WEBVTT 00:00:00.000 --> 00:00:03.944 align:middle line:90% [SQUEAKING][RUSTLING][CLICKING] 00:00:03.944 --> 00:00:18.760 align:middle line:90% 00:00:18.760 --> 00:00:25.020 align:middle line:84% CASEY RODRIGUEZ: OK, so the name of the game today 00:00:25.020 --> 00:00:26.520 align:middle line:90% is compactness. 00:00:26.520 --> 00:00:33.000 align:middle line:90% 00:00:33.000 --> 00:00:36.360 align:middle line:84% And so last time I recalled that the definition 00:00:36.360 --> 00:00:48.390 align:middle line:84% of a subset of A metric space x is compact 00:00:48.390 --> 00:01:04.080 align:middle line:84% if every sequence xn in k, so every sequence of elements in k 00:01:04.080 --> 00:01:14.000 align:middle line:84% has a subsequence converging in K, 00:01:14.000 --> 00:01:18.230 align:middle line:84% has a subsequence which converges to an element in k. 00:01:18.230 --> 00:01:24.290 align:middle line:84% And so for example by the Heiner-Borel theorem, 00:01:24.290 --> 00:01:29.510 align:middle line:84% closed and bounded subsets of Rn or Cn are compact. 00:01:29.510 --> 00:01:36.280 align:middle line:90% This is from 18100. 00:01:36.280 --> 00:01:40.520 align:middle line:84% And general closed and bounded subsets of a Hilbert space 00:01:40.520 --> 00:01:41.720 align:middle line:90% are not necessarily compact. 00:01:41.720 --> 00:01:48.470 align:middle line:84% Last time we looked at the closed unit ball in the Hilbert 00:01:48.470 --> 00:01:51.380 align:middle line:84% space, or let's say in little l2, 00:01:51.380 --> 00:01:55.490 align:middle line:84% this is not compact because you can take or orthonormal 00:01:55.490 --> 00:02:00.680 align:middle line:84% basis vectors which are the sequences that have zero 00:02:00.680 --> 00:02:02.900 align:middle line:90% except for one in the n-th spot. 00:02:02.900 --> 00:02:05.270 align:middle line:90% These are all of unit linked. 00:02:05.270 --> 00:02:10.910 align:middle line:84% And the sequence consisting of these basis vectors, which 00:02:10.910 --> 00:02:12.950 align:middle line:84% are sequences, so the sequence of sequences 00:02:12.950 --> 00:02:16.370 align:middle line:84% does not converge in little l2 or have a subsequence which 00:02:16.370 --> 00:02:18.200 align:middle line:90% converges in little l2. 00:02:18.200 --> 00:02:20.930 align:middle line:84% So we were looking for an additional condition 00:02:20.930 --> 00:02:22.820 align:middle line:90% that would ensure compactness. 00:02:22.820 --> 00:02:25.430 align:middle line:84% And as a warm up to this, last time 00:02:25.430 --> 00:02:32.375 align:middle line:84% we proved the following that at-- 00:02:32.375 --> 00:02:37.020 align:middle line:90% 00:02:37.020 --> 00:02:39.000 align:middle line:84% so we're always working in a Hilbert space that 00:02:39.000 --> 00:02:41.665 align:middle line:84% may be separable or not depending on if I say it is 00:02:41.665 --> 00:02:42.165 align:middle line:90% or not-- 00:02:42.165 --> 00:02:45.930 align:middle line:90% 00:02:45.930 --> 00:03:05.190 align:middle line:84% that if we take a sequence, a convergent sequence in H, 00:03:05.190 --> 00:03:10.710 align:middle line:84% then the conclusion is, 1, that the set k consisting 00:03:10.710 --> 00:03:19.680 align:middle line:84% of all the elements of this sequence and the limit 00:03:19.680 --> 00:03:20.445 align:middle line:90% is compact. 00:03:20.445 --> 00:03:23.280 align:middle line:90% 00:03:23.280 --> 00:03:30.060 align:middle line:84% And the set k has what are called equi-small tails with 00:03:30.060 --> 00:03:37.060 align:middle line:84% respect to any orthonormal subset of H. 00:03:37.060 --> 00:03:55.370 align:middle line:84% If e sub k is a countable orthonormal subset of H, 00:03:55.370 --> 00:04:00.380 align:middle line:84% then the following holds, this condition we called K having 00:04:00.380 --> 00:04:06.330 align:middle line:84% equi-small tails, then for every epsilon positive, 00:04:06.330 --> 00:04:12.770 align:middle line:84% there exists a natural number N such that for all-- 00:04:12.770 --> 00:04:23.090 align:middle line:84% let's call it-- use a different letter, v tilde in K, 00:04:23.090 --> 00:04:27.900 align:middle line:84% so either an element of the sequence or v, 00:04:27.900 --> 00:04:39.900 align:middle line:84% we have that the sum over K bigger than N squared is less 00:04:39.900 --> 00:04:41.710 align:middle line:90% than epsilon squared. 00:04:41.710 --> 00:04:43.470 align:middle line:84% So what this says in this condition 00:04:43.470 --> 00:04:53.950 align:middle line:84% we called K has equi-small tails with respect 00:04:53.950 --> 00:04:55.480 align:middle line:90% to this orthonormal subset. 00:04:55.480 --> 00:05:02.050 align:middle line:84% So of course for any fixed v, because this entire some-- 00:05:02.050 --> 00:05:04.210 align:middle line:84% forget just K over N-- is bounded 00:05:04.210 --> 00:05:08.620 align:middle line:84% by the norm of this v tilde squared by Bessel's inequality. 00:05:08.620 --> 00:05:14.360 align:middle line:84% We can always choose an N for any individual v tilde. 00:05:14.360 --> 00:05:17.120 align:middle line:84% Now equi-small tails means that I 00:05:17.120 --> 00:05:22.190 align:middle line:84% can choose this independent of the element in K. 00:05:22.190 --> 00:05:24.950 align:middle line:84% So if you like uniformly small tails is another way 00:05:24.950 --> 00:05:27.870 align:middle line:90% you could think about it. 00:05:27.870 --> 00:05:32.300 align:middle line:84% But the terminology equi has been used instead. 00:05:32.300 --> 00:05:34.850 align:middle line:84% But another way is to think we can uniformly 00:05:34.850 --> 00:05:39.740 align:middle line:84% make the tails with respect to an orthonormal sequence, 00:05:39.740 --> 00:05:42.140 align:middle line:84% or orthonormal subset of H small. 00:05:42.140 --> 00:05:48.260 align:middle line:90% 00:05:48.260 --> 00:05:51.110 align:middle line:84% Now, what we're going to prove is 00:05:51.110 --> 00:05:56.180 align:middle line:84% that, in fact, this condition here of having equi-small tails 00:05:56.180 --> 00:06:01.100 align:middle line:84% with respect to an orthonormal basis on top of the set being 00:06:01.100 --> 00:06:04.940 align:middle line:84% closed and bounded, suffice to prove 00:06:04.940 --> 00:06:07.170 align:middle line:90% that the subset is compact. 00:06:07.170 --> 00:06:11.890 align:middle line:84% And that's the following theorem. 00:06:11.890 --> 00:06:22.795 align:middle line:84% So let H be a separable Hilbert space. 00:06:22.795 --> 00:06:33.110 align:middle line:90% 00:06:33.110 --> 00:06:54.740 align:middle line:84% And let ek be an orthonormal basis of H. So I said suffice, 00:06:54.740 --> 00:06:56.150 align:middle line:90% but not just suffice. 00:06:56.150 --> 00:06:58.160 align:middle line:84% It's also necessary, as we'll show. 00:06:58.160 --> 00:07:16.650 align:middle line:84% Then K subset of H is compact if and only if K is closed, 00:07:16.650 --> 00:07:32.880 align:middle line:84% bounded, and has equi-small tails with respect 00:07:32.880 --> 00:07:38.670 align:middle line:84% to this orthonormal basis, so in particular 00:07:38.670 --> 00:07:39.855 align:middle line:90% for any orthonormal basis. 00:07:39.855 --> 00:07:50.110 align:middle line:90% 00:07:50.110 --> 00:07:53.110 align:middle line:84% So this is a complete description 00:07:53.110 --> 00:07:55.300 align:middle line:84% of compact subsets of a Hilbert space, 00:07:55.300 --> 00:07:57.850 align:middle line:84% it's those subsets which are closed, bounded, 00:07:57.850 --> 00:08:00.940 align:middle line:84% and have equi-small details with respect 00:08:00.940 --> 00:08:05.170 align:middle line:90% to one orthonormal basis. 00:08:05.170 --> 00:08:06.910 align:middle line:84% And what this theorem also says is 00:08:06.910 --> 00:08:09.520 align:middle line:84% that if it has equi-small details with respect 00:08:09.520 --> 00:08:12.670 align:middle line:84% to one orthonormal basis and is closed and bounded, 00:08:12.670 --> 00:08:15.265 align:middle line:84% then the same holds with respect to any orthonormal basis. 00:08:15.265 --> 00:08:18.630 align:middle line:90% 00:08:18.630 --> 00:08:20.460 align:middle line:90% OK, so let's prove-- 00:08:20.460 --> 00:08:21.645 align:middle line:90% so it's a two way street. 00:08:21.645 --> 00:08:26.830 align:middle line:90% 00:08:26.830 --> 00:08:30.550 align:middle line:84% Well, it doesn't make sense to say one street is shorter, 00:08:30.550 --> 00:08:33.010 align:middle line:84% one side of the street is shorter-- 00:08:33.010 --> 00:08:35.740 align:middle line:90% maybe wider, easier to use. 00:08:35.740 --> 00:08:37.870 align:middle line:84% But let's prove that if K is compact, 00:08:37.870 --> 00:08:39.289 align:middle line:90% then it implies all of this. 00:08:39.289 --> 00:08:56.840 align:middle line:84% So suppose K is compact and K is closed and bounded 00:08:56.840 --> 00:08:58.430 align:middle line:90% by general metric space theory. 00:08:58.430 --> 00:09:05.220 align:middle line:90% 00:09:05.220 --> 00:09:13.170 align:middle line:84% So look this up in the 18100's material, 00:09:13.170 --> 00:09:16.950 align:middle line:90% or intro to analysis material. 00:09:16.950 --> 00:09:27.130 align:middle line:84% So let's-- this also gives us an opportunity to understand this 00:09:27.130 --> 00:09:29.960 align:middle line:84% equi-small tail condition as well by having to negate it. 00:09:29.960 --> 00:09:32.890 align:middle line:84% So we're also going to prove that K has equi-small details 00:09:32.890 --> 00:09:34.310 align:middle line:90% by contradiction. 00:09:34.310 --> 00:09:49.410 align:middle line:84% So suppose K does not have equi-small tails with respect 00:09:49.410 --> 00:09:55.360 align:middle line:84% to this basis, or this orthonormal basis. 00:09:55.360 --> 00:09:58.130 align:middle line:90% Then what's that mean? 00:09:58.130 --> 00:10:02.520 align:middle line:84% So this is a for all there exists in a statement. 00:10:02.520 --> 00:10:09.130 align:middle line:84% So then there exists some bad epsilon 00:10:09.130 --> 00:10:28.280 align:middle line:84% so that for all N, a natural number, there exist a uN in K 00:10:28.280 --> 00:10:29.840 align:middle line:90% such that-- 00:10:29.840 --> 00:10:34.700 align:middle line:84% so these N's are increasing such that u-- 00:10:34.700 --> 00:10:38.090 align:middle line:84% so we have the negation of the condition that 00:10:38.090 --> 00:10:46.070 align:middle line:84% satisfied K bigger than N of uN eK squared 00:10:46.070 --> 00:10:48.425 align:middle line:84% is bigger than or equal to this epsilon 0 squared. 00:10:48.425 --> 00:11:00.690 align:middle line:90% 00:11:00.690 --> 00:11:10.200 align:middle line:84% So note that this is a sequence in this compact set K. 00:11:10.200 --> 00:11:13.560 align:middle line:84% And so by the defining property of a compact set, 00:11:13.560 --> 00:11:20.350 align:middle line:84% it must have a convergent subsequence, which 00:11:20.350 --> 00:11:35.580 align:middle line:84% implies there exists subsequence which you usually 00:11:35.580 --> 00:11:40.320 align:middle line:84% denote use of N, capital N sub m, some other letter. 00:11:40.320 --> 00:11:45.646 align:middle line:84% But I'll just call it, to make it 00:11:45.646 --> 00:11:50.090 align:middle line:84% a little connection back to what we wrote a minute ago, v sub n. 00:11:50.090 --> 00:11:58.670 align:middle line:84% So this is a subsequence of u sub capital N and v in K such 00:11:58.670 --> 00:12:11.660 align:middle line:84% that v sub n converges to v. Now then for all N, natural number, 00:12:11.660 --> 00:12:22.070 align:middle line:84% we have that sum over K bigger than n of vN eK squared 00:12:22.070 --> 00:12:27.230 align:middle line:84% is bigger than epsilon 0 squared because this 00:12:27.230 --> 00:12:30.080 align:middle line:84% is a subsequence of the original sequence which 00:12:30.080 --> 00:12:31.430 align:middle line:90% satisfied this condition. 00:12:31.430 --> 00:12:35.660 align:middle line:90% 00:12:35.660 --> 00:12:38.450 align:middle line:84% Now this would actually be capital N sub n. 00:12:38.450 --> 00:12:41.150 align:middle line:84% But if I throw in some more, capital N sub n 00:12:41.150 --> 00:12:45.990 align:middle line:84% would be an increasing sequence of numbers. 00:12:45.990 --> 00:12:47.240 align:middle line:90% And it's always bigger than n. 00:12:47.240 --> 00:12:49.790 align:middle line:84% So I can replace that capital N sub n with this. 00:12:49.790 --> 00:12:51.485 align:middle line:90% All right, so this is fine. 00:12:51.485 --> 00:12:54.070 align:middle line:90% 00:12:54.070 --> 00:12:59.680 align:middle line:84% Then that says the subset consisting of the elements v 00:12:59.680 --> 00:13:07.990 align:middle line:84% sub n in the natural number union v 00:13:07.990 --> 00:13:13.110 align:middle line:84% does not have equi-small tails, which 00:13:13.110 --> 00:13:17.760 align:middle line:84% is a contradiction to the theorem we proved last time. 00:13:17.760 --> 00:13:20.450 align:middle line:90% 00:13:20.450 --> 00:13:22.670 align:middle line:84% This contradicts the previous theorem that I stated 00:13:22.670 --> 00:13:25.490 align:middle line:90% and we proved last time. 00:13:25.490 --> 00:13:29.420 align:middle line:84% So any convergent sequence along with its limit is a compact set 00:13:29.420 --> 00:13:32.160 align:middle line:90% and has equi-small tails. 00:13:32.160 --> 00:13:36.240 align:middle line:90% So that's a contradiction. 00:13:36.240 --> 00:13:41.520 align:middle line:84% And what was the assumption that brought us to this road forking 00:13:41.520 --> 00:13:46.200 align:middle line:84% itself is the assumption that K is not compact. 00:13:46.200 --> 00:13:47.550 align:middle line:90% So therefore K-- 00:13:47.550 --> 00:13:48.720 align:middle line:90% I mean, not K's compact. 00:13:48.720 --> 00:13:59.840 align:middle line:84% K has equi-small tails with respect 00:13:59.840 --> 00:14:03.690 align:middle line:90% to this orthonormal basis. 00:14:03.690 --> 00:14:08.570 align:middle line:84% So that's one direction that if K is compact then 00:14:08.570 --> 00:14:10.828 align:middle line:84% it has to have equi-small tails with respect 00:14:10.828 --> 00:14:11.870 align:middle line:90% to the orthonormal basis. 00:14:11.870 --> 00:14:14.920 align:middle line:90% 00:14:14.920 --> 00:14:17.750 align:middle line:84% We didn't use anything about it being a basis, 00:14:17.750 --> 00:14:21.760 align:middle line:84% so this also says that if you have a compact set, then 00:14:21.760 --> 00:14:23.890 align:middle line:84% with respect to any orthonormal subset, 00:14:23.890 --> 00:14:25.450 align:middle line:84% it has equi-small tails with respect 00:14:25.450 --> 00:14:28.480 align:middle line:90% to that orthonormal subset. 00:14:28.480 --> 00:14:35.940 align:middle line:84% So now let's use this condition that K is closed and bounded 00:14:35.940 --> 00:14:41.850 align:middle line:84% and has equi-small tails to prove that K is compact. 00:14:41.850 --> 00:14:56.480 align:middle line:84% So suppose K is closed, bounded, and has 00:14:56.480 --> 00:15:07.860 align:middle line:84% equi-small tails with respect to this orthonormal basis. 00:15:07.860 --> 00:15:11.760 align:middle line:90% 00:15:11.760 --> 00:15:23.320 align:middle line:84% Now let uN be a sequence in K. So we 00:15:23.320 --> 00:15:27.460 align:middle line:84% want to show that it has a subsequence which converges 00:15:27.460 --> 00:15:30.280 align:middle line:84% to an element in K. Now, K is closed, 00:15:30.280 --> 00:15:34.450 align:middle line:84% which means that if I take any sequence which converges in K, 00:15:34.450 --> 00:15:37.790 align:middle line:84% or which any sequence of elements in K that converges, 00:15:37.790 --> 00:15:40.390 align:middle line:84% the limit has to be in K. That's the definition 00:15:40.390 --> 00:15:41.860 align:middle line:90% of a set being closed. 00:15:41.860 --> 00:15:45.320 align:middle line:84% So I just need to show that this sequence, in fact, 00:15:45.320 --> 00:15:47.527 align:middle line:90% has a convergence subsequence. 00:15:47.527 --> 00:15:48.610 align:middle line:90% That's all I have to show. 00:15:48.610 --> 00:16:10.300 align:middle line:84% So since K is closed, we just need to show uN 00:16:10.300 --> 00:16:15.840 align:middle line:90% as a convergent subsequence. 00:16:15.840 --> 00:16:19.860 align:middle line:90% 00:16:19.860 --> 00:16:32.000 align:middle line:84% So the idea is to use what we know about bounded subsets 00:16:32.000 --> 00:16:33.060 align:middle line:90% of complex numbers. 00:16:33.060 --> 00:16:38.970 align:middle line:84% So we know about bounded subsets of complex numbers. 00:16:38.970 --> 00:16:41.570 align:middle line:90% If I have a bounded sequence-- 00:16:41.570 --> 00:16:43.880 align:middle line:84% I shouldn't say subsets, but if I 00:16:43.880 --> 00:16:46.820 align:middle line:84% have a bounded sequence of complex numbers, 00:16:46.820 --> 00:16:48.290 align:middle line:90% it has a convergent subsequence. 00:16:48.290 --> 00:16:50.900 align:middle line:90% This is a Bolzano-Weierstrass. 00:16:50.900 --> 00:16:53.420 align:middle line:84% Usually it's stated in terms of subsets of R. 00:16:53.420 --> 00:16:58.170 align:middle line:84% But sequences of complex numbers converge if and only 00:16:58.170 --> 00:17:01.260 align:middle line:84% if the real and imaginary part converge, and are 00:17:01.260 --> 00:17:03.780 align:middle line:84% bounded if and only if the real and imaginary part, which 00:17:03.780 --> 00:17:06.960 align:middle line:84% are sequences of real numbers, are bounded. 00:17:06.960 --> 00:17:09.690 align:middle line:84% So Bolzano-Weierstrass immediately implies 00:17:09.690 --> 00:17:12.180 align:middle line:84% a similar statement for complex numbers, 00:17:12.180 --> 00:17:15.089 align:middle line:84% that every bounded sequence of complex numbers 00:17:15.089 --> 00:17:18.470 align:middle line:90% has a convergent subsequence. 00:17:18.470 --> 00:17:19.010 align:middle line:90% so. 00:17:19.010 --> 00:17:20.240 align:middle line:90% How would we use that? 00:17:20.240 --> 00:17:24.577 align:middle line:84% What we're going to do is we're going to expand uN's. 00:17:24.577 --> 00:17:26.119 align:middle line:84% And this is where we'll use that this 00:17:26.119 --> 00:17:29.850 align:middle line:84% as an orthonormal basis in terms of this orthonormal basis. 00:17:29.850 --> 00:17:33.290 align:middle line:84% So for each K, we'll have a sequence 00:17:33.290 --> 00:17:38.240 align:middle line:84% of complex numbers given by the entry, the K-th entry, 00:17:38.240 --> 00:17:41.495 align:middle line:84% of course the K-th entry of this sequence. 00:17:41.495 --> 00:17:44.060 align:middle line:90% 00:17:44.060 --> 00:17:47.510 align:middle line:84% And that will be a bounded sequence of complex numbers 00:17:47.510 --> 00:17:50.600 align:middle line:84% because we're assuming K is bounded, 00:17:50.600 --> 00:17:53.570 align:middle line:84% and therefore this sequence is bounded in K. 00:17:53.570 --> 00:17:56.510 align:middle line:84% So what we're going to end up having is entry by entry, 00:17:56.510 --> 00:17:59.900 align:middle line:84% a bounded sequence of complex numbers 00:17:59.900 --> 00:18:04.220 align:middle line:84% where these complex numbers represent the K-th entry 00:18:04.220 --> 00:18:08.940 align:middle line:90% of the sequence of vectors. 00:18:08.940 --> 00:18:11.790 align:middle line:84% And we're going to then take a subsequence that 00:18:11.790 --> 00:18:13.962 align:middle line:90% converges for the first entry. 00:18:13.962 --> 00:18:16.170 align:middle line:84% We're going to take a subsequence of that subsequence 00:18:16.170 --> 00:18:19.410 align:middle line:84% to get a subsequence which converges 00:18:19.410 --> 00:18:22.710 align:middle line:84% for the first and second entry, and then so on and so on, 00:18:22.710 --> 00:18:25.230 align:middle line:90% a diagonalization argument. 00:18:25.230 --> 00:18:29.740 align:middle line:84% And that subsequence we'll show converges. 00:18:29.740 --> 00:18:33.270 align:middle line:84% Now, it's not that simple because if you can do something 00:18:33.270 --> 00:18:37.710 align:middle line:84% 10 times it doesn't mean you can get some uniform control. 00:18:37.710 --> 00:18:42.210 align:middle line:84% We just have control for any finite number of entries. 00:18:42.210 --> 00:18:46.710 align:middle line:84% What saves us in the end is this equi-small tail condition, 00:18:46.710 --> 00:18:53.160 align:middle line:84% which allows us to basically throw away 00:18:53.160 --> 00:18:54.790 align:middle line:90% infinitely many entries. 00:18:54.790 --> 00:18:58.210 align:middle line:90% 00:18:58.210 --> 00:19:01.120 align:middle line:84% That's what this says is that you can always choose an N, 00:19:01.120 --> 00:19:05.120 align:middle line:84% so that the tail end of the entries don't matter. 00:19:05.120 --> 00:19:10.360 align:middle line:84% And so that, with having control over the first large block 00:19:10.360 --> 00:19:13.660 align:middle line:84% of entries, will be enough to conclude. 00:19:13.660 --> 00:19:15.130 align:middle line:90% So that was the roadmap. 00:19:15.130 --> 00:19:16.910 align:middle line:84% If you didn't follow the roadmap, fine. 00:19:16.910 --> 00:19:18.202 align:middle line:90% Hopefully you follow the proof. 00:19:18.202 --> 00:19:23.290 align:middle line:90% 00:19:23.290 --> 00:19:26.200 align:middle line:90% But it's a diagonal argument. 00:19:26.200 --> 00:19:33.280 align:middle line:84% And those are relatively easy to understand, 00:19:33.280 --> 00:19:38.110 align:middle line:84% somewhat difficult to write down, so I will do my best. 00:19:38.110 --> 00:19:52.000 align:middle line:84% So since K is bounded, that implies 00:19:52.000 --> 00:20:00.520 align:middle line:84% there exists a c non-negative such that for all n, 00:20:00.520 --> 00:20:04.930 align:middle line:84% norm of u sub n is less than or equal to C. 00:20:04.930 --> 00:20:06.790 align:middle line:90% The C is independent of n. 00:20:06.790 --> 00:20:11.970 align:middle line:90% It just depends on the set K. 00:20:11.970 --> 00:20:18.060 align:middle line:84% And therefore that sets for all K and n-- 00:20:18.060 --> 00:20:27.560 align:middle line:84% I should say for all K, for all n, if I look at the K-th-- 00:20:27.560 --> 00:20:30.650 align:middle line:84% if you let me call this the Fourier coefficient 00:20:30.650 --> 00:20:35.870 align:middle line:84% in this basis in this orthonormal basis of ek, 00:20:35.870 --> 00:20:39.320 align:middle line:84% if I look at the K-th Fourier coefficient, 00:20:39.320 --> 00:20:46.510 align:middle line:84% this is bounded by Cauchy-Schwartz, u n ek. 00:20:46.510 --> 00:20:48.920 align:middle line:84% And these are orthonormal so this has unit length. 00:20:48.920 --> 00:20:51.220 align:middle line:90% This is bounded by c. 00:20:51.220 --> 00:20:55.300 align:middle line:84% So for each fixed K, this is a sequence 00:20:55.300 --> 00:20:57.235 align:middle line:84% in N of complex numbers which is bounded. 00:20:57.235 --> 00:21:04.380 align:middle line:90% 00:21:04.380 --> 00:21:09.370 align:middle line:84% The sequence consisting of the K-th Fourier coefficient 00:21:09.370 --> 00:21:17.830 align:middle line:84% of the elements u sub n, so n is the thing that's changing, 00:21:17.830 --> 00:21:25.455 align:middle line:84% is the bounded sequence of complex numbers. 00:21:25.455 --> 00:21:28.520 align:middle line:90% 00:21:28.520 --> 00:21:32.960 align:middle line:84% So now this is where we're going to start choosing subsequences 00:21:32.960 --> 00:21:36.080 align:middle line:84% of subsequences so that we get convergence along the entries. 00:21:36.080 --> 00:21:39.470 align:middle line:90% 00:21:39.470 --> 00:21:53.420 align:middle line:84% So since u sub is bounded, this implies that there exists 00:21:53.420 --> 00:21:56.390 align:middle line:84% by Bolzano-Weierstrass, a subsequence, 00:21:56.390 --> 00:21:59.960 align:middle line:84% which I'll denote u sub n sub 1 of k-- 00:21:59.960 --> 00:22:03.170 align:middle line:90% 00:22:03.170 --> 00:22:04.190 align:middle line:90% let me not use k. 00:22:04.190 --> 00:22:07.520 align:middle line:84% Let's use-- what do I use here? j-- 00:22:07.520 --> 00:22:10.560 align:middle line:90% 00:22:10.560 --> 00:22:12.795 align:middle line:90% e1, so 1, 1-- 00:22:12.795 --> 00:22:20.740 align:middle line:90% 00:22:20.740 --> 00:22:34.980 align:middle line:90% in n, which converges in c. 00:22:34.980 --> 00:22:48.520 align:middle line:90% 00:22:48.520 --> 00:22:54.500 align:middle line:84% OK, so I now have a subsequence of the original sequence 00:22:54.500 --> 00:23:02.960 align:middle line:84% of the un's so that the first entry converges here. 00:23:02.960 --> 00:23:05.220 align:middle line:90% Or this sequence converges. 00:23:05.220 --> 00:23:06.410 align:middle line:90% Now I'm going to take-- 00:23:06.410 --> 00:23:21.500 align:middle line:84% OK, so since un1 of j e2, this is still a bounded sequence 00:23:21.500 --> 00:23:30.850 align:middle line:84% by what I showed here, bounded sequence. 00:23:30.850 --> 00:23:36.050 align:middle line:84% This implies there exists a subsequence. 00:23:36.050 --> 00:23:49.270 align:middle line:84% Now I'll call this subsequence un2 of j e2 j. 00:23:49.270 --> 00:23:52.060 align:middle line:84% So in 2 of j, this is a sequence of integers. 00:23:52.060 --> 00:23:54.460 align:middle line:84% That is a subset of these guys that's 00:23:54.460 --> 00:24:06.870 align:middle line:90% increasing --of un1 of j. 00:24:06.870 --> 00:24:07.755 align:middle line:90% So I should say-- 00:24:07.755 --> 00:24:15.730 align:middle line:90% 00:24:15.730 --> 00:24:19.290 align:middle line:90% which converges. 00:24:19.290 --> 00:24:26.290 align:middle line:84% So now note, this sequence converges. 00:24:26.290 --> 00:24:37.520 align:middle line:84% But the n2 of j, so this is a subsequence of integers 00:24:37.520 --> 00:24:39.040 align:middle line:90% of the n1 of j's. 00:24:39.040 --> 00:24:44.860 align:middle line:90% 00:24:44.860 --> 00:24:47.800 align:middle line:84% Since the n2 of j's a subsequence 00:24:47.800 --> 00:24:57.640 align:middle line:84% of the n1 of j's, we get that limit j 00:24:57.640 --> 00:25:02.920 align:middle line:90% goes to infinity of un1 of j-- 00:25:02.920 --> 00:25:05.215 align:middle line:90% or should say n2 of j-- 00:25:05.215 --> 00:25:06.550 align:middle line:90% e1. 00:25:06.550 --> 00:25:12.630 align:middle line:84% So this is a subsequence of this sequence. 00:25:12.630 --> 00:25:16.910 align:middle line:84% So the n2 of j is just a subsequence of n1 of j's. 00:25:16.910 --> 00:25:21.800 align:middle line:84% So since this sequence in j converges, 00:25:21.800 --> 00:25:23.900 align:middle line:90% this subsequence also converges. 00:25:23.900 --> 00:25:26.730 align:middle line:90% 00:25:26.730 --> 00:25:35.300 align:middle line:84% But now we've chosen the two so that we also have not just 00:25:35.300 --> 00:25:37.280 align:middle line:84% convergence along the first entry, 00:25:37.280 --> 00:25:42.725 align:middle line:84% but convergence along the second entry. 00:25:42.725 --> 00:25:46.670 align:middle line:90% 00:25:46.670 --> 00:25:48.260 align:middle line:90% But now you see what-- 00:25:48.260 --> 00:25:51.800 align:middle line:84% so this is how the argument goes, that now I 00:25:51.800 --> 00:25:54.500 align:middle line:84% will take a subsequence of the n2 of j 00:25:54.500 --> 00:25:58.760 align:middle line:84% so that the subsequence converges 00:25:58.760 --> 00:26:01.940 align:middle line:90% when I pair it with e3. 00:26:01.940 --> 00:26:05.450 align:middle line:84% And then I take a subsequence of that subsequence 00:26:05.450 --> 00:26:07.560 align:middle line:84% so that I get e4 and so on, and so on. 00:26:07.560 --> 00:26:12.020 align:middle line:84% And all of these subsequences are subsequences of-- 00:26:12.020 --> 00:26:13.670 align:middle line:84% or these subsequences of integers 00:26:13.670 --> 00:26:16.080 align:middle line:84% are subsequences of the previous set of integers, 00:26:16.080 --> 00:26:20.105 align:middle line:84% which are again a subsequence of the original integers n. 00:26:20.105 --> 00:26:31.370 align:middle line:90% 00:26:31.370 --> 00:26:40.385 align:middle line:90% So then what do we get? 00:26:40.385 --> 00:26:43.130 align:middle line:90% 00:26:43.130 --> 00:26:52.370 align:middle line:84% For all l there exists a subsequence of integers 00:26:52.370 --> 00:27:00.310 align:middle line:84% in l of j of the previous set of integers in, 00:27:00.310 --> 00:27:17.280 align:middle line:84% let's say, l minus 1 of j, such that for all k between 1 and l 00:27:17.280 --> 00:27:30.595 align:middle line:84% limit as j goes to infinity of u in l of j ek exists. 00:27:30.595 --> 00:27:34.880 align:middle line:90% 00:27:34.880 --> 00:27:38.450 align:middle line:90% So I hope this is clear. 00:27:38.450 --> 00:27:41.190 align:middle line:84% So you just take a subsequence of the original sequence 00:27:41.190 --> 00:27:42.800 align:middle line:84% so that the first entry converges, 00:27:42.800 --> 00:27:45.020 align:middle line:84% take a subsequence of that so that the second entry 00:27:45.020 --> 00:27:46.920 align:middle line:90% converges, and so on and so on. 00:27:46.920 --> 00:27:54.260 align:middle line:84% So then you have this nested subsequences so 00:27:54.260 --> 00:28:00.115 align:middle line:84% that at each fixed l, you have convergence of the first l 00:28:00.115 --> 00:28:00.615 align:middle line:90% entries. 00:28:00.615 --> 00:28:03.470 align:middle line:90% 00:28:03.470 --> 00:28:05.850 align:middle line:84% And so what you now do is that you pick-- 00:28:05.850 --> 00:28:18.600 align:middle line:84% so let me write or pick vl to be the diagonal 00:28:18.600 --> 00:28:21.090 align:middle line:90% along this sequence. 00:28:21.090 --> 00:28:35.460 align:middle line:84% So u in l of l, where l equals 1, 2, 3, and so on. 00:28:35.460 --> 00:28:38.140 align:middle line:90% 00:28:38.140 --> 00:28:46.390 align:middle line:84% Then what we get is since the l-th subsequence converges-- 00:28:46.390 --> 00:28:52.330 align:middle line:90% or the k-th entry for one-- 00:28:52.330 --> 00:28:56.740 align:middle line:84% OK, so let me back up and say this one more time. 00:28:56.740 --> 00:29:02.680 align:middle line:84% Since four fixed l here I have convergence of the first l 00:29:02.680 --> 00:29:05.620 align:middle line:84% entries, by picking along the diagonal 00:29:05.620 --> 00:29:30.090 align:middle line:84% I get a subsequence of un such that for all k 00:29:30.090 --> 00:29:38.780 align:middle line:84% I have convergence for the k-th entry as l goes to infinity, 00:29:38.780 --> 00:29:39.780 align:middle line:90% converges. 00:29:39.780 --> 00:29:46.150 align:middle line:90% 00:29:46.150 --> 00:29:50.940 align:middle line:84% All right, so I can take subsequences of subsequences 00:29:50.940 --> 00:29:52.770 align:middle line:84% and obtain a subsequence in the end 00:29:52.770 --> 00:29:57.600 align:middle line:84% so that for this subsequence of the original sequence un's, 00:29:57.600 --> 00:30:02.100 align:middle line:84% I have convergence of the entries entry by entry, 00:30:02.100 --> 00:30:04.830 align:middle line:84% or I have convergence of the Fourier coefficients 00:30:04.830 --> 00:30:08.410 align:middle line:90% as l goes to infinity. 00:30:08.410 --> 00:30:12.510 align:middle line:84% And now we're going to use this equi-small tails to prove 00:30:12.510 --> 00:30:16.245 align:middle line:84% that the sequence v sub l converges. 00:30:16.245 --> 00:30:18.120 align:middle line:84% We're just actually going to show its Cauchy. 00:30:18.120 --> 00:30:20.010 align:middle line:84% Since H is a Hilbert space, we then 00:30:20.010 --> 00:30:25.420 align:middle line:90% conclude that it must converge. 00:30:25.420 --> 00:30:36.020 align:middle line:90% So claim vl is Cauchy. 00:30:36.020 --> 00:30:38.300 align:middle line:84% And then that will conclude the proof because then it 00:30:38.300 --> 00:30:42.350 align:middle line:84% must converge in H. And by what I said at the beginning, 00:30:42.350 --> 00:30:45.530 align:middle line:84% since this is a sequence of elements 00:30:45.530 --> 00:30:48.920 align:middle line:84% in k, which is a closed set, the limit has to be in k. 00:30:48.920 --> 00:30:51.080 align:middle line:84% All right, so we just need to show it's Cauchy, 00:30:51.080 --> 00:30:52.500 align:middle line:90% and then the proof is done. 00:30:52.500 --> 00:30:54.410 align:middle line:90% So let epsilon be positive. 00:30:54.410 --> 00:30:59.470 align:middle line:90% 00:30:59.470 --> 00:31:13.220 align:middle line:84% So since k has equi-small tails-- 00:31:13.220 --> 00:31:15.880 align:middle line:90% let's go over here-- 00:31:15.880 --> 00:31:21.930 align:middle line:84% there exists natural number capital N 00:31:21.930 --> 00:31:31.250 align:middle line:84% so that for all l we have that the sum of squares k 00:31:31.250 --> 00:31:38.330 align:middle line:84% bigger than N of the tails, vl, ek squared, 00:31:38.330 --> 00:31:41.270 align:middle line:84% is less than epsilon squared over 16. 00:31:41.270 --> 00:31:42.260 align:middle line:90% Why 16? 00:31:42.260 --> 00:31:48.410 align:middle line:84% Because I want everything to come out correct in the end. 00:31:48.410 --> 00:32:01.870 align:middle line:84% So since the N sequences given by the first N 00:32:01.870 --> 00:32:10.690 align:middle line:84% entries of the sequence vl, so since the N sequences vl, e1 l 00:32:10.690 --> 00:32:15.970 align:middle line:90% up to vl, vn, l-- 00:32:15.970 --> 00:32:17.060 align:middle line:90% so these are only-- 00:32:17.060 --> 00:32:22.030 align:middle line:84% so this is the bit where I said equi-small tails allows 00:32:22.030 --> 00:32:28.840 align:middle line:84% us to throw away the tail end of these entries, 00:32:28.840 --> 00:32:33.130 align:middle line:84% and just control over the first finitely many entries gives us 00:32:33.130 --> 00:32:36.320 align:middle line:84% the condition or the convergence that we want. 00:32:36.320 --> 00:32:40.010 align:middle line:90% This is where that's happening. 00:32:40.010 --> 00:32:50.210 align:middle line:84% So since these converge, there exists a natural number M 00:32:50.210 --> 00:32:58.950 align:middle line:84% so that for all l, m bigger than capital M, 00:32:58.950 --> 00:33:07.170 align:middle line:84% I have that the sum n equals 1 to capital N of vl, ek 00:33:07.170 --> 00:33:18.760 align:middle line:84% minus vm, ek squared is less than epsilon squared over 4. 00:33:18.760 --> 00:33:24.370 align:middle line:84% So each of these are convergent sequences. 00:33:24.370 --> 00:33:27.130 align:middle line:84% So each of these are Cauchy sequences. 00:33:27.130 --> 00:33:31.270 align:middle line:84% So for each one, I can find a capital M sub 1 00:33:31.270 --> 00:33:36.480 align:middle line:84% so that just one of these entries is very small for l 00:33:36.480 --> 00:33:38.820 align:middle line:84% and m bigger than that capital M sub 1. 00:33:38.820 --> 00:33:42.540 align:middle line:84% And then I can do the next one so that for-- 00:33:42.540 --> 00:33:44.490 align:middle line:90% so this should be k. 00:33:44.490 --> 00:33:47.300 align:middle line:84% I don't know what that looks like, but k. 00:33:47.300 --> 00:33:49.940 align:middle line:84% So that for k equals 2, it's very small, and then up to k 00:33:49.940 --> 00:33:50.600 align:middle line:90% equals n. 00:33:50.600 --> 00:33:58.060 align:middle line:84% And then I choose capital M to be the maximum over all those M 00:33:58.060 --> 00:34:00.550 align:middle line:90% sub N's. 00:34:00.550 --> 00:34:03.430 align:middle line:84% So I can always choose this because, again, there's 00:34:03.430 --> 00:34:07.879 align:middle line:84% only finitely many conditions I have to verify, all right. 00:34:07.879 --> 00:34:14.239 align:middle line:90% 00:34:14.239 --> 00:34:20.510 align:middle line:84% So now I claim this capital M works. 00:34:20.510 --> 00:34:24.380 align:middle line:84% Then for all lm bigger than or equal to M, 00:34:24.380 --> 00:34:30.199 align:middle line:84% if we look at the norm of vl minus vm, 00:34:30.199 --> 00:34:39.080 align:middle line:90% so this is equal to sum-- 00:34:39.080 --> 00:34:46.670 align:middle line:84% it should say k equals 1 to n of vl minus vm. 00:34:46.670 --> 00:34:51.469 align:middle line:90% 00:34:51.469 --> 00:34:58.430 align:middle line:90% So by k equals-- 00:34:58.430 --> 00:35:07.700 align:middle line:84% k bigger than N, vl minus vm, ek 1/2. 00:35:07.700 --> 00:35:13.370 align:middle line:84% So here I'm using the fact that we have an orthonormal basis. 00:35:13.370 --> 00:35:18.230 align:middle line:84% Then in that case, the norm is equal to the sum of the squares 00:35:18.230 --> 00:35:21.540 align:middle line:90% of the Fourier coefficients. 00:35:21.540 --> 00:35:23.630 align:middle line:84% Bessel's inequality would say this is less than 00:35:23.630 --> 00:35:24.770 align:middle line:90% or equal to this side. 00:35:24.770 --> 00:35:27.380 align:middle line:84% It wouldn't give us any control we want over the actual norm. 00:35:27.380 --> 00:35:30.080 align:middle line:84% But because we're working in orthonormal basis, 00:35:30.080 --> 00:35:32.930 align:middle line:84% the norm of that is equal to the sum of squares, 00:35:32.930 --> 00:35:33.980 align:middle line:90% take the square root. 00:35:33.980 --> 00:35:39.760 align:middle line:90% 00:35:39.760 --> 00:35:43.290 align:middle line:84% And so now the square root of A plus B, 00:35:43.290 --> 00:35:46.030 align:middle line:84% that's always less than or equal to the square root of A plus 00:35:46.030 --> 00:35:53.020 align:middle line:84% the square root of B. So that's less than A equals 1 00:35:53.020 --> 00:36:15.880 align:middle line:84% to N of vl minus vm squared 1/2 plus k bigger than n vl, ek. 00:36:15.880 --> 00:36:21.280 align:middle line:84% So I'm just going to write this out a little differently, vm, 00:36:21.280 --> 00:36:26.170 align:middle line:90% ek, 1/2. 00:36:26.170 --> 00:36:28.990 align:middle line:84% Now this part we know is less than-- 00:36:28.990 --> 00:36:32.260 align:middle line:84% by how we've chosen capital M, this finite part is small. 00:36:32.260 --> 00:36:34.840 align:middle line:90% It's less than epsilon over 2. 00:36:34.840 --> 00:36:39.870 align:middle line:90% 00:36:39.870 --> 00:36:44.070 align:middle line:84% So that's less than epsilon over 2. 00:36:44.070 --> 00:36:50.250 align:middle line:84% Plus now we use the triangle inequality now for little l2. 00:36:50.250 --> 00:36:55.530 align:middle line:84% I can think of this as the sum in little l2-- 00:36:55.530 --> 00:36:57.960 align:middle line:84% or I can think of this as the norm 00:36:57.960 --> 00:37:03.960 align:middle line:84% in little l2 of this sequence in k minus this sequence in k. 00:37:03.960 --> 00:37:14.470 align:middle line:84% So this is less than or equal to k 00:37:14.470 --> 00:37:43.430 align:middle line:84% bigger than N, vl, ek, 1/2 plus sum k bigger than vm, ek 1/2 00:37:43.430 --> 00:37:44.790 align:middle line:90% All right, we're almost done. 00:37:44.790 --> 00:37:48.290 align:middle line:84% So this is less than, the square part is less 00:37:48.290 --> 00:37:50.700 align:middle line:90% than epsilon squared over 16. 00:37:50.700 --> 00:37:56.530 align:middle line:84% So taking the square root of that, I get epsilon over 4 00:37:56.530 --> 00:38:00.910 align:middle line:84% by how we chose N. So this is based on how we chose N, plus-- 00:38:00.910 --> 00:38:02.740 align:middle line:84% which came from the equi-small tails part. 00:38:02.740 --> 00:38:08.680 align:middle line:84% And this part also because of N is less than epsilon over 4 00:38:08.680 --> 00:38:09.680 align:middle line:90% equals epsilon. 00:38:09.680 --> 00:38:13.450 align:middle line:84% So we've now shown the claim that the subsequence v sub l 00:38:13.450 --> 00:38:16.735 align:middle line:84% is Cauchy, and therefore it converges since H is complete. 00:38:16.735 --> 00:38:25.080 align:middle line:90% 00:38:25.080 --> 00:38:34.600 align:middle line:84% Now, so for example, one can readily verify now 00:38:34.600 --> 00:38:35.725 align:middle line:90% that the following subset. 00:38:35.725 --> 00:38:38.650 align:middle line:90% 00:38:38.650 --> 00:38:45.790 align:middle line:84% Let k be the i set of all sequences n little l2 00:38:45.790 --> 00:38:52.120 align:middle line:84% with the property that ak is less than or equal to 2 00:38:52.120 --> 00:38:53.170 align:middle line:90% to the minus k. 00:38:53.170 --> 00:38:55.495 align:middle line:84% So this is not a subspace, it's a subset-- 00:38:55.495 --> 00:39:01.980 align:middle line:90% 00:39:01.980 --> 00:39:03.960 align:middle line:90% is compact. 00:39:03.960 --> 00:39:06.540 align:middle line:84% This subset is known as the Hilbert cube. 00:39:06.540 --> 00:39:10.970 align:middle line:90% 00:39:10.970 --> 00:39:15.570 align:middle line:84% All right, now maybe it's a bit unwieldy 00:39:15.570 --> 00:39:20.430 align:middle line:84% that our condition for being compact 00:39:20.430 --> 00:39:25.380 align:middle line:84% is phrased in terms of an orthonormal basis. 00:39:25.380 --> 00:39:28.350 align:middle line:84% Maybe that's not so simple to verify 00:39:28.350 --> 00:39:36.190 align:middle line:84% or definitely not necessarily, what's 00:39:36.190 --> 00:39:38.470 align:middle line:84% the word I'm looking for, canonical-- 00:39:38.470 --> 00:39:40.830 align:middle line:84% I guess that's the word I'm looking for-- 00:39:40.830 --> 00:39:42.580 align:middle line:84% in the sense that we have to make a choice 00:39:42.580 --> 00:39:43.750 align:middle line:90% to verify compactness. 00:39:43.750 --> 00:39:45.910 align:middle line:84% But a different way of characterizing 00:39:45.910 --> 00:39:48.500 align:middle line:84% compact subsets of a Hilbert space is the following. 00:39:48.500 --> 00:39:50.960 align:middle line:84% Now, I'm not going to give the proof of this. 00:39:50.960 --> 00:39:54.190 align:middle line:84% You can look it up in Melrose's notes. 00:39:54.190 --> 00:39:56.380 align:middle line:84% Maybe I'll say a word about why you should 00:39:56.380 --> 00:39:58.960 align:middle line:90% believe or think it's true. 00:39:58.960 --> 00:40:02.570 align:middle line:84% But, again, it involves a diagonal argument, 00:40:02.570 --> 00:40:05.590 align:middle line:84% which that was tough enough to do here, 00:40:05.590 --> 00:40:08.122 align:middle line:84% or at least painful enough to write out. 00:40:08.122 --> 00:40:09.580 align:middle line:84% And so I don't want to do it again. 00:40:09.580 --> 00:40:11.770 align:middle line:90% So we have the following that-- 00:40:11.770 --> 00:40:18.400 align:middle line:84% and this theorem also holds in a nonseparable Hilbert space, 00:40:18.400 --> 00:40:22.450 align:middle line:84% basically by a trick of reducing it to a separable Hilbert 00:40:22.450 --> 00:40:24.070 align:middle line:90% space, but that's OK. 00:40:24.070 --> 00:40:33.020 align:middle line:84% A subset K of a Hilbert space H is compact if and only 00:40:33.020 --> 00:40:39.060 align:middle line:90% if K is closed, bounded. 00:40:39.060 --> 00:40:41.610 align:middle line:90% 00:40:41.610 --> 00:40:43.020 align:middle line:84% And the following condition holds 00:40:43.020 --> 00:40:46.860 align:middle line:84% what you can think of as K can be approximated 00:40:46.860 --> 00:40:50.350 align:middle line:90% by finite dimensional subspaces. 00:40:50.350 --> 00:40:53.530 align:middle line:90% 00:40:53.530 --> 00:40:58.670 align:middle line:84% And for all epsilon positive, there 00:40:58.670 --> 00:41:16.170 align:middle line:84% exists a finite dimensional subspace W in K such 00:41:16.170 --> 00:41:24.540 align:middle line:84% that for all u in K, if I look at the distance from u 00:41:24.540 --> 00:41:27.953 align:middle line:84% to this finite dimensional subspace-- which 00:41:27.953 --> 00:41:29.370 align:middle line:84% I think I said at one point, maybe 00:41:29.370 --> 00:41:31.770 align:middle line:84% I didn't, but a finite dimensional subspace is always 00:41:31.770 --> 00:41:39.768 align:middle line:90% a closed subspace in H. 00:41:39.768 --> 00:41:42.000 align:middle line:84% A finite dimensional subspace is always closed. 00:41:42.000 --> 00:41:46.970 align:middle line:84% But if I look at the distance from this point 00:41:46.970 --> 00:41:52.020 align:middle line:84% to the subspace, this finite dimensional subspace, 00:41:52.020 --> 00:41:55.150 align:middle line:90% this is less than epsilon. 00:41:55.150 --> 00:41:59.040 align:middle line:84% So a subset in the Hilbert space is compact if and only 00:41:59.040 --> 00:42:00.880 align:middle line:84% if it's closed, bounded, and if you like 00:42:00.880 --> 00:42:04.500 align:middle line:84% can be approximated by compact subsets 00:42:04.500 --> 00:42:11.130 align:middle line:84% of finite dimensional subspaces, another way to think about it. 00:42:11.130 --> 00:42:15.240 align:middle line:84% Now, why is this believable in the first place? 00:42:15.240 --> 00:42:17.720 align:middle line:90% Well, what this-- where is it? 00:42:17.720 --> 00:42:20.950 align:middle line:90% 00:42:20.950 --> 00:42:29.070 align:middle line:84% What the equi-small tail condition says is basically 00:42:29.070 --> 00:42:36.270 align:middle line:84% that all elements in K can be approximated by-- 00:42:36.270 --> 00:42:39.000 align:middle line:84% if you let epsilon be positive-- can 00:42:39.000 --> 00:42:42.630 align:middle line:84% be approximated by the subspace consisting 00:42:42.630 --> 00:42:49.260 align:middle line:84% of the span of the first one up to N basis vectors 00:42:49.260 --> 00:42:51.360 align:middle line:90% or orthonormal basis vectors. 00:42:51.360 --> 00:42:52.620 align:middle line:90% That's what this is. 00:42:52.620 --> 00:42:54.760 align:middle line:84% For all epsilon the tails are small. 00:42:54.760 --> 00:42:58.770 align:middle line:84% Another way of saying that is that for all epsilon, 00:42:58.770 --> 00:43:00.870 align:middle line:84% you can approximate any element in the set 00:43:00.870 --> 00:43:07.510 align:middle line:84% K in the span of the first N basis vectors. 00:43:07.510 --> 00:43:18.980 align:middle line:84% So K has to be close to finite dimensional subspace. 00:43:18.980 --> 00:43:22.020 align:middle line:84% But one can go the opposite direction as well, 00:43:22.020 --> 00:43:28.010 align:middle line:84% showing that finite dimensional subspace-- 00:43:28.010 --> 00:43:30.350 align:middle line:84% being able to be approximated by those guys 00:43:30.350 --> 00:43:31.640 align:middle line:90% implies that K is compact. 00:43:31.640 --> 00:43:34.010 align:middle line:84% And it's another diagonal argument that I just 00:43:34.010 --> 00:43:36.080 align:middle line:84% don't want to write out and rather just start 00:43:36.080 --> 00:43:39.802 align:middle line:84% using these conditions instead to start saying things 00:43:39.802 --> 00:43:41.510 align:middle line:84% about when we can solve certain equations 00:43:41.510 --> 00:43:43.520 align:middle line:84% and certain interesting operators 00:43:43.520 --> 00:43:47.450 align:middle line:90% that arise quite naturally. 00:43:47.450 --> 00:43:50.400 align:middle line:90% 00:43:50.400 --> 00:43:57.860 align:middle line:84% So let's start talking about certain classes of operators. 00:43:57.860 --> 00:43:59.540 align:middle line:84% And we should start with the simplest, 00:43:59.540 --> 00:44:05.110 align:middle line:90% which are finite rank operator. 00:44:05.110 --> 00:44:12.450 align:middle line:84% So the only linear operators you came into this class knowing 00:44:12.450 --> 00:44:13.950 align:middle line:90% probably-- 00:44:13.950 --> 00:44:17.760 align:middle line:84% if you didn't and you were more sophisticated, that's good-- 00:44:17.760 --> 00:44:22.830 align:middle line:90% but were matrices. 00:44:22.830 --> 00:44:26.340 align:middle line:84% Now a matrix is just a linear transformation 00:44:26.340 --> 00:44:29.510 align:middle line:84% defined in terms of an orthonormal-- or not 00:44:29.510 --> 00:44:35.597 align:middle line:84% orthonormal basis, but a basis in both the domain and target. 00:44:35.597 --> 00:44:37.180 align:middle line:84% We picked the target to be the domain. 00:44:37.180 --> 00:44:38.680 align:middle line:84% We just fix a basis, and then we can 00:44:38.680 --> 00:44:43.950 align:middle line:84% express a linear transformation as an array of numbers. 00:44:43.950 --> 00:44:48.390 align:middle line:84% Finite rank operators are the generalization 00:44:48.390 --> 00:44:54.780 align:middle line:84% of what you think of matrices now for to Hilbert spaces. 00:44:54.780 --> 00:44:59.730 align:middle line:84% And we'll see that in just a minute. 00:44:59.730 --> 00:45:07.490 align:middle line:84% And what follows H is a Hilbert space. 00:45:07.490 --> 00:45:10.270 align:middle line:90% 00:45:10.270 --> 00:45:17.790 align:middle line:84% And instead of writing the space of bounded linear operators, 00:45:17.790 --> 00:45:23.460 align:middle line:84% B H, H, I'm just going to concatenate this by dropping 00:45:23.460 --> 00:45:24.390 align:middle line:90% one of those H's. 00:45:24.390 --> 00:45:27.240 align:middle line:84% So the space of bounded linear operators from H to itself 00:45:27.240 --> 00:45:36.200 align:middle line:90% will be denoted by B of H. 00:45:36.200 --> 00:45:40.790 align:middle line:84% So finite rank operators, these are, 00:45:40.790 --> 00:45:43.205 align:middle line:84% like I said, these will be the analog of matrices. 00:45:43.205 --> 00:45:45.920 align:middle line:90% 00:45:45.920 --> 00:45:54.470 align:middle line:84% But let me give a invariant definition first. 00:45:54.470 --> 00:46:06.030 align:middle line:84% So a bounded linear operator is a finite rank operator 00:46:06.030 --> 00:46:27.260 align:middle line:84% if the range of T which is a subspace of H 00:46:27.260 --> 00:46:32.570 align:middle line:90% is finite dimensional, OK. 00:46:32.570 --> 00:46:37.680 align:middle line:90% 00:46:37.680 --> 00:46:42.970 align:middle line:84% So, for example, let me give you the simple-- 00:46:42.970 --> 00:46:48.630 align:middle line:84% well of course the zero operator is a simple one. 00:46:48.630 --> 00:46:52.350 align:middle line:84% If H is a finite dimensional Hilbert space, [INAUDIBLE],, 00:46:52.350 --> 00:46:55.290 align:middle line:84% then every bounded linear operator 00:46:55.290 --> 00:46:57.930 align:middle line:84% is a finite rank operator because the range is always 00:46:57.930 --> 00:47:02.520 align:middle line:84% contained in Cn, which is finite dimensional. 00:47:02.520 --> 00:47:06.720 align:middle line:84% For example on little l2, if I define 00:47:06.720 --> 00:47:20.100 align:middle line:84% Ta to be now the sequence given by a1 over 1, 00:47:20.100 --> 00:47:30.990 align:middle line:84% a2 over 2, up to an over n for some n and then 0 afterwards, 00:47:30.990 --> 00:47:39.530 align:middle line:84% and this is for a equals sequence in little l2, 00:47:39.530 --> 00:47:50.520 align:middle line:84% then T is a finite rank operator because the range 00:47:50.520 --> 00:47:56.280 align:middle line:84% of this operator-- so here N is a fixed number 00:47:56.280 --> 00:47:57.480 align:middle line:90% acting on sequences. 00:47:57.480 --> 00:47:59.370 align:middle line:84% And little l2 just spits out the first-- 00:47:59.370 --> 00:48:02.760 align:middle line:90% 00:48:02.760 --> 00:48:04.890 align:middle line:84% let's make this even more explicit. 00:48:04.890 --> 00:48:07.680 align:middle line:90% Let's say that's 5. 00:48:07.680 --> 00:48:09.720 align:middle line:84% This is a finite rank operator because it's 00:48:09.720 --> 00:48:11.580 align:middle line:84% contained in the subspace consisting 00:48:11.580 --> 00:48:17.740 align:middle line:84% of those sequences which are 0 after the fifth entry. 00:48:17.740 --> 00:48:18.990 align:middle line:90% And that's finite dimensional. 00:48:18.990 --> 00:48:22.860 align:middle line:84% A basis is given by the sequence consisting of 0's with a 1 00:48:22.860 --> 00:48:26.980 align:middle line:84% here, 0's with a 1 here, 0's with a 1 here, and so on. 00:48:26.980 --> 00:48:28.650 align:middle line:84% So this is a finite rank operator. 00:48:28.650 --> 00:48:32.300 align:middle line:90% 00:48:32.300 --> 00:48:37.550 align:middle line:84% And so let me just put over here that as far as notation goes, 00:48:37.550 --> 00:48:45.865 align:middle line:84% we write T is in RH, R for rank, finite rank. 00:48:45.865 --> 00:48:49.860 align:middle line:90% 00:48:49.860 --> 00:48:52.500 align:middle line:84% All right, so R of H, this is the set 00:48:52.500 --> 00:48:53.910 align:middle line:90% of all finite rank operators. 00:48:53.910 --> 00:48:55.005 align:middle line:90% Now, it's not just a set. 00:48:55.005 --> 00:49:07.190 align:middle line:90% 00:49:07.190 --> 00:49:14.270 align:middle line:84% It's easy to see that this is a subspace of the Banach space 00:49:14.270 --> 00:49:20.190 align:middle line:84% consisting of all bounded linear operators from H to itself. 00:49:20.190 --> 00:49:21.180 align:middle line:90% Why is that? 00:49:21.180 --> 00:49:25.400 align:middle line:84% Well, so if I take an operator that 00:49:25.400 --> 00:49:29.000 align:middle line:84% has finite dimensional range and multiply it by a scalar, 00:49:29.000 --> 00:49:32.930 align:middle line:84% then the range of the scalar multiple of that operator 00:49:32.930 --> 00:49:36.110 align:middle line:84% is equal to the range of the original operator 00:49:36.110 --> 00:49:38.150 align:middle line:90% unless it's the scalar 0. 00:49:38.150 --> 00:49:40.340 align:middle line:84% So that will be finite dimensional. 00:49:40.340 --> 00:49:43.860 align:middle line:84% On the other hand, if I take two finite rank operators 00:49:43.860 --> 00:49:48.050 align:middle line:84% and I consider their sum, the range of the sum 00:49:48.050 --> 00:49:55.520 align:middle line:84% is going to be contained in the direct sum of the ranges. 00:49:55.520 --> 00:50:01.520 align:middle line:84% And the direct sum of two finite dimensional spaces, subspaces, 00:50:01.520 --> 00:50:04.200 align:middle line:90% is again finite dimensional. 00:50:04.200 --> 00:50:08.120 align:middle line:84% So if one has dimension 5 and the other has dimension 6, 00:50:08.120 --> 00:50:12.920 align:middle line:84% then the direct sum of these two will be of dimension 11. 00:50:12.920 --> 00:50:16.720 align:middle line:84% So that is me talking you through the proof 00:50:16.720 --> 00:50:18.020 align:middle line:90% that this is a subspace. 00:50:18.020 --> 00:50:23.150 align:middle line:84% So take a minute to write it down. 00:50:23.150 --> 00:50:35.920 align:middle line:84% But now let me prove that these finite rank operators really 00:50:35.920 --> 00:50:39.625 align:middle line:90% are essentially matrices. 00:50:39.625 --> 00:50:50.740 align:middle line:90% 00:50:50.740 --> 00:50:53.590 align:middle line:84% I have the following theorem characterizing finite rank 00:50:53.590 --> 00:50:54.700 align:middle line:90% operators. 00:50:54.700 --> 00:50:59.890 align:middle line:84% So T is a finite rank operator if and only 00:50:59.890 --> 00:51:13.502 align:middle line:84% if there exists an orthonormal, finite orthonormal subset ek, 00:51:13.502 --> 00:51:25.590 align:middle line:84% k equals 1 to some integer L. And complex numbers 00:51:25.590 --> 00:51:30.920 align:middle line:90% cij ij equals 1 to l. 00:51:30.920 --> 00:51:39.420 align:middle line:84% So this is subsequence-- or a subset of complex numbers 00:51:39.420 --> 00:51:44.490 align:middle line:84% such that if I want to compute what T applied to u 00:51:44.490 --> 00:51:56.010 align:middle line:84% is, it's just going to be sum i equals j, ij goes from 1 to l 00:51:56.010 --> 00:52:07.290 align:middle line:84% of these numbers Cij times the j entry of u times ei. 00:52:07.290 --> 00:52:12.270 align:middle line:84% So, in other words, T corresponds to a matrix where 00:52:12.270 --> 00:52:15.420 align:middle line:84% the coefficients are these c's, basically, 00:52:15.420 --> 00:52:19.935 align:middle line:84% acting on this finite dimensional subspace of ek's. 00:52:19.935 --> 00:52:28.390 align:middle line:90% 00:52:28.390 --> 00:52:31.340 align:middle line:90% So let's give the proof of this. 00:52:31.340 --> 00:52:34.580 align:middle line:90% One direction is immediate. 00:52:34.580 --> 00:52:38.390 align:middle line:84% If T has such a representation, then 00:52:38.390 --> 00:52:41.420 align:middle line:84% it has finite rank because then the image is 00:52:41.420 --> 00:52:45.470 align:middle line:84% contained in the span of the ei's for i going from 1 to l. 00:52:45.470 --> 00:52:47.820 align:middle line:84% And that's a finite dimensional subspace. 00:52:47.820 --> 00:52:54.460 align:middle line:84% So this is clear that this condition implies that T 00:52:54.460 --> 00:52:55.850 align:middle line:90% is a finite rank operator. 00:52:55.850 --> 00:53:02.680 align:middle line:84% So let's go the opposite direction. 00:53:02.680 --> 00:53:16.880 align:middle line:84% So since the range of T is finite dimensional, we 00:53:16.880 --> 00:53:30.800 align:middle line:84% can find an orthonormal basis, call it ek, 00:53:30.800 --> 00:53:42.540 align:middle line:84% k equals 1 to N such that T applied to u is equal to-- 00:53:42.540 --> 00:53:46.440 align:middle line:84% so it's an element in the span of-- 00:53:46.440 --> 00:53:49.500 align:middle line:90% so this spans the range. 00:53:49.500 --> 00:53:56.060 align:middle line:84% So I can write this as some numbers times the ek's. 00:53:56.060 --> 00:53:58.575 align:middle line:90% 00:53:58.575 --> 00:53:59.700 align:middle line:90% And what are these numbers? 00:53:59.700 --> 00:54:01.742 align:middle line:84% Well, these numbers, since these are orthonormal, 00:54:01.742 --> 00:54:06.100 align:middle line:84% have to be T applied to u in a product ek. 00:54:06.100 --> 00:54:09.460 align:middle line:90% 00:54:09.460 --> 00:54:11.095 align:middle line:90% All right, so Tu-- 00:54:11.095 --> 00:54:13.660 align:middle line:90% 00:54:13.660 --> 00:54:17.590 align:middle line:84% so the ranges spanned by these orthonormal vectors, 00:54:17.590 --> 00:54:21.040 align:middle line:84% and anything in the span of this orthonormal vectors 00:54:21.040 --> 00:54:25.205 align:middle line:84% has to be written as that thing in a product ek times ek. 00:54:25.205 --> 00:54:27.940 align:middle line:90% 00:54:27.940 --> 00:54:30.250 align:middle line:84% But let me now make an observation 00:54:30.250 --> 00:54:39.710 align:middle line:84% that this thing right here we can 00:54:39.710 --> 00:54:53.880 align:middle line:84% write as using the adjoint as u in a product T star ek, ek. 00:54:53.880 --> 00:54:56.490 align:middle line:90% 00:54:56.490 --> 00:55:00.990 align:middle line:84% The adjoint operator, so T star which 00:55:00.990 --> 00:55:03.870 align:middle line:84% goes from H to H, remember satisfies this property 00:55:03.870 --> 00:55:07.200 align:middle line:90% that given any two vectors a-- 00:55:07.200 --> 00:55:10.320 align:middle line:84% or u and v, T applied to u in a product v 00:55:10.320 --> 00:55:13.440 align:middle line:84% is equal to u in a product T star v. That's 00:55:13.440 --> 00:55:17.580 align:middle line:84% the defining property of the adjoint 00:55:17.580 --> 00:55:19.090 align:middle line:90% So I can write it like this. 00:55:19.090 --> 00:55:24.680 align:middle line:84% And-- which is k equals 1 to N of u in a product, 00:55:24.680 --> 00:55:31.040 align:middle line:84% call this vk, ek bar, where here vk is simply 00:55:31.040 --> 00:55:46.240 align:middle line:84% defined to be T star ek now, let e1 up 00:55:46.240 --> 00:56:07.470 align:middle line:84% to el be the orthonormal subset of H obtained 00:56:07.470 --> 00:56:20.530 align:middle line:84% by applying the Gram-Schmidt process to vectors E1, e2 00:56:20.530 --> 00:56:26.290 align:middle line:90% up to eL vL, up to vL. 00:56:26.290 --> 00:56:31.210 align:middle line:84% So applying it to this subset, I keep the e1 up to eL's. 00:56:31.210 --> 00:56:32.740 align:middle line:84% And then I might pick up some more 00:56:32.740 --> 00:56:35.380 align:middle line:84% from some more orthonormal vectors 00:56:35.380 --> 00:56:39.160 align:middle line:84% that are the normal to these guys when 00:56:39.160 --> 00:56:40.480 align:middle line:90% I hit the v1's up to vl. 00:56:40.480 --> 00:56:44.020 align:middle line:90% 00:56:44.020 --> 00:56:47.120 align:middle line:84% I should have gone to N, I'm sorry. 00:56:47.120 --> 00:56:55.110 align:middle line:90% So not L here, but N. 00:56:55.110 --> 00:56:58.440 align:middle line:84% So I obtained an orthonormal subset 00:56:58.440 --> 00:57:02.740 align:middle line:84% that spans the same span of these guys. 00:57:02.740 --> 00:57:05.910 align:middle line:90% And so what does that mean? 00:57:05.910 --> 00:57:09.940 align:middle line:90% 00:57:09.940 --> 00:57:20.590 align:middle line:84% Then there exists numbers akj, bkj-- 00:57:20.590 --> 00:57:21.430 align:middle line:90% let's call it i-- 00:57:21.430 --> 00:57:25.460 align:middle line:90% 00:57:25.460 --> 00:57:33.180 align:middle line:84% such that ek bar is in the span of these orthonormal vectors. 00:57:33.180 --> 00:57:44.500 align:middle line:84% So I can write it as aki, ei k equals 1 to l-- 00:57:44.500 --> 00:57:46.990 align:middle line:84% no, no, no, sums going the wrong way-- 00:57:46.990 --> 00:57:58.550 align:middle line:84% i equals 1 to l, and also the same for the v's, some vk 00:57:58.550 --> 00:58:09.500 align:middle line:90% equals j equals 1 to l bkj kj. 00:58:09.500 --> 00:58:14.810 align:middle line:84% So these are obtained by the Gram-Schmidt process applied 00:58:14.810 --> 00:58:16.280 align:middle line:90% to this collection of vectors. 00:58:16.280 --> 00:58:18.890 align:middle line:84% And therefore I can write each one of these vectors 00:58:18.890 --> 00:58:22.020 align:middle line:84% as a linear combination of these orthonormal vectors. 00:58:22.020 --> 00:58:25.580 align:middle line:90% So that's all I'm saying here. 00:58:25.580 --> 00:58:32.110 align:middle line:84% Now we just plug this in to this computation. 00:58:32.110 --> 00:58:36.040 align:middle line:84% And we have Tu which we computed was equal to sum from k 00:58:36.040 --> 00:58:42.280 align:middle line:84% equals 1 to N of u in a product vk, ek bar. 00:58:42.280 --> 00:58:43.540 align:middle line:90% This is equal to-- 00:58:43.540 --> 00:58:47.360 align:middle line:84% now if I stick in these sums here, 00:58:47.360 --> 00:58:53.080 align:middle line:84% I get sum ij equals 1 to L of some k 00:58:53.080 --> 00:59:06.290 align:middle line:84% equals 1 to N of aki, bkj complex conjugate. 00:59:06.290 --> 00:59:12.040 align:middle line:90% 00:59:12.040 --> 00:59:14.175 align:middle line:90% Now, u in a product ej-- 00:59:14.175 --> 00:59:18.610 align:middle line:90% 00:59:18.610 --> 00:59:21.970 align:middle line:84% this is coming from when I stick in the vk's-- 00:59:21.970 --> 00:59:23.260 align:middle line:90% ei. 00:59:23.260 --> 00:59:24.760 align:middle line:84% And this is coming from when I stick 00:59:24.760 --> 00:59:29.200 align:middle line:90% in what I have for the ek bar. 00:59:29.200 --> 00:59:33.430 align:middle line:84% And this number right here is my number cij. 00:59:33.430 --> 00:59:38.370 align:middle line:90% 00:59:38.370 --> 00:59:39.240 align:middle line:90% And that's the end. 00:59:39.240 --> 00:59:44.370 align:middle line:84% So finite rank operators are really just matrices. 00:59:44.370 --> 00:59:46.350 align:middle line:84% You can just compute these numbers 00:59:46.350 --> 00:59:50.730 align:middle line:84% and they completely characterize the linear operator. 00:59:50.730 --> 00:59:55.760 align:middle line:90% 00:59:55.760 --> 00:59:59.350 align:middle line:84% So in particular, the kernel, the null space of T 00:59:59.350 --> 01:00:06.370 align:middle line:84% is the subspace that's orthogonal to the e1 up to eL, 01:00:06.370 --> 01:00:07.690 align:middle line:90% or at least includes that. 01:00:07.690 --> 01:00:20.660 align:middle line:90% 01:00:20.660 --> 01:00:31.670 align:middle line:84% Now from this we can conclude a nice property about finite rank 01:00:31.670 --> 01:00:36.260 align:middle line:84% operators, namely the first is that if T is a finite rank 01:00:36.260 --> 01:00:40.160 align:middle line:84% operator, then its adjoint is also a finite rank operator. 01:00:40.160 --> 01:00:47.200 align:middle line:90% 01:00:47.200 --> 01:00:53.790 align:middle line:84% And if T is a finite rank operator 01:00:53.790 --> 01:01:02.280 align:middle line:84% and A and B are just bounded linear operators on H, then 01:01:02.280 --> 01:01:09.520 align:middle line:84% A times T times B, this is also a finite rank operator. 01:01:09.520 --> 01:01:12.640 align:middle line:84% In fancy or in short form you would 01:01:12.640 --> 01:01:20.590 align:middle line:84% say that this is a star closed two-sided ideal in the space 01:01:20.590 --> 01:01:26.110 align:middle line:84% of bounded linear operators, or its closed undertaking adjoints 01:01:26.110 --> 01:01:29.590 align:middle line:84% in closed undertaking the two-sided multiplication 01:01:29.590 --> 01:01:33.600 align:middle line:90% by bounded linear operators. 01:01:33.600 --> 01:01:36.910 align:middle line:90% So 2 I'll leave as an exercise. 01:01:36.910 --> 01:01:42.790 align:middle line:90% 01:01:42.790 --> 01:01:45.310 align:middle line:90% What's the point? 01:01:45.310 --> 01:01:49.270 align:middle line:84% The T has finite dimensional range. 01:01:49.270 --> 01:01:52.270 align:middle line:84% A then acts on that finite dimensional range. 01:01:52.270 --> 01:01:56.970 align:middle line:84% And therefore its range, the range of A hitting T 01:01:56.970 --> 01:01:59.130 align:middle line:90% must then be finite dimensional. 01:01:59.130 --> 01:02:03.200 align:middle line:84% And what you do with B really doesn't matter. 01:02:03.200 --> 01:02:05.150 align:middle line:90% So 1 is an exercise. 01:02:05.150 --> 01:02:05.905 align:middle line:90% So let's prove 1. 01:02:05.905 --> 01:02:10.690 align:middle line:84% 1 we'll prove using this characterization of finite rink 01:02:10.690 --> 01:02:11.810 align:middle line:90% operators. 01:02:11.810 --> 01:02:17.470 align:middle line:84% So we're assuming T is finite rank. 01:02:17.470 --> 01:02:29.650 align:middle line:84% So we can write T as some constant cij, u and a product 01:02:29.650 --> 01:02:37.530 align:middle line:84% ej, ei, u in H. So we can express it in that way 01:02:37.530 --> 01:02:41.680 align:middle line:84% where these are just some fixed constants not depending on u. 01:02:41.680 --> 01:02:46.950 align:middle line:84% Now let's compute how T star x on vectors 01:02:46.950 --> 01:02:50.280 align:middle line:84% by using this defining property of the adjoint. 01:02:50.280 --> 01:02:53.410 align:middle line:90% 01:02:53.410 --> 01:02:56.720 align:middle line:90% Then u, T star v-- 01:02:56.720 --> 01:02:58.720 align:middle line:84% what we're going to do is we're going to come up 01:02:58.720 --> 01:03:03.700 align:middle line:84% with an expression for this as u in a product something 01:03:03.700 --> 01:03:06.820 align:middle line:84% and from which we can conclude that T star v has to equal 01:03:06.820 --> 01:03:12.040 align:middle line:90% that something for all v. 01:03:12.040 --> 01:03:16.120 align:middle line:84% Now this is equal to, by the defining 01:03:16.120 --> 01:03:20.980 align:middle line:84% property of the adjoint, T applied to u, v. So this 01:03:20.980 --> 01:03:21.625 align:middle line:90% is equal to-- 01:03:21.625 --> 01:03:25.840 align:middle line:90% 01:03:25.840 --> 01:03:28.300 align:middle line:84% i and j will be understood to go from 1 to l. 01:03:28.300 --> 01:03:31.960 align:middle line:84% So this is just finite, a finite sum, OK-- 01:03:31.960 --> 01:03:39.590 align:middle line:90% cij, u, ei-- or ej, sorry-- 01:03:39.590 --> 01:03:42.850 align:middle line:90% ei, inner product v. 01:03:42.850 --> 01:03:50.000 align:middle line:84% Now taking the inner product is linear in the first entry. 01:03:50.000 --> 01:03:52.070 align:middle line:90% So this is equal to sum ij. 01:03:52.070 --> 01:03:58.530 align:middle line:84% Again, this is a finite sum ij, u, ej, inner product 01:03:58.530 --> 01:04:08.990 align:middle line:84% ei, v. And now I'm just going to switch what I'm summing. 01:04:08.990 --> 01:04:25.880 align:middle line:84% This I can write as now u inner product sum ij cij bar. 01:04:25.880 --> 01:04:28.100 align:middle line:90% So if I want to-- 01:04:28.100 --> 01:04:41.430 align:middle line:84% this I can write as bar, bar, but ei, v, complex conjugate, 01:04:41.430 --> 01:04:41.930 align:middle line:90% ej. 01:04:41.930 --> 01:04:46.350 align:middle line:90% 01:04:46.350 --> 01:04:48.570 align:middle line:84% We can check this as correct because-- 01:04:48.570 --> 01:04:50.640 align:middle line:84% so u will then inner product with each 01:04:50.640 --> 01:04:53.070 align:middle line:84% of the ej's and these constants when they come out 01:04:53.070 --> 01:04:56.310 align:middle line:84% of the inner product, get hit by a complex conjugate, which 01:04:56.310 --> 01:05:01.190 align:middle line:84% turns them back into cij and ei, v. 01:05:01.190 --> 01:05:05.630 align:middle line:84% Now, the complex conjugate of the inner product 01:05:05.630 --> 01:05:11.370 align:middle line:84% is just the same as if I flip the entry. 01:05:11.370 --> 01:05:19.235 align:middle line:84% So this is equal to cij times v, ei, ej. 01:05:19.235 --> 01:05:24.070 align:middle line:90% 01:05:24.070 --> 01:05:27.040 align:middle line:84% Now, this holds for all u and v. So what have I shown? 01:05:27.040 --> 01:05:34.570 align:middle line:84% I've shown that u T star v minus this finite sum, so I actually 01:05:34.570 --> 01:05:46.210 align:middle line:84% put it in here, cij v, ei, vj, this equals 0 for all u, 01:05:46.210 --> 01:05:48.190 align:middle line:90% for all v. 01:05:48.190 --> 01:05:51.790 align:middle line:84% Now, for a fixed v this quantity here 01:05:51.790 --> 01:05:56.410 align:middle line:84% is orthogonal to everything in H. So it has to be 0. 01:05:56.410 --> 01:05:58.300 align:middle line:84% And therefore I conclude that T star 01:05:58.300 --> 01:06:04.690 align:middle line:84% v has to equal sum from ij equals 1 to l, 01:06:04.690 --> 01:06:11.350 align:middle line:84% the complex conjugates of cij, v inner product ei, 01:06:11.350 --> 01:06:16.450 align:middle line:84% ej for all v in H. And that proves 01:06:16.450 --> 01:06:24.350 align:middle line:84% that T star is a finite rank, and also 01:06:24.350 --> 01:06:28.220 align:middle line:84% gives you the expression for the coefficients in terms 01:06:28.220 --> 01:06:29.150 align:middle line:90% of the old ones. 01:06:29.150 --> 01:06:32.120 align:middle line:90% I could write this-- 01:06:32.120 --> 01:06:36.110 align:middle line:84% now if I wanted to write it in terms of the i 01:06:36.110 --> 01:06:39.740 align:middle line:84% being in front of the bases vector that I'm having here, 01:06:39.740 --> 01:06:42.254 align:middle line:90% I write it as sum ij. 01:06:42.254 --> 01:06:48.870 align:middle line:90% 01:06:48.870 --> 01:06:52.920 align:middle line:84% So we see that the matrix corresponding to the adjoint is 01:06:52.920 --> 01:06:55.830 align:middle line:84% what we computed last class basically, 01:06:55.830 --> 01:07:01.080 align:middle line:84% is the complex conjugate of the transpose of this matrix, 01:07:01.080 --> 01:07:02.390 align:middle line:90% of the original matrix cij's. 01:07:02.390 --> 01:07:06.520 align:middle line:90% 01:07:06.520 --> 01:07:09.775 align:middle line:84% All right, so that proves adjoints are also finite rank. 01:07:09.775 --> 01:07:13.330 align:middle line:90% 01:07:13.330 --> 01:07:25.270 align:middle line:84% Now, so taking the adjoint of a finite rank, 01:07:25.270 --> 01:07:31.570 align:middle line:84% operator leaves you in the space of finite rank operators. 01:07:31.570 --> 01:07:34.750 align:middle line:84% If you take finite rank operators 01:07:34.750 --> 01:07:37.750 align:middle line:84% and compose them on the left and on the right 01:07:37.750 --> 01:07:42.295 align:middle line:84% with bounded operators, you remain finite rank. 01:07:42.295 --> 01:07:47.610 align:middle line:90% 01:07:47.610 --> 01:07:49.470 align:middle line:84% The subset of finite rank operators 01:07:49.470 --> 01:07:54.600 align:middle line:84% also form a subspace in the Banach space 01:07:54.600 --> 01:07:58.830 align:middle line:84% of bounded linear operators on a Hilbert space. 01:07:58.830 --> 01:08:01.650 align:middle line:84% So the space of bounded linear operators on a Hilbert space 01:08:01.650 --> 01:08:07.620 align:middle line:84% naturally come with a norm as well, the operator norm. 01:08:07.620 --> 01:08:17.109 align:middle line:84% So the next obvious question is, is the subspace of finite rank 01:08:17.109 --> 01:08:18.990 align:middle line:90% operators, is this closed? 01:08:18.990 --> 01:08:38.490 align:middle line:90% 01:08:38.490 --> 01:08:41.979 align:middle line:84% Closed not meaning-- not as a closed undertaking 01:08:41.979 --> 01:08:42.939 align:middle line:90% linear combinations. 01:08:42.939 --> 01:08:44.606 align:middle line:84% Of course that holds for every subspace. 01:08:44.606 --> 01:08:56.770 align:middle line:84% But if I take a sequence of finite rank operators 01:08:56.770 --> 01:09:11.460 align:middle line:84% that are converging in operator norm to something, 01:09:11.460 --> 01:09:18.200 align:middle line:84% then is the limit also a finite rank operator? 01:09:18.200 --> 01:09:23.090 align:middle line:84% So that's the definition of being a closed subspace, 01:09:23.090 --> 01:09:29.830 align:middle line:84% or let me say subset so that we're now asking 01:09:29.830 --> 01:09:31.870 align:middle line:90% about this in a metric sense. 01:09:31.870 --> 01:09:37.210 align:middle line:84% Is it a closed subset in the space 01:09:37.210 --> 01:09:39.290 align:middle line:90% of bounded linear operators? 01:09:39.290 --> 01:09:40.749 align:middle line:90% Now, the answer to this is no. 01:09:40.749 --> 01:09:45.540 align:middle line:90% 01:09:45.540 --> 01:09:48.000 align:middle line:90% Why? 01:09:48.000 --> 01:09:54.180 align:middle line:90% Let's, for example, take Tn. 01:09:54.180 --> 01:09:59.280 align:middle line:84% So this will be a sequence of operators from little l2 01:09:59.280 --> 01:10:01.740 align:middle line:90% to little l2. 01:10:01.740 --> 01:10:12.610 align:middle line:84% And for an element a in little l2, 01:10:12.610 --> 01:10:14.920 align:middle line:84% this should give me another sequence in little l2. 01:10:14.920 --> 01:10:19.720 align:middle line:84% And it'll be the sequence given by a1 over 1, a2 over 2, 01:10:19.720 --> 01:10:26.290 align:middle line:84% up to an over n, and then zeros after that. 01:10:26.290 --> 01:10:38.780 align:middle line:84% All right, so T1, the operator T1 01:10:38.780 --> 01:10:41.210 align:middle line:84% takes the sequence in little l2 and just spits out 01:10:41.210 --> 01:10:44.180 align:middle line:84% the sequence that has the first entry divided by 1. 01:10:44.180 --> 01:10:45.950 align:middle line:90% And 0's followed after that. 01:10:45.950 --> 01:10:47.990 align:middle line:90% That's the operator T1. 01:10:47.990 --> 01:10:50.840 align:middle line:84% T2 takes a sequence in little l2 and spits 01:10:50.840 --> 01:10:53.150 align:middle line:84% out the first entry divided by 1, 01:10:53.150 --> 01:11:01.970 align:middle line:84% the second entry divided by 2, and o's after that, T3, so on. 01:11:01.970 --> 01:11:09.260 align:middle line:90% Now, this is just a picture. 01:11:09.260 --> 01:11:09.980 align:middle line:90% This is formal. 01:11:09.980 --> 01:11:12.050 align:middle line:84% This is not meant to mean anything. 01:11:12.050 --> 01:11:15.590 align:middle line:84% If I were to express this as an infinite matrix 01:11:15.590 --> 01:11:18.830 align:middle line:84% times an infinite vector, or infinite length vector where 01:11:18.830 --> 01:11:22.400 align:middle line:84% the infinite length is the sequence ak's, this 01:11:22.400 --> 01:11:33.100 align:middle line:84% is 1, 1/2, 1/3, 1 over n, 0, 0, 0, 0. 01:11:33.100 --> 01:11:54.130 align:middle line:84% This infinite matrix times a1, a2, a3, and so on. 01:11:54.130 --> 01:12:00.810 align:middle line:84% So I'm just multiplying a1 by 1, a2 by 1/2, a3 by a 1/3, 01:12:00.810 --> 01:12:06.150 align:middle line:84% and then up to a n over n, multiplying a n by 1 over n. 01:12:06.150 --> 01:12:09.720 align:middle line:84% And then the rest of the entries get spat out to give me 0. 01:12:09.720 --> 01:12:12.960 align:middle line:84% So these are the finite rank operators. 01:12:12.960 --> 01:12:21.870 align:middle line:90% 01:12:21.870 --> 01:12:36.930 align:middle line:84% And you can check that the Tn minus T in operator norm 01:12:36.930 --> 01:12:42.240 align:middle line:90% goes to 0, where what is T? 01:12:42.240 --> 01:12:45.990 align:middle line:84% T is the operator that takes a spits out-- 01:12:45.990 --> 01:12:53.320 align:middle line:84% so a will be an element in little l2. 01:12:53.320 --> 01:12:55.780 align:middle line:84% Now as n goes to infinity, you can kind of 01:12:55.780 --> 01:12:58.720 align:middle line:84% guess what's going to happen is I end up multiplying everything 01:12:58.720 --> 01:13:03.730 align:middle line:84% by 1 over wherever entry I'm at, a1 over 1, 01:13:03.730 --> 01:13:10.600 align:middle line:84% a2 over 2, a3 over 3, a4 over 4, and so on, and on. 01:13:10.600 --> 01:13:15.390 align:middle line:90% 01:13:15.390 --> 01:13:20.600 align:middle line:84% So as an exercise you can show that the operator 01:13:20.600 --> 01:13:22.850 align:middle line:84% norm of T minus Tn, this is less than 01:13:22.850 --> 01:13:30.180 align:middle line:84% or equal to v1 over n plus 1, something like that, OK. 01:13:30.180 --> 01:13:34.880 align:middle line:90% 01:13:34.880 --> 01:13:39.620 align:middle line:84% Now, this operator here does not have finite rank 01:13:39.620 --> 01:13:52.920 align:middle line:84% because then I can find infinitely many linearly 01:13:52.920 --> 01:13:54.360 align:middle line:84% independent vectors in the range. 01:13:54.360 --> 01:13:58.590 align:middle line:84% For example T of e1 where this here is 01:13:58.590 --> 01:14:01.580 align:middle line:90% the basis vector gives me the-- 01:14:01.580 --> 01:14:07.760 align:middle line:90% 01:14:07.760 --> 01:14:10.340 align:middle line:84% this is the first basis vector in little l2. 01:14:10.340 --> 01:14:13.130 align:middle line:84% If this is the second basis vector in little l2, 01:14:13.130 --> 01:14:13.955 align:middle line:90% this is just-- 01:14:13.955 --> 01:14:20.470 align:middle line:90% 01:14:20.470 --> 01:14:21.500 align:middle line:90% and so on. 01:14:21.500 --> 01:14:26.690 align:middle line:90% If I applied T to en, This. 01:14:26.690 --> 01:14:30.560 align:middle line:90% Is 1 over n, 0, 0, and so on. 01:14:30.560 --> 01:14:31.760 align:middle line:90% This is in the n spot. 01:14:31.760 --> 01:14:36.040 align:middle line:90% 01:14:36.040 --> 01:14:39.720 align:middle line:84% So it just takes each basis vector and-- 01:14:39.720 --> 01:14:47.070 align:middle line:84% so if you like, this is e1, this is 1/2 e2, this is 1 over n en, 01:14:47.070 --> 01:14:52.560 align:middle line:84% and multiplies each of these basis vectors by 1 over 01:14:52.560 --> 01:14:56.600 align:middle line:90% the integer that's-- 01:14:56.600 --> 01:15:00.080 align:middle line:90% 01:15:00.080 --> 01:15:04.080 align:middle line:84% not delineating-- denoting, I'm sorry. 01:15:04.080 --> 01:15:11.990 align:middle line:84% All right, it's the end of a day so some of my thinking 01:15:11.990 --> 01:15:12.680 align:middle line:90% has slowed down. 01:15:12.680 --> 01:15:15.230 align:middle line:90% 01:15:15.230 --> 01:15:19.010 align:middle line:84% So finite rank operators are not closed-- 01:15:19.010 --> 01:15:22.580 align:middle line:84% is not a closed subset of the space 01:15:22.580 --> 01:15:24.770 align:middle line:90% of bounded linear operators. 01:15:24.770 --> 01:15:27.870 align:middle line:90% So what's the closure? 01:15:27.870 --> 01:15:32.120 align:middle line:90% 01:15:32.120 --> 01:15:34.520 align:middle line:84% And finite rank operators we'd like to think we 01:15:34.520 --> 01:15:36.770 align:middle line:90% know how to solve equations for. 01:15:36.770 --> 01:15:41.690 align:middle line:84% I mean, just based on this expression for a finite rank 01:15:41.690 --> 01:15:48.630 align:middle line:84% operator, well, if we want to solve Tu equals v, well, first 01:15:48.630 --> 01:15:55.240 align:middle line:84% we know v has to be in the range of or in the span 01:15:55.240 --> 01:15:57.610 align:middle line:84% of these basis vectors that appear here. 01:15:57.610 --> 01:16:00.610 align:middle line:84% And now once we've restricted to that, 01:16:00.610 --> 01:16:05.620 align:middle line:84% then we can solve the equation by looking at the properties 01:16:05.620 --> 01:16:06.550 align:middle line:90% of this matrix. 01:16:06.550 --> 01:16:09.880 align:middle line:84% And the null space will be not only the null space 01:16:09.880 --> 01:16:12.730 align:middle line:84% that sits inside the span of these basis vectors, 01:16:12.730 --> 01:16:16.870 align:middle line:84% but it will also include the orthogonal complement 01:16:16.870 --> 01:16:20.170 align:middle line:90% in H of these basis vectors. 01:16:20.170 --> 01:16:25.950 align:middle line:84% So solving equations involving finite rank operators 01:16:25.950 --> 01:16:29.490 align:middle line:84% ought to be fairly straightforward 01:16:29.490 --> 01:16:30.785 align:middle line:90% if we remember linear algebra. 01:16:30.785 --> 01:16:34.050 align:middle line:90% 01:16:34.050 --> 01:16:37.847 align:middle line:84% Unfortunately, they're not closed upon taking 01:16:37.847 --> 01:16:38.930 align:middle line:90% some subsequential limits. 01:16:38.930 --> 01:16:44.750 align:middle line:84% But we can identify the closure of-- 01:16:44.750 --> 01:16:54.440 align:middle line:84% so now the question is, what is the closure 01:16:54.440 --> 01:16:59.990 align:middle line:84% of finite rank operators in the space 01:16:59.990 --> 01:17:02.060 align:middle line:84% of bounded linear operators on H? 01:17:02.060 --> 01:17:04.640 align:middle line:84% So something in the closure has to be a limit of finite rank 01:17:04.640 --> 01:17:05.510 align:middle line:90% operators. 01:17:05.510 --> 01:17:09.140 align:middle line:84% And since we know how to solve finite-- 01:17:09.140 --> 01:17:11.150 align:middle line:84% equations involving finite rank operators, 01:17:11.150 --> 01:17:13.670 align:middle line:84% hopefully we know how to solve equations use 01:17:13.670 --> 01:17:15.980 align:middle line:90% with these operators as well. 01:17:15.980 --> 01:17:18.440 align:middle line:84% Is there a simple way we can characterize-- 01:17:18.440 --> 01:17:23.160 align:middle line:84% or not simple, but at least a more useful way 01:17:23.160 --> 01:17:26.430 align:middle line:84% to characterize them than just being the limit of finite rank 01:17:26.430 --> 01:17:27.330 align:middle line:90% operators? 01:17:27.330 --> 01:17:30.810 align:middle line:90% 01:17:30.810 --> 01:17:33.030 align:middle line:84% So the answer to this question is-- 01:17:33.030 --> 01:17:34.290 align:middle line:90% so I won't even write-- 01:17:34.290 --> 01:17:36.360 align:middle line:90% OK, I'll write answer-- 01:17:36.360 --> 01:17:40.800 align:middle line:84% is what are called compact operators, so 01:17:40.800 --> 01:17:42.810 align:middle line:84% the subspace of compact operators. 01:17:42.810 --> 01:17:48.340 align:middle line:90% 01:17:48.340 --> 01:17:50.780 align:middle line:90% So let me make a definition. 01:17:50.780 --> 01:18:06.040 align:middle line:84% So k, a bounded linear operator is compact operator 01:18:06.040 --> 01:18:24.770 align:middle line:84% if the closure of the image of the closed unit ball in H-- 01:18:24.770 --> 01:18:28.340 align:middle line:84% so the closure of the image by k-- 01:18:28.340 --> 01:18:30.590 align:middle line:84% or the closure of the image of the closed unit ball 01:18:30.590 --> 01:18:31.145 align:middle line:90% is compact. 01:18:31.145 --> 01:18:35.010 align:middle line:90% 01:18:35.010 --> 01:18:41.520 align:middle line:84% I can also write this as a closure of the subset ku norm 01:18:41.520 --> 01:18:43.140 align:middle line:90% of u less than or equal to 1. 01:18:43.140 --> 01:18:51.550 align:middle line:90% 01:18:51.550 --> 01:18:56.520 align:middle line:84% So I won't have time to prove this theorem in this class, 01:18:56.520 --> 01:18:58.440 align:middle line:90% but we'll do it next time. 01:18:58.440 --> 01:19:02.200 align:middle line:84% And we'll actually show that, in fact, 01:19:02.200 --> 01:19:17.430 align:middle line:84% this class of compact operators which are defined in this way, 01:19:17.430 --> 01:19:21.930 align:middle line:84% this is a compact operator according to this definition 01:19:21.930 --> 01:19:31.750 align:middle line:84% if and only if k is in the closure of finite rank 01:19:31.750 --> 01:19:34.570 align:middle line:84% operators, meaning there exists a sequence of finite rank 01:19:34.570 --> 01:19:38.920 align:middle line:84% operators converging in operator norm to k. 01:19:38.920 --> 01:19:41.350 align:middle line:84% And, again, why are we interested 01:19:41.350 --> 01:19:44.830 align:middle line:90% in compact operators? 01:19:44.830 --> 01:19:47.320 align:middle line:84% Well, we know how to solve finite rank 01:19:47.320 --> 01:19:49.990 align:middle line:84% or equations involving finite rank operators. 01:19:49.990 --> 01:19:57.040 align:middle line:84% We just look at the subspace which that contains the range. 01:19:57.040 --> 01:20:01.120 align:middle line:84% And then we look at the matrix that appears there 01:20:01.120 --> 01:20:03.870 align:middle line:90% in that decomposition. 01:20:03.870 --> 01:20:11.580 align:middle line:84% Compact operators there are something completely new. 01:20:11.580 --> 01:20:16.620 align:middle line:84% When we move from finite dimensional linear algebra 01:20:16.620 --> 01:20:20.850 align:middle line:84% now to functional analysis because these 01:20:20.850 --> 01:20:25.740 align:middle line:84% are not exactly necessarily equal to finite rank operators. 01:20:25.740 --> 01:20:27.600 align:middle line:84% I mean, we just did this example a minute 01:20:27.600 --> 01:20:34.090 align:middle line:84% ago of this operator given here by where 01:20:34.090 --> 01:20:38.440 align:middle line:84% I take the entry of the sequence and divide it by the place 01:20:38.440 --> 01:20:42.400 align:middle line:84% that it occurred in the sequence for a sequence given in l2. 01:20:42.400 --> 01:20:47.230 align:middle line:84% So if you believe the theorem, this is a finite rank-- 01:20:47.230 --> 01:20:48.040 align:middle line:90% I mean, not finite. 01:20:48.040 --> 01:20:49.630 align:middle line:90% This is a compact operator. 01:20:49.630 --> 01:20:56.940 align:middle line:84% It's the limit of this sequence of finite rank operators. 01:20:56.940 --> 01:21:01.170 align:middle line:84% Perhaps you want to take a try at trying 01:21:01.170 --> 01:21:03.090 align:middle line:84% to prove this as a compact operator 01:21:03.090 --> 01:21:07.140 align:middle line:84% directly from the definition, or not. 01:21:07.140 --> 01:21:10.710 align:middle line:84% you can wait till next time and just use this theorem. 01:21:10.710 --> 01:21:15.110 align:middle line:84% But many other operators, many interesting operators, 01:21:15.110 --> 01:21:18.530 align:middle line:84% for example, inverses of differentiable operators 01:21:18.530 --> 01:21:21.020 align:middle line:84% in fact turn out to be compact operators. 01:21:21.020 --> 01:21:22.880 align:middle line:84% That's also why we're interested in them. 01:21:22.880 --> 01:21:27.440 align:middle line:90% 01:21:27.440 --> 01:21:32.480 align:middle line:84% So we got these by taking the closure of finite rank 01:21:32.480 --> 01:21:33.170 align:middle line:90% operators. 01:21:33.170 --> 01:21:39.490 align:middle line:84% And then after-- so now we have this interesting subspace 01:21:39.490 --> 01:21:43.720 align:middle line:84% of operators that we hope to be able to solve equations 01:21:43.720 --> 01:21:44.850 align:middle line:90% involving. 01:21:44.850 --> 01:21:50.630 align:middle line:84% And that'll be basically what takes up the remainder 01:21:50.630 --> 01:21:52.385 align:middle line:90% of our time in this course. 01:21:52.385 --> 01:21:56.500 align:middle line:90% 01:21:56.500 --> 01:21:59.870 align:middle line:84% So you just staring at this definition, 01:21:59.870 --> 01:22:01.240 align:middle line:90% you can kind of see that-- 01:22:01.240 --> 01:22:05.350 align:middle line:84% well, you can see that if k is a finite rank operator, 01:22:05.350 --> 01:22:07.390 align:middle line:90% then it's a compact operator. 01:22:07.390 --> 01:22:10.630 align:middle line:84% So we're not going to prove this theorem, like I said, but just 01:22:10.630 --> 01:22:17.390 align:middle line:84% to see or do a sanity check that finite rank operators ought 01:22:17.390 --> 01:22:18.590 align:middle line:90% to be compact operators. 01:22:18.590 --> 01:22:24.830 align:middle line:84% Well, a finite rank operator, the image of this 01:22:24.830 --> 01:22:28.400 align:middle line:84% closed unit ball of a finite by a finite rank operator, 01:22:28.400 --> 01:22:32.240 align:middle line:84% that's going to be a bounded subset 01:22:32.240 --> 01:22:35.540 align:middle line:84% of a finite dimensional subspace. 01:22:35.540 --> 01:22:39.260 align:middle line:84% Any bounded operator takes bounded sets to bounded sets. 01:22:39.260 --> 01:22:43.880 align:middle line:84% So if it's finite rank, it takes this bounded set 01:22:43.880 --> 01:22:51.270 align:middle line:84% to a bounded subset of a finite dimensional subspace of H. 01:22:51.270 --> 01:22:52.470 align:middle line:90% And then I take the closure. 01:22:52.470 --> 01:22:55.770 align:middle line:84% I get a bounded and closed subset 01:22:55.770 --> 01:22:58.120 align:middle line:84% of a finite dimensional subspace. 01:22:58.120 --> 01:23:04.290 align:middle line:84% And we know by Heine-Borel that closed and bounded subsets 01:23:04.290 --> 01:23:07.650 align:middle line:84% of finite dimensional subspaces are compact. 01:23:07.650 --> 01:23:11.100 align:middle line:84% So that shows why at the very minimum 01:23:11.100 --> 01:23:13.830 align:middle line:84% you should believe that finite rank operators are compact. 01:23:13.830 --> 01:23:16.430 align:middle line:90% OK, so we'll stop there. 01:23:16.430 --> 01:23:22.000 align:middle line:90%