line:84% are separable or not or even over a-- 00:39:57.945 --> 00:40:09.850 align:middle line:90% 00:40:09.850 --> 00:40:11.680 align:middle line:90% or even over a-- 00:40:11.680 --> 00:40:15.160 align:middle line:84% so let's stick to either a closed and bounded interval 00:40:15.160 --> 00:40:18.120 align:middle line:90% or r. 00:40:18.120 --> 00:40:23.010 align:middle line:84% Now, why are these spaces separable, 00:40:23.010 --> 00:40:27.990 align:middle line:84% meaning they have a countably dense subset, so 00:40:27.990 --> 00:40:29.460 align:middle line:90% including big L2? 00:40:29.460 --> 00:40:32.880 align:middle line:90% 00:40:32.880 --> 00:40:35.550 align:middle line:84% The reason is the following, so what 00:40:35.550 --> 00:40:44.490 align:middle line:84% you proved in the assignment is that the continuous functions 00:40:44.490 --> 00:40:48.780 align:middle line:84% are dense and little and big LP for p between p 00:40:48.780 --> 00:40:53.010 align:middle line:84% equals 1 to infinity, strictly less than infinity. 00:40:53.010 --> 00:40:55.600 align:middle line:84% OK, they're not dense in l infinity. 00:40:55.600 --> 00:40:58.640 align:middle line:84% So as long as you stay away from that, they're dense, OK. 00:40:58.640 --> 00:41:04.230 align:middle line:90% 00:41:04.230 --> 00:41:06.870 align:middle line:84% So continuous functions are dense in lp. 00:41:06.870 --> 00:41:09.330 align:middle line:84% Now, what's one way to approximate 00:41:09.330 --> 00:41:11.910 align:middle line:90% a continuous function? 00:41:11.910 --> 00:41:14.850 align:middle line:84% Going back to your introductory analysis class, 00:41:14.850 --> 00:41:17.130 align:middle line:84% hopefully you covered what's called the Weierstrauss 00:41:17.130 --> 00:41:18.930 align:middle line:84% approximation theorem, which says 00:41:18.930 --> 00:41:21.100 align:middle line:84% that for every continuous function, 00:41:21.100 --> 00:41:23.910 align:middle line:84% you can approximate it uniformly on the interval 00:41:23.910 --> 00:41:29.660 align:middle line:90% by a polynomial, OK. 00:41:29.660 --> 00:41:32.720 align:middle line:84% So that shows that polynomials are 00:41:32.720 --> 00:41:39.410 align:middle line:84% dense and all the lp spaces, of course, not l infinity. 00:41:39.410 --> 00:41:44.150 align:middle line:84% Now how do you go from the set of polynomials, 00:41:44.150 --> 00:41:47.590 align:middle line:84% which is uncountable to a count, which is dense in LP, 00:41:47.590 --> 00:41:51.340 align:middle line:84% but is uncountable to a countable dense subset, which 00:41:51.340 --> 00:41:53.200 align:middle line:90% is dense in LP. 00:41:53.200 --> 00:41:54.940 align:middle line:90% Make everything rational. 00:41:54.940 --> 00:41:59.230 align:middle line:84% OK, the set of polynomials with rational coefficients 00:41:59.230 --> 00:42:00.700 align:middle line:90% is, in fact, countable. 00:42:00.700 --> 00:42:03.670 align:middle line:90% 00:42:03.670 --> 00:42:06.460 align:middle line:84% And you can approximate every polynomial 00:42:06.460 --> 00:42:11.470 align:middle line:84% with real coefficients on a closed and bounded interval 00:42:11.470 --> 00:42:15.730 align:middle line:84% by a polynomial with rational coefficients. 00:42:15.730 --> 00:42:18.840 align:middle line:84% It's not too difficult to believe. 00:42:18.840 --> 00:42:23.550 align:middle line:84% And therefore, the polynomials with rational coefficients 00:42:23.550 --> 00:42:29.770 align:middle line:84% are dense in the lp spaces as long as I'm not in l infinity. 00:42:29.770 --> 00:42:33.460 align:middle line:84% And therefore all the lp spaces are separable. 00:42:33.460 --> 00:42:40.550 align:middle line:84% All right, now the little lp spaces are also 00:42:40.550 --> 00:42:45.230 align:middle line:84% separable because except for L infinity also, 00:42:45.230 --> 00:42:48.760 align:middle line:90% which is not separable-- 00:42:48.760 --> 00:42:54.990 align:middle line:84% because, first off, a dense subset of the, let's say, 00:42:54.990 --> 00:42:57.390 align:middle line:84% little l2-- let's make things definite-- 00:42:57.390 --> 00:43:00.660 align:middle line:84% is the subspace consisting of all sequences, which 00:43:00.660 --> 00:43:03.850 align:middle line:90% terminate after some entry. 00:43:03.850 --> 00:43:07.350 align:middle line:84% In other words, it's 0 after that, OK. 00:43:07.350 --> 00:43:09.000 align:middle line:84% Convince yourself that this subspace 00:43:09.000 --> 00:43:11.380 align:middle line:84% of all finitely terminating sequences, 00:43:11.380 --> 00:43:14.790 align:middle line:84% this is dense in little lp for every-- 00:43:14.790 --> 00:43:19.630 align:middle line:84% for p between one an infinity, not equal to infinity. 00:43:19.630 --> 00:43:22.450 align:middle line:84% So finitely terminating sequences 00:43:22.450 --> 00:43:25.870 align:middle line:90% are dense and little lp. 00:43:25.870 --> 00:43:31.920 align:middle line:84% Unfortunately, that's, again, an uncountable-- 00:43:31.920 --> 00:43:33.580 align:middle line:90% I mean, it's a subspace. 00:43:33.580 --> 00:43:34.980 align:middle line:90% So it's going to be uncountable. 00:43:34.980 --> 00:43:37.530 align:middle line:84% So how do I get now a countable thing again? 00:43:37.530 --> 00:43:39.310 align:middle line:84% I replace everything by rational numbers. 00:43:39.310 --> 00:43:43.740 align:middle line:84% So if I can approximate every sequence which 00:43:43.740 --> 00:43:48.390 align:middle line:84% terminates after a certain point consisting of real numbers 00:43:48.390 --> 00:43:51.720 align:middle line:84% by a sequence full of just rationals terminating 00:43:51.720 --> 00:43:55.320 align:middle line:84% after a certain point by just choosing the rationals very 00:43:55.320 --> 00:43:57.480 align:middle line:90% close to those real numbers. 00:43:57.480 --> 00:44:02.010 align:middle line:84% And I didn't say this explicitly a minute ago that we still 00:44:02.010 --> 00:44:04.080 align:middle line:84% have the density of the rationals and the reals. 00:44:04.080 --> 00:44:06.810 align:middle line:84% For every real, I can find a rational very close to it. 00:44:06.810 --> 00:44:09.810 align:middle line:90% 00:44:09.810 --> 00:44:13.530 align:middle line:84% So this is thinking that goes on. 00:44:13.530 --> 00:44:17.190 align:middle line:84% But now the set of all finitely terminating 00:44:17.190 --> 00:44:19.720 align:middle line:84% sequences with rational coefficients. 00:44:19.720 --> 00:44:22.530 align:middle line:90% This is a countable set. 00:44:22.530 --> 00:44:26.040 align:middle line:84% And that countable set is dense in little lp. 00:44:26.040 --> 00:44:27.630 align:middle line:84% And therefore little lp is separable 00:44:27.630 --> 00:44:32.550 align:middle line:84% as long as p is between 1 and infinity, excluding infinity. 00:44:32.550 --> 00:44:36.940 align:middle line:84% OK, so I said at the beginning that we're 00:44:36.940 --> 00:44:40.810 align:middle line:84% going to be mainly interested in separable spaces 00:44:40.810 --> 00:44:43.690 align:middle line:84% without actually saying why little lp 00:44:43.690 --> 00:44:46.540 align:middle line:84% or little l2 and big L2 are separable. 00:44:46.540 --> 00:44:49.060 align:middle line:90% 00:44:49.060 --> 00:44:53.200 align:middle line:84% But I just gave you the argument by word of mouth now instead 00:44:53.200 --> 00:44:54.735 align:middle line:90% of actually writing it down. 00:44:54.735 --> 00:45:03.000 align:middle line:84% OK, so we have the following Bessel's inequality equality 00:45:03.000 --> 00:45:12.920 align:middle line:84% for countable orthonormal subsets. 00:45:12.920 --> 00:45:20.170 align:middle line:90% So if the n is-- 00:45:20.170 --> 00:45:22.210 align:middle line:90% well, let me just put n here. 00:45:22.210 --> 00:45:32.090 align:middle line:84% If this is a countable, meaning it's either finite 00:45:32.090 --> 00:45:43.910 align:middle line:84% or it's countably infinite, when it's 00:45:43.910 --> 00:46:11.650 align:middle line:84% a countable orthonormal subset of a pre-Hilbert space, h, 00:46:11.650 --> 00:46:18.590 align:middle line:84% and for all u [? en ?] h, if I look at the sum of squares 00:46:18.590 --> 00:46:22.720 align:middle line:84% u in a product e sub n, this is less than 00:46:22.720 --> 00:46:29.700 align:middle line:84% or equal to the norm of u squared, OK. 00:46:29.700 --> 00:46:33.280 align:middle line:90% 00:46:33.280 --> 00:46:37.120 align:middle line:84% So our discussion here of orthonormal subsets 00:46:37.120 --> 00:46:40.510 align:middle line:84% is taking place within a pre-Hilbert space. 00:46:40.510 --> 00:46:43.565 align:middle line:84% We don't need Hilbert spaces to talk about these concepts. 00:46:43.565 --> 00:46:45.190 align:middle line:84% Yeah, but when we're in a Hilbert space 00:46:45.190 --> 00:46:47.710 align:middle line:84% and we have a certain or the normal subset, 00:46:47.710 --> 00:46:52.726 align:middle line:84% that would be important that we're in a Hilbert space. 00:46:52.726 --> 00:46:59.690 align:middle line:90% OK, so the proof is-- 00:46:59.690 --> 00:47:01.310 align:middle line:90% let's do the finite case first. 00:47:01.310 --> 00:47:07.190 align:middle line:90% 00:47:07.190 --> 00:47:15.370 align:middle line:84% So suppose I have a finite collection of orthonormal 00:47:15.370 --> 00:47:18.370 align:middle line:84% or a finite orthonormal subset of h or a finite-- 00:47:18.370 --> 00:47:20.110 align:middle line:84% or I'll often say finite collection 00:47:20.110 --> 00:47:21.685 align:middle line:90% of all normal vectors in h. 00:47:21.685 --> 00:47:25.300 align:middle line:90% 00:47:25.300 --> 00:47:32.148 align:middle line:84% And subset of h in on, standing in for orthonormal. 00:47:32.148 --> 00:47:40.460 align:middle line:90% 00:47:40.460 --> 00:47:42.920 align:middle line:84% Then let me just record a few identities, which 00:47:42.920 --> 00:47:48.170 align:middle line:84% are pretty easy to verify sum from n equals 1 to n 00:47:48.170 --> 00:47:51.290 align:middle line:90% of u inner product en, en. 00:47:51.290 --> 00:47:53.990 align:middle line:90% 00:47:53.990 --> 00:47:55.835 align:middle line:84% If I take the norm of this thing squared-- 00:47:55.835 --> 00:47:58.730 align:middle line:90% 00:47:58.730 --> 00:47:59.880 align:middle line:90% let's compute this out. 00:47:59.880 --> 00:48:04.230 align:middle line:84% This is the inner product of with the understanding 00:48:04.230 --> 00:48:10.380 align:middle line:84% that n is going from 1 to capital N. Let's 00:48:10.380 --> 00:48:11.760 align:middle line:90% use a different index here. 00:48:11.760 --> 00:48:17.180 align:middle line:90% 00:48:17.180 --> 00:48:18.940 align:middle line:90% And this is equal to nmuen. 00:48:18.940 --> 00:48:23.950 align:middle line:90% 00:48:23.950 --> 00:48:28.060 align:middle line:90% Now I have this sum here of-- 00:48:28.060 --> 00:48:38.430 align:middle line:84% oh, I'm leaving out em of u, vm, the complex conjugate 00:48:38.430 --> 00:48:41.707 align:middle line:84% times the inner product of en with em. 00:48:41.707 --> 00:48:43.290 align:middle line:84% That's the two vectors that are taking 00:48:43.290 --> 00:48:45.450 align:middle line:90% the complex conjugate of. 00:48:45.450 --> 00:48:47.010 align:middle line:90% This is a number here. 00:48:47.010 --> 00:48:47.760 align:middle line:90% It just comes out. 00:48:47.760 --> 00:48:50.040 align:middle line:84% This number here gets hit with a complex conjugate 00:48:50.040 --> 00:48:50.970 align:middle line:90% when it comes out. 00:48:50.970 --> 00:48:54.750 align:middle line:84% Now the inner product of en with em is 0. 00:48:54.750 --> 00:48:57.300 align:middle line:84% When n does not equal m, it is equal to 1 00:48:57.300 --> 00:49:00.870 align:middle line:84% because it's equal to the norm of en squared when n equals m. 00:49:00.870 --> 00:49:03.210 align:middle line:84% So all I pick up from this double sum, 00:49:03.210 --> 00:49:05.250 align:middle line:84% which is just a finite double sum, 00:49:05.250 --> 00:49:13.540 align:middle line:84% is when n equals m and n going from 1 to n. 00:49:13.540 --> 00:49:18.850 align:middle line:90% 00:49:18.850 --> 00:49:30.070 align:middle line:84% And therefore, this is equal to uen squared, OK. 00:49:30.070 --> 00:49:37.070 align:middle line:90% 00:49:37.070 --> 00:49:43.720 align:middle line:84% And that's one formula I want to have. 00:49:43.720 --> 00:49:48.730 align:middle line:84% Another one is that if I take the inner product of u with m 00:49:48.730 --> 00:49:55.260 align:middle line:90% equals 1 to the n of the sum. 00:49:55.260 --> 00:50:00.870 align:middle line:84% And maybe you recognize what this sum here actually is. 00:50:00.870 --> 00:50:02.140 align:middle line:90% I'll say so in a minute. 00:50:02.140 --> 00:50:07.320 align:middle line:84% So this is equal to sum from n equals 1 to n of-- 00:50:07.320 --> 00:50:12.480 align:middle line:84% so u inner product e sub n times this number. 00:50:12.480 --> 00:50:15.170 align:middle line:84% This number comes out and gives me a complex conjugate. 00:50:15.170 --> 00:50:28.610 align:middle line:90% 00:50:28.610 --> 00:50:29.110 align:middle line:90% OK. 00:50:29.110 --> 00:50:41.210 align:middle line:90% 00:50:41.210 --> 00:50:46.580 align:middle line:84% And therefore, zero, which is bigger than 00:50:46.580 --> 00:50:57.750 align:middle line:84% or equal to the norm of u minus n, equals 1 to n of uen, en. 00:50:57.750 --> 00:51:01.260 align:middle line:84% Now if you remember back from linear algebra or from calculus 00:51:01.260 --> 00:51:07.690 align:middle line:84% that if I have orthonormal vectors, and I have a vector u, 00:51:07.690 --> 00:51:10.510 align:middle line:84% this quantity is nothing but the projection of u 00:51:10.510 --> 00:51:14.360 align:middle line:84% onto the span of those orthonormal vectors. 00:51:14.360 --> 00:51:18.340 align:middle line:84% So what I'm looking at here is, if you 00:51:18.340 --> 00:51:23.260 align:middle line:84% like, the norm part of u that's orthogonal 00:51:23.260 --> 00:51:25.570 align:middle line:90% to these finitely many vectors. 00:51:25.570 --> 00:51:31.529 align:middle line:90% 00:51:31.529 --> 00:51:34.100 align:middle line:84% OK, so this thing is bigger than or equal to zero. 00:51:34.100 --> 00:51:37.970 align:middle line:84% We use that formula of how to compute the norm of something 00:51:37.970 --> 00:51:38.600 align:middle line:90% plus something. 00:51:38.600 --> 00:51:43.190 align:middle line:84% And this is equal to norm of u squared plus norm n 00:51:43.190 --> 00:51:50.540 align:middle line:84% equals 1 to nuen, en squared minus 2 times 00:51:50.540 --> 00:51:56.660 align:middle line:84% the real part of u inner product sum from n equals 1 to n 00:51:56.660 --> 00:51:59.870 align:middle line:90% of uen, en. 00:51:59.870 --> 00:52:02.440 align:middle line:90% 00:52:02.440 --> 00:52:05.510 align:middle line:84% OK, and now we know what all of these things are. 00:52:05.510 --> 00:52:07.730 align:middle line:90% This is equal to-- 00:52:07.730 --> 00:52:10.940 align:middle line:84% I'm not even going to go over-- this is equal to this quantity 00:52:10.940 --> 00:52:13.580 align:middle line:90% here as is this inner product. 00:52:13.580 --> 00:52:15.270 align:middle line:84% It's also equal to this thing here. 00:52:15.270 --> 00:52:17.900 align:middle line:84% So the real part is equal to this thing 00:52:17.900 --> 00:52:21.290 align:middle line:84% here because this is a real number. 00:52:21.290 --> 00:52:24.480 align:middle line:84% Since this comes with a 2, I cancel one of those. 00:52:24.480 --> 00:52:30.740 align:middle line:84% So I get norm of u squared minus sum from n equals 1 to n 00:52:30.740 --> 00:52:39.110 align:middle line:84% of u, en squared, which is exactly what I wanted 00:52:39.110 --> 00:52:40.460 align:middle line:90% to prove for the finit case. 00:52:40.460 --> 00:52:47.000 align:middle line:90% 00:52:47.000 --> 00:52:50.105 align:middle line:84% But the infinite case then follows from the finite case. 00:52:50.105 --> 00:53:00.050 align:middle line:90% 00:53:00.050 --> 00:53:05.270 align:middle line:84% And by letting capital N go to infinity-- so infinite case, 00:53:05.270 --> 00:53:16.310 align:middle line:84% suppose en equals 1 to infinity is an orthonormal normal subset 00:53:16.310 --> 00:53:24.150 align:middle line:84% of H, then we know that for all N, capital N, 00:53:24.150 --> 00:53:28.840 align:middle line:84% we have that sum over n equals 1 to n of uen. 00:53:28.840 --> 00:53:32.470 align:middle line:90% 00:53:32.470 --> 00:53:36.110 align:middle line:84% This is less than or equal to the norm of u squared. 00:53:36.110 --> 00:53:40.240 align:middle line:84% So I can just send capital N to infinity 00:53:40.240 --> 00:53:48.880 align:middle line:84% to get that the sum equals 1 to infinity of norm of un squared 00:53:48.880 --> 00:53:52.435 align:middle line:84% is less than or equal to the norm of u squared. 00:53:52.435 --> 00:53:55.110 align:middle line:90% 00:53:55.110 --> 00:53:55.610 align:middle line:90% OK. 00:53:55.610 --> 00:54:18.095 align:middle line:90% 00:54:18.095 --> 00:54:18.595 align:middle line:90% OK. 00:54:18.595 --> 00:54:21.130 align:middle line:90% 00:54:21.130 --> 00:54:26.620 align:middle line:84% So orthonormal subsets we can define 00:54:26.620 --> 00:54:30.340 align:middle line:84% as a collection of vectors that have unit length 00:54:30.340 --> 00:54:34.945 align:middle line:84% and are mutually orthogonal to each other. 00:54:34.945 --> 00:54:43.330 align:middle line:90% 00:54:43.330 --> 00:54:46.600 align:middle line:90% Now just orthonormal subsets-- 00:54:46.600 --> 00:54:50.920 align:middle line:84% just any old orthonormal subset is not really the most useful 00:54:50.920 --> 00:55:02.160 align:middle line:84% thing, if we're trying to study the entire Hilbert's, 00:55:02.160 --> 00:55:08.720 align:middle line:84% or pre-Hilbert space h, because we 00:55:08.720 --> 00:55:10.760 align:middle line:84% may miss something if we leave out 00:55:10.760 --> 00:55:13.385 align:middle line:90% certain orthonromal vectors. 00:55:13.385 --> 00:55:29.510 align:middle line:90% 00:55:29.510 --> 00:55:36.790 align:middle line:84% But a more useful type orthonormal set 00:55:36.790 --> 00:55:40.390 align:middle line:84% is a maximal orthonormal set, which 00:55:40.390 --> 00:55:47.270 align:middle line:84% is defined as follows, an orthonormal subset 00:55:47.270 --> 00:55:57.800 align:middle line:84% e lambda, lambda and capital lambda of a pre-Hilbert space 00:55:57.800 --> 00:56:06.670 align:middle line:90% H is maximal. 00:56:06.670 --> 00:56:11.310 align:middle line:84% So again, if having a possibly uncountable collection 00:56:11.310 --> 00:56:16.070 align:middle line:84% of orthonormal subsets indexed by some indexing set 00:56:16.070 --> 00:56:18.560 align:middle line:84% makes you uncomfortable, replace this 00:56:18.560 --> 00:56:22.130 align:middle line:84% with n, where in is going from 1 to capital 00:56:22.130 --> 00:56:25.080 align:middle line:84% N or n is going from 1 to infinity, so 00:56:25.080 --> 00:56:27.330 align:middle line:84% a countable collection, if you like. 00:56:27.330 --> 00:56:33.320 align:middle line:84% But I'm stating this so that you know that something 00:56:33.320 --> 00:56:34.680 align:middle line:90% more general is true. 00:56:34.680 --> 00:56:39.910 align:middle line:90% So this is maximal if what? 00:56:39.910 --> 00:56:44.030 align:middle line:90% The following holds. 00:56:44.030 --> 00:56:51.590 align:middle line:84% If u is an h and u is orthogonal to everything 00:56:51.590 --> 00:56:58.785 align:middle line:84% in this orthonormal subset, this implies that u is 0. 00:56:58.785 --> 00:57:01.540 align:middle line:90% 00:57:01.540 --> 00:57:04.010 align:middle line:90% OK. 00:57:04.010 --> 00:57:15.270 align:middle line:84% So an example, of course, is you can 00:57:15.270 --> 00:57:26.490 align:middle line:84% check that this collection here is maximal orthonormal subset 00:57:26.490 --> 00:57:31.320 align:middle line:90% of c2, a non-example. 00:57:31.320 --> 00:57:36.120 align:middle line:84% You can maybe see this coming, the one we had a minute ago. 00:57:36.120 --> 00:57:40.440 align:middle line:90% 00:57:40.440 --> 00:57:48.260 align:middle line:84% This is not maximal since there's 00:57:48.260 --> 00:57:51.210 align:middle line:84% a vector that has inner product zero with both of these 00:57:51.210 --> 00:57:56.990 align:middle line:84% but is not zero, since this should be c3. 00:57:56.990 --> 00:58:08.830 align:middle line:90% 00:58:08.830 --> 00:58:13.630 align:middle line:84% You just write it loosely this way since this vector 010 is 00:58:13.630 --> 00:58:16.930 align:middle line:84% orthogonal to both of these, but it's not 0. 00:58:16.930 --> 00:58:19.480 align:middle line:84% Maximal means if you're orthogonal to everything 00:58:19.480 --> 00:58:22.300 align:middle line:84% in your collection, then it has to be zero. 00:58:22.300 --> 00:58:29.700 align:middle line:84% But this is non-zero, and it's orthogonal to everything 00:58:29.700 --> 00:58:33.030 align:middle line:90% to these two vectors there. 00:58:33.030 --> 00:58:43.590 align:middle line:84% Another example is, again, with the notation from before. 00:58:43.590 --> 00:58:48.210 align:middle line:84% This is where this is the sequence that 00:58:48.210 --> 00:58:51.930 align:middle line:84% is 0, except for the end spot where it is 1. 00:58:51.930 --> 00:59:03.110 align:middle line:84% This is maximal orthonormal subset of little l2. 00:59:03.110 --> 00:59:03.610 align:middle line:90% OK. 00:59:03.610 --> 00:59:09.700 align:middle line:90% 00:59:09.700 --> 00:59:12.430 align:middle line:84% Now, what we're going to see very shortly 00:59:12.430 --> 00:59:20.880 align:middle line:84% is that if we have a countably infinite maximal subset 00:59:20.880 --> 00:59:23.670 align:middle line:84% of a Hilbert space, then that set 00:59:23.670 --> 00:59:26.580 align:middle line:84% serves kind of the same purpose as a basis 00:59:26.580 --> 00:59:31.110 align:middle line:84% as an orthonormal basis does in linear algebra. 00:59:31.110 --> 00:59:36.090 align:middle line:84% I mean, if you look at here, already this orthonormal set 00:59:36.090 --> 00:59:36.870 align:middle line:90% is maximal. 00:59:36.870 --> 00:59:40.140 align:middle line:84% And it also forms the basis for c2. 00:59:40.140 --> 00:59:41.790 align:middle line:90% This was not maximal. 00:59:41.790 --> 00:59:45.900 align:middle line:84% And you see it doesn't form a basis for c3. 00:59:45.900 --> 00:59:55.270 align:middle line:84% So maximal is going to give us a condition, which is equally 00:59:55.270 --> 00:59:56.800 align:middle line:90% useful as being a basis. 00:59:56.800 --> 00:59:59.140 align:middle line:90% But it won't be a Hamel base. 00:59:59.140 --> 01:00:02.230 align:middle line:84% These subsets won't be a Hamel basis in the sense 01:00:02.230 --> 01:00:05.560 align:middle line:84% that every vector can be written as a finite linear combination 01:00:05.560 --> 01:00:09.970 align:middle line:84% of the elements of a maximal orthonormal subset. 01:00:09.970 --> 01:00:12.430 align:middle line:84% But what is true is that we can write it 01:00:12.430 --> 01:00:18.230 align:middle line:84% as possibly an infinite sum involving 01:00:18.230 --> 01:00:20.450 align:middle line:84% the maximal orthonormal normal subset, which 01:00:20.450 --> 01:00:24.960 align:middle line:84% is, in most cases, just as good if you want to use that, 01:00:24.960 --> 01:00:25.460 align:middle line:90% all right. 01:00:25.460 --> 01:00:28.540 align:middle line:90% 01:00:28.540 --> 01:00:33.420 align:middle line:90% OK, so first off, when-- 01:00:33.420 --> 01:00:41.420 align:middle line:84% does every pre-Hilbert space have a maximal, a subset? 01:00:41.420 --> 01:00:43.805 align:middle line:84% So let me state this as a theorem. 01:00:43.805 --> 01:00:45.180 align:middle line:84% In fact, I'll state two theorems. 01:00:45.180 --> 01:00:50.526 align:middle line:90% The first is every-- 01:00:50.526 --> 01:00:59.460 align:middle line:84% I should say non-trivial because we could have the Hilbert 01:00:59.460 --> 01:01:01.320 align:middle line:90% space-- 01:01:01.320 --> 01:01:07.060 align:middle line:84% this be the 0 vector, pre-Hilbert space 01:01:07.060 --> 01:01:17.960 align:middle line:90% as a maximal orthonormal subset. 01:01:17.960 --> 01:01:20.720 align:middle line:84% Whether it's separable or not, it 01:01:20.720 --> 01:01:22.500 align:middle line:84% has a maximal orthonormal subset. 01:01:22.500 --> 01:01:24.500 align:middle line:84% And the way you prove this is using-- so I'm not 01:01:24.500 --> 01:01:25.917 align:middle line:84% going to give a proof of this, I'm 01:01:25.917 --> 01:01:28.310 align:middle line:84% going to give you a proof of something a little less 01:01:28.310 --> 01:01:30.815 align:middle line:84% strong but about as useful as we'll need. 01:01:30.815 --> 01:01:33.400 align:middle line:90% 01:01:33.400 --> 01:01:39.670 align:middle line:84% One proves this by using Zorn's Lemma, 01:01:39.670 --> 01:01:43.078 align:middle line:84% by taking a set, your set that you're 01:01:43.078 --> 01:01:44.620 align:middle line:84% going to put a partial order on to be 01:01:44.620 --> 01:01:48.070 align:middle line:84% the collections of orthonormal basis, or not 01:01:48.070 --> 01:01:51.070 align:middle line:84% orthonormal basis, max of orthonormal subsets 01:01:51.070 --> 01:01:53.350 align:middle line:90% and then ordered by inclusion. 01:01:53.350 --> 01:01:57.280 align:middle line:84% And then one can do a Zorn's Lemma argument 01:01:57.280 --> 01:02:03.490 align:middle line:84% and apply Zorn's Lemma to obtain a maximal orthonormal normal 01:02:03.490 --> 01:02:04.840 align:middle line:90% subset. 01:02:04.840 --> 01:02:06.130 align:middle line:90% But that's kind of hands off. 01:02:06.130 --> 01:02:07.713 align:middle line:84% And maybe that scares you a little bit 01:02:07.713 --> 01:02:12.040 align:middle line:84% because Zorn's Lemma is equivalent to the axiom 01:02:12.040 --> 01:02:12.620 align:middle line:90% of choice. 01:02:12.620 --> 01:02:18.420 align:middle line:84% So if you don't like using the axiom of choice, 01:02:18.420 --> 01:02:20.010 align:middle line:84% maybe you have a problem with using 01:02:20.010 --> 01:02:23.580 align:middle line:84% it to construct a maximal orthonormal subset 01:02:23.580 --> 01:02:25.830 align:middle line:90% of a pre-Hilbert. 01:02:25.830 --> 01:02:30.610 align:middle line:84% But we can actually do this by hand, 01:02:30.610 --> 01:02:33.420 align:middle line:84% if the Hilbert space is separable. 01:02:33.420 --> 01:02:35.760 align:middle line:84% So this is a theorem we'll actually 01:02:35.760 --> 01:02:46.580 align:middle line:84% prove every non-trivial separable, 01:02:46.580 --> 01:02:52.160 align:middle line:84% meaning the pre-Hilbert space has a countably dense subset. 01:02:52.160 --> 01:03:03.470 align:middle line:84% Every non-trivial separable pre-Hilbert space 01:03:03.470 --> 01:03:16.120 align:middle line:84% as a countable maximal orthonormal subset, 01:03:16.120 --> 01:03:23.470 align:middle line:84% OK, which as I said, this is the main types 01:03:23.470 --> 01:03:26.260 align:middle line:84% of orthonormal subsets we'll be interested in just 01:03:26.260 --> 01:03:29.950 align:middle line:84% because defining infinite sums is easier 01:03:29.950 --> 01:03:33.460 align:middle line:84% to do over a countable index than it 01:03:33.460 --> 01:03:35.050 align:middle line:90% is an uncountable index. 01:03:35.050 --> 01:03:39.400 align:middle line:84% But anyways, so we're going to prove and actually 01:03:39.400 --> 01:03:47.200 align:middle line:84% construct this essentially by hand, using the process that's 01:03:47.200 --> 01:03:49.750 align:middle line:84% the name of this little section, the Gram-Schmidt process. 01:03:49.750 --> 01:03:52.520 align:middle line:90% 01:03:52.520 --> 01:03:55.090 align:middle line:84% So if you remember from linear algebra, 01:03:55.090 --> 01:03:58.600 align:middle line:84% if you have a collection of vectors, 01:03:58.600 --> 01:04:01.900 align:middle line:84% you can always find an orthonormal collection 01:04:01.900 --> 01:04:07.990 align:middle line:84% of vectors that span the vector space that is spanned 01:04:07.990 --> 01:04:09.700 align:middle line:90% by the original set of vectors. 01:04:09.700 --> 01:04:11.178 align:middle line:90% OK. 01:04:11.178 --> 01:04:12.220 align:middle line:90% And that's what we'll do. 01:04:12.220 --> 01:04:18.020 align:middle line:90% 01:04:18.020 --> 01:04:44.920 align:middle line:84% So since H is separable, let be a countable dense subset of H. 01:04:44.920 --> 01:04:48.610 align:middle line:84% And this is a non-trivial pre-Hilbert space. 01:04:48.610 --> 01:04:54.170 align:middle line:84% So we can always make sure that the first one 01:04:54.170 --> 01:04:55.640 align:middle line:90% is a non-zero vector. 01:04:55.640 --> 01:04:56.270 align:middle line:90% OK. 01:04:56.270 --> 01:04:59.780 align:middle line:84% So you know what countable means? 01:04:59.780 --> 01:05:02.480 align:middle line:84% Dense, remember, means that for any element in h 01:05:02.480 --> 01:05:06.950 align:middle line:84% there exists an element from this sequence, 01:05:06.950 --> 01:05:09.380 align:middle line:84% or from this collection, that's within epsilon 01:05:09.380 --> 01:05:22.410 align:middle line:90% of that vector from H. 01:05:22.410 --> 01:05:24.570 align:middle line:84% So now I'm going to make the following claim. 01:05:24.570 --> 01:05:38.560 align:middle line:84% This is essentially the Gram-Schmidt process, 01:05:38.560 --> 01:05:42.010 align:middle line:90% which we'll prove by induction. 01:05:42.010 --> 01:05:50.390 align:middle line:84% For all n, natural number, there exists 01:05:50.390 --> 01:06:09.120 align:middle line:84% another integer natural number m of n and an orthonormal subset 01:06:09.120 --> 01:06:17.610 align:middle line:84% e1 to em such that following is true. 01:06:17.610 --> 01:06:20.670 align:middle line:90% 01:06:20.670 --> 01:06:37.010 align:middle line:84% First is the span of e1 up to em of n equals the span of v1 01:06:37.010 --> 01:06:37.580 align:middle line:90% up to vn. 01:06:37.580 --> 01:06:43.350 align:middle line:90% 01:06:43.350 --> 01:06:48.630 align:middle line:90% And so you see n is changing. 01:06:48.630 --> 01:06:52.110 align:middle line:84% So maybe for each time I change n, 01:06:52.110 --> 01:06:56.220 align:middle line:84% I get a different orthonormal subset or a wildly different 01:06:56.220 --> 01:07:01.830 align:middle line:84% orthonormal subset from the integer before it. 01:07:01.830 --> 01:07:04.350 align:middle line:84% So the property of these subsets are that I'm just 01:07:04.350 --> 01:07:09.770 align:middle line:90% simply adding a vector or not. 01:07:09.770 --> 01:07:16.150 align:middle line:84% And e1 up to em of n, so this collection 01:07:16.150 --> 01:07:26.250 align:middle line:84% is equal to the previous collection, union 01:07:26.250 --> 01:07:33.420 align:middle line:84% either the empty set if vn is in the span 01:07:33.420 --> 01:07:43.350 align:middle line:84% of the 1 up to vn minus 1 and some new vector e of m sub n. 01:07:43.350 --> 01:07:48.210 align:middle line:84% And I'll tell you what em of m sub n is otherwise. 01:07:48.210 --> 01:07:52.154 align:middle line:90% 01:07:52.154 --> 01:07:54.150 align:middle line:84% OK, so what I'm saying here is that I 01:07:54.150 --> 01:07:59.040 align:middle line:84% have this countable infinite list of vs. And for each n, 01:07:59.040 --> 01:08:04.740 align:middle line:84% I can come up with a finite orthonormal subset that spans 01:08:04.740 --> 01:08:07.410 align:middle line:90% the same span as v1 up to vn. 01:08:07.410 --> 01:08:10.080 align:middle line:84% And at each stage, all I do is add a vector 01:08:10.080 --> 01:08:12.600 align:middle line:84% or not depending on if the next v is 01:08:12.600 --> 01:08:14.980 align:middle line:84% in the span of the previous or not. 01:08:14.980 --> 01:08:16.215 align:middle line:90% OK, I hope that's clear. 01:08:16.215 --> 01:08:20.342 align:middle line:90% 01:08:20.342 --> 01:08:24.979 align:middle line:84% OK, so I proved this by induction. 01:08:24.979 --> 01:08:31.810 align:middle line:84% So proof of claim, this is by induction. 01:08:31.810 --> 01:08:37.305 align:middle line:84% So let's do the base case n equals 1. 01:08:37.305 --> 01:08:43.220 align:middle line:90% 01:08:43.220 --> 01:08:52.020 align:middle line:84% We take e1 to be e1 over length of v. All right. 01:08:52.020 --> 01:08:54.240 align:middle line:90% So now we're started. 01:08:54.240 --> 01:08:58.080 align:middle line:84% Now we've got our first vector in this list 01:08:58.080 --> 01:09:07.740 align:middle line:84% that we're building up, inductive step. 01:09:07.740 --> 01:09:13.990 align:middle line:84% So let's call this, what I want to prove, star. 01:09:13.990 --> 01:09:21.579 align:middle line:90% So suppose stars hold for a-- 01:09:21.579 --> 01:09:24.200 align:middle line:90% 01:09:24.200 --> 01:09:26.930 align:middle line:84% so this whole claim here, I shouldn't say just a 01:09:26.930 --> 01:09:31.880 align:middle line:84% and b, but the whole claim holds for n equals k. 01:09:31.880 --> 01:09:35.359 align:middle line:84% And now I want to prove it holds for n equals k plus 1. 01:09:35.359 --> 01:09:36.470 align:middle line:90% So I want to-- 01:09:36.470 --> 01:09:38.210 align:middle line:84% what kind of vector do I need to add 01:09:38.210 --> 01:09:42.350 align:middle line:84% to the previous collection of vectors to now span v1 01:09:42.350 --> 01:09:44.320 align:middle line:90% up to vk plus 1? 01:09:44.320 --> 01:09:44.820 align:middle line:90% OK. 01:09:44.820 --> 01:09:51.800 align:middle line:90% 01:09:51.800 --> 01:10:04.340 align:middle line:84% So if vk plus 1 is in the span of v1 up to vk, 01:10:04.340 --> 01:10:10.370 align:middle line:90% then e1 up to em of k-- 01:10:10.370 --> 01:10:12.230 align:middle line:84% I should say the span of these vectors-- 01:10:12.230 --> 01:10:16.840 align:middle line:90% 01:10:16.840 --> 01:10:25.750 align:middle line:84% equals span of v1 up to vk equals because vk plus 1 01:10:25.750 --> 01:10:26.950 align:middle line:90% is in this span. 01:10:26.950 --> 01:10:33.760 align:middle line:84% This span is also equal to vk plus 1. 01:10:33.760 --> 01:10:41.600 align:middle line:84% OK, and therefore, this case is handled. 01:10:41.600 --> 01:10:45.110 align:middle line:84% And we're in this spot where we don't add anything 01:10:45.110 --> 01:10:47.935 align:middle line:90% to the previous collection, OK. 01:10:47.935 --> 01:11:01.040 align:middle line:90% 01:11:01.040 --> 01:11:07.260 align:middle line:84% This proves what we wanted for n equals k plus 1 01:11:07.260 --> 01:11:08.820 align:middle line:90% by not adding anything. 01:11:08.820 --> 01:11:12.600 align:middle line:84% So this was in the case that vk plus 1 is in the span of v1 01:11:12.600 --> 01:11:13.380 align:middle line:90% up to vk. 01:11:13.380 --> 01:11:17.130 align:middle line:84% So now let's do the more interesting case of v1 01:11:17.130 --> 01:11:21.390 align:middle line:84% not equal or not in the span of v1 up to vk. 01:11:21.390 --> 01:11:31.975 align:middle line:84% So now suppose vk plus 1 is not in the span of v1 to vk. 01:11:31.975 --> 01:11:35.520 align:middle line:90% 01:11:35.520 --> 01:11:49.070 align:middle line:84% I define wk plus 1 to be vk plus 1 minus its projection 01:11:49.070 --> 01:11:52.860 align:middle line:84% onto the previous list of orthonormal vectors. 01:11:52.860 --> 01:12:04.420 align:middle line:84% So sum from j equals 1 m of k will be k plus 1 ej, ej. 01:12:04.420 --> 01:12:11.360 align:middle line:90% 01:12:11.360 --> 01:12:17.480 align:middle line:84% Then first note that this vector cannot be-- 01:12:17.480 --> 01:12:21.080 align:middle line:84% so wk plus 1 this vector cannot be zero. 01:12:21.080 --> 01:12:23.240 align:middle line:84% Otherwise, this would imply that vk plus 1 01:12:23.240 --> 01:12:25.400 align:middle line:90% is equal to this quantity here. 01:12:25.400 --> 01:12:28.130 align:middle line:90% 01:12:28.130 --> 01:12:30.590 align:middle line:84% And therefore, vk plus 1 is in the span 01:12:30.590 --> 01:12:35.400 align:middle line:84% of the ejs for j between one of between 1 and imps of k, 01:12:35.400 --> 01:12:40.460 align:middle line:84% which is equal to the span of the vjs for 1 up to k. 01:12:40.460 --> 01:12:43.040 align:middle line:84% But we're assuming this is not in the span. 01:12:43.040 --> 01:12:46.710 align:middle line:90% 01:12:46.710 --> 01:12:53.115 align:middle line:84% I should say this is equal to the span of e1 up to em sub k. 01:12:53.115 --> 01:12:54.240 align:middle line:90% That's what we're assuming. 01:12:54.240 --> 01:12:57.940 align:middle line:84% That's the inductive hypothesis, right? 01:12:57.940 --> 01:13:05.195 align:middle line:84% So then this vector does not equal 0. 01:13:05.195 --> 01:13:07.810 align:middle line:90% 01:13:07.810 --> 01:13:16.880 align:middle line:84% And define e n sub k plus 1 or m to be the vector 01:13:16.880 --> 01:13:22.015 align:middle line:90% wk over its length for plus 1. 01:13:22.015 --> 01:13:23.595 align:middle line:90% Now, this is a unit vector. 01:13:23.595 --> 01:13:36.472 align:middle line:90% 01:13:36.472 --> 01:13:37.240 align:middle line:90% And what? 01:13:37.240 --> 01:13:45.290 align:middle line:90% 01:13:45.290 --> 01:13:46.790 align:middle line:90% And if I take-- 01:13:46.790 --> 01:13:50.720 align:middle line:84% so now I claim this is orthogonal to e1 up 01:13:50.720 --> 01:13:52.670 align:middle line:90% to e of m sub k. 01:13:52.670 --> 01:13:58.505 align:middle line:84% If I take emk plus 1 inner product e sub n sub j-- 01:13:58.505 --> 01:14:06.430 align:middle line:90% 01:14:06.430 --> 01:14:12.220 align:middle line:84% this is equal to, simply by definition, 1 over length of mk 01:14:12.220 --> 01:14:17.110 align:middle line:84% plus 1 times the inner product of vk plus 1, 01:14:17.110 --> 01:14:21.340 align:middle line:84% minus the sum from j equals 1 to m sub k. 01:14:21.340 --> 01:14:24.190 align:middle line:84% So remember, this is supposed to be the projection 01:14:24.190 --> 01:14:26.770 align:middle line:90% onto the e1 up to em sub k. 01:14:26.770 --> 01:14:30.100 align:middle line:84% And vk plus 1 minus that is supposed 01:14:30.100 --> 01:14:31.540 align:middle line:90% to be orthogonal to those guys. 01:14:31.540 --> 01:14:34.060 align:middle line:84% So we'll see that this ends up being 0. 01:14:34.060 --> 01:14:37.110 align:middle line:90% 01:14:37.110 --> 01:14:40.770 align:middle line:84% And let me change this j to an l since I'm-- 01:14:40.770 --> 01:14:46.130 align:middle line:90% 01:14:46.130 --> 01:14:46.630 align:middle line:90% or no. 01:14:46.630 --> 01:14:52.500 align:middle line:90% 01:14:52.500 --> 01:15:06.920 align:middle line:84% And I get 1 over time the sum, times vk plus 1, e sub l, 01:15:06.920 --> 01:15:14.800 align:middle line:84% minus now e sub l inner product with the sum. 01:15:14.800 --> 01:15:17.800 align:middle line:84% Now this is the sum from j equals 1 to m sub k 01:15:17.800 --> 01:15:21.220 align:middle line:84% of this number here times the inner product of e sub j 01:15:21.220 --> 01:15:24.790 align:middle line:90% inner product e sub l. 01:15:24.790 --> 01:15:27.460 align:middle line:84% Now, the inner product of e sub j and e sub l is 01:15:27.460 --> 01:15:32.920 align:middle line:84% zero unless j equals and 1 if j equals l. 01:15:32.920 --> 01:15:36.910 align:middle line:84% So I only pick up the j equals l part 01:15:36.910 --> 01:15:40.105 align:middle line:84% of this sum when this inner product hits e sub l. 01:15:40.105 --> 01:15:43.050 align:middle line:90% 01:15:43.050 --> 01:15:45.000 align:middle line:84% In particular, I only pick up the coefficient 01:15:45.000 --> 01:15:46.630 align:middle line:84% in front since these have unit length. 01:15:46.630 --> 01:15:52.440 align:middle line:84% So I get v sub k plus 1, v sub l equals 0. 01:15:52.440 --> 01:15:59.050 align:middle line:90% 01:15:59.050 --> 01:16:12.205 align:middle line:90% Then this guy now does. 01:16:12.205 --> 01:16:15.060 align:middle line:90% 01:16:15.060 --> 01:16:15.560 align:middle line:90% the job. 01:16:15.560 --> 01:16:17.310 align:middle line:84% So I would write a little bit more, 01:16:17.310 --> 01:16:18.560 align:middle line:90% but I'm running short on time. 01:16:18.560 --> 01:16:28.140 align:middle line:90% 01:16:28.140 --> 01:16:29.550 align:middle line:90% OK. 01:16:29.550 --> 01:16:33.120 align:middle line:90% So we proved the claim. 01:16:33.120 --> 01:16:41.530 align:middle line:84% And now let's use this to conclude 01:16:41.530 --> 01:16:53.990 align:middle line:84% that this collection of all of these els are maximal. 01:16:53.990 --> 01:17:05.250 align:middle line:90% 01:17:05.250 --> 01:17:17.280 align:middle line:90% So we let s to be v sub n. 01:17:17.280 --> 01:17:25.330 align:middle line:90% 01:17:25.330 --> 01:17:40.560 align:middle line:84% And s is orthonormal subset of H. 01:17:40.560 --> 01:17:45.930 align:middle line:84% So I may not be adding any more vectors after a certain point 01:17:45.930 --> 01:17:48.000 align:middle line:84% and therefore just have a finite collection. 01:17:48.000 --> 01:17:50.320 align:middle line:90% That's also possible. 01:17:50.320 --> 01:17:53.700 align:middle line:84% But I may also have a countably infinite collection. 01:17:53.700 --> 01:17:58.400 align:middle line:90% 01:17:58.400 --> 01:18:02.980 align:middle line:84% So this is an orthonormal set by v. 01:18:02.980 --> 01:18:16.870 align:middle line:84% And so we now show x is maximal, OK. 01:18:16.870 --> 01:18:26.760 align:middle line:90% 01:18:26.760 --> 01:18:29.070 align:middle line:90% All right. 01:18:29.070 --> 01:18:31.500 align:middle line:84% So we haven't used anything about the nature 01:18:31.500 --> 01:18:36.520 align:middle line:84% of this subset of H. All we view is that it's countable. 01:18:36.520 --> 01:18:38.100 align:middle line:84% So we could do this step by step, 01:18:38.100 --> 01:18:40.710 align:middle line:84% where we construct these orthonormal vectors that 01:18:40.710 --> 01:18:43.830 align:middle line:90% span each finite-- 01:18:43.830 --> 01:18:46.920 align:middle line:84% equal the span of each finite collection of vectors. 01:18:46.920 --> 01:18:48.960 align:middle line:84% Now, to show it's maximal, this is 01:18:48.960 --> 01:18:51.885 align:middle line:84% where we use the fact that this collection of vs were dense 01:18:51.885 --> 01:18:53.370 align:middle line:90% or are dense in h. 01:18:53.370 --> 01:19:09.340 align:middle line:84% So suppose u is in H and for all l, u of e sub l equals 0. 01:19:09.340 --> 01:19:10.790 align:middle line:90% So this is either a finite set. 01:19:10.790 --> 01:19:12.280 align:middle line:90% So l is between 1 and n. 01:19:12.280 --> 01:19:17.560 align:middle line:84% Or it's a countable set, and l is going from 1 to infinity. 01:19:17.560 --> 01:19:25.820 align:middle line:84% Either this is equal to e1 up to e m sub n for some n. 01:19:25.820 --> 01:19:32.055 align:middle line:90% Or e1 is countably infinite? 01:19:32.055 --> 01:19:32.555 align:middle line:90% OK. 01:19:32.555 --> 01:19:38.550 align:middle line:90% 01:19:38.550 --> 01:19:50.465 align:middle line:84% So since vj-- this is a dense subset of h-- 01:19:50.465 --> 01:19:53.030 align:middle line:90% 01:19:53.030 --> 01:20:01.810 align:middle line:84% we can find a sequence of elements from this collection. 01:20:01.810 --> 01:20:08.040 align:middle line:84% So v of j sub k, k equals 1 to infinity 01:20:08.040 --> 01:20:16.000 align:middle line:84% such that vj of k converges to u as k goes to infinity. 01:20:16.000 --> 01:20:18.900 align:middle line:90% 01:20:18.900 --> 01:20:29.400 align:middle line:84% OK, now by property a, the fact that this span 01:20:29.400 --> 01:20:39.360 align:middle line:84% is equal to that span, v of j sub k is in the span of e1 01:20:39.360 --> 01:20:43.255 align:middle line:90% up to em of j sub k. 01:20:43.255 --> 01:20:47.870 align:middle line:90% 01:20:47.870 --> 01:20:52.430 align:middle line:84% And therefore, by Bessel's inequality, 01:20:52.430 --> 01:20:55.460 align:middle line:84% and the fact that u is orthogonal to all 01:20:55.460 --> 01:21:12.230 align:middle line:84% of these orthonormal vectors, I get that v of j sub k squared-- 01:21:12.230 --> 01:21:18.040 align:middle line:84% now v sub j of k, this is in this span. 01:21:18.040 --> 01:21:26.730 align:middle line:84% So you can show that, in fact, this norm 01:21:26.730 --> 01:21:33.030 align:middle line:84% is in fact equal to the sum of the coefficients 01:21:33.030 --> 01:21:35.250 align:middle line:84% that I get from taking the inner product of-- 01:21:35.250 --> 01:21:42.420 align:middle line:90% 01:21:42.420 --> 01:21:45.120 align:middle line:84% OK, if I have a vector in the span 01:21:45.120 --> 01:21:47.760 align:middle line:84% of a collection of orthonormal vectors, 01:21:47.760 --> 01:21:49.950 align:middle line:84% a finite collection of all the normal vectors, then 01:21:49.950 --> 01:21:52.950 align:middle line:84% the norm squared is equal to the sum of the coefficients, 01:21:52.950 --> 01:21:55.500 align:middle line:90% just like an rn. 01:21:55.500 --> 01:22:01.524 align:middle line:84% And since u is orthogonal to each of these, 01:22:01.524 --> 01:22:02.940 align:middle line:90% this I can write as-- 01:22:02.940 --> 01:22:07.460 align:middle line:90% 01:22:07.460 --> 01:22:10.610 align:middle line:84% And by Bessel's inequality, this is less than or equal to-- 01:22:10.610 --> 01:22:15.460 align:middle line:90% 01:22:15.460 --> 01:22:17.560 align:middle line:84% the sum of these things squared is 01:22:17.560 --> 01:22:26.030 align:middle line:84% less than or equal to squared, and this goes to 0 01:22:26.030 --> 01:22:31.250 align:middle line:84% as k goes to infinity since the v of j sub k is converted to u. 01:22:31.250 --> 01:22:33.350 align:middle line:84% But we started off with the norm of e 01:22:33.350 --> 01:22:40.920 align:middle line:90% of j sub k, which implies zero. 01:22:40.920 --> 01:22:44.580 align:middle line:84% And therefore u, which is the limit of these, 01:22:44.580 --> 01:22:50.196 align:middle line:84% must be zero, proving that this is a maximal subset. 01:22:50.196 --> 01:22:50.696 align:middle line:90% OK. 01:22:50.696 --> 01:22:53.940 align:middle line:90% 01:22:53.940 --> 01:22:59.880 align:middle line:84% So next time we'll prove that these maximal orthonormal, 01:22:59.880 --> 01:23:05.910 align:middle line:84% these maximal countable orthonormal subsets, in fact, 01:23:05.910 --> 01:23:10.890 align:middle line:84% form a pretty good analog of basis 01:23:10.890 --> 01:23:14.432 align:middle line:84% that you find in finite dimensional linear algebra. 01:23:14.432 --> 01:23:16.750 align:middle line:90% And we'll stop there. 01:23:16.750 --> 01:23:23.000 align:middle line:90%