WEBVTT 00:00:00.000 --> 00:00:18.780 align:middle line:84% [SQUEAKING] [RUSTLING] [CLICKING] 00:00:18.780 --> 00:00:21.390 align:middle line:84% PROFESSOR: All right, so let's continue our discussion 00:00:21.390 --> 00:00:22.905 align:middle line:90% of Banach spaces. 00:00:22.905 --> 00:00:25.500 align:middle line:90% 00:00:25.500 --> 00:00:39.870 align:middle line:84% So let V be a norm space, meaning a vector space 00:00:39.870 --> 00:00:41.520 align:middle line:90% with a norm on it. 00:00:41.520 --> 00:00:43.440 align:middle line:84% And last time, a Banach space was 00:00:43.440 --> 00:00:49.150 align:middle line:84% defined to be a norm space such that the metric induced 00:00:49.150 --> 00:00:51.580 align:middle line:90% by this norm is complete-- 00:00:51.580 --> 00:00:55.040 align:middle line:90% all Cauchy sequences converge. 00:00:55.040 --> 00:00:58.510 align:middle line:84% So if you want to check that a norm space is a Banach space, 00:00:58.510 --> 00:01:01.060 align:middle line:84% you have to take a Cauchy sequence 00:01:01.060 --> 00:01:04.330 align:middle line:84% and show that it converges in the space, which 00:01:04.330 --> 00:01:08.540 align:middle line:84% we did last time for the space of bounded continuous functions 00:01:08.540 --> 00:01:10.970 align:middle line:90% on a metric space. 00:01:10.970 --> 00:01:15.920 align:middle line:84% Now, there's an alternative, useful way 00:01:15.920 --> 00:01:18.770 align:middle line:84% of checking to see if a space is a Banach space, which 00:01:18.770 --> 00:01:21.480 align:middle line:90% we'll use in a minute. 00:01:21.480 --> 00:01:28.595 align:middle line:84% But to state it, I need to introduce 00:01:28.595 --> 00:01:30.440 align:middle line:90% a definition real quick. 00:01:30.440 --> 00:01:41.480 align:middle line:84% So let vn be a sequence in V. And I'm going to abuse notation 00:01:41.480 --> 00:01:46.460 align:middle line:84% and write subset of V, even though that's a sequence. 00:01:46.460 --> 00:01:48.260 align:middle line:90% I'm using this notation, though. 00:01:48.260 --> 00:01:56.690 align:middle line:84% So this means let vn an a sequence in V. 00:01:56.690 --> 00:02:02.480 align:middle line:84% The series, which is just right now an expression, 00:02:02.480 --> 00:02:08.530 align:middle line:84% just chalk on a board right now, it's just a symbol-- 00:02:08.530 --> 00:02:19.500 align:middle line:84% we say this series is summable if the sequence 00:02:19.500 --> 00:02:29.630 align:middle line:84% of partial sums, which are now elements in the norm space-- 00:02:29.630 --> 00:02:32.900 align:middle line:84% so if this sequence of partial sums converges. 00:02:32.900 --> 00:02:39.140 align:middle line:90% 00:02:39.140 --> 00:02:53.780 align:middle line:84% We say that the series vn is absolutely summable 00:02:53.780 --> 00:03:05.320 align:middle line:84% if the series of involving now these non-negative numbers 00:03:05.320 --> 00:03:06.250 align:middle line:90% converges. 00:03:06.250 --> 00:03:11.585 align:middle line:90% 00:03:11.585 --> 00:03:15.480 align:middle line:84% OK, so this is just like the definition of the convergence 00:03:15.480 --> 00:03:18.780 align:middle line:84% of a series of real numbers, which 00:03:18.780 --> 00:03:21.450 align:middle line:84% you dealt with in earlier analysis 00:03:21.450 --> 00:03:23.550 align:middle line:84% and absolute convergence, only here I'm 00:03:23.550 --> 00:03:25.920 align:middle line:84% using the terminology absolutely summable 00:03:25.920 --> 00:03:28.990 align:middle line:84% because this is the terminology Richard Melrose used, 00:03:28.990 --> 00:03:34.020 align:middle line:84% so I want to stick to what he's using. 00:03:34.020 --> 00:03:35.430 align:middle line:84% He's Australian, so I think maybe 00:03:35.430 --> 00:03:36.805 align:middle line:84% that has something to do with it. 00:03:36.805 --> 00:03:41.295 align:middle line:90% 00:03:41.295 --> 00:03:43.420 align:middle line:84% So that's one of the unfortunate things about this, 00:03:43.420 --> 00:03:44.680 align:middle line:84% is I can't tell if you're laughing, 00:03:44.680 --> 00:03:46.347 align:middle line:84% but I'm going to assume you're laughing. 00:03:46.347 --> 00:03:49.202 align:middle line:90% 00:03:49.202 --> 00:03:51.440 align:middle line:84% OK, so absolutely sum of all means 00:03:51.440 --> 00:03:57.800 align:middle line:84% the sum of the absolute-- of the norms converges. 00:03:57.800 --> 00:04:05.010 align:middle line:84% And you have this theorem, just like from real analysis, 00:04:05.010 --> 00:04:12.680 align:middle line:84% which you saw at one point, that if vn-- 00:04:12.680 --> 00:04:20.440 align:middle line:90% so if this series is summable-- 00:04:20.440 --> 00:04:26.875 align:middle line:84% is-- I'm missing an adjective there. 00:04:26.875 --> 00:04:30.500 align:middle line:90% 00:04:30.500 --> 00:04:40.550 align:middle line:84% If I have an absolutely summable series, then 00:04:40.550 --> 00:04:43.805 align:middle line:90% the sequence of partial sums-- 00:04:43.805 --> 00:04:48.880 align:middle line:90% 00:04:48.880 --> 00:05:00.700 align:middle line:84% this is a Cauchy sequence in the space V. So again, we're 00:05:00.700 --> 00:05:04.580 align:middle line:84% working through in a normal space V. 00:05:04.580 --> 00:05:09.070 align:middle line:84% All right, and the proof is the same as in the real numbers 00:05:09.070 --> 00:05:09.570 align:middle line:90% case. 00:05:09.570 --> 00:05:10.820 align:middle line:90% So proof I'll leave to you. 00:05:10.820 --> 00:05:14.800 align:middle line:90% 00:05:14.800 --> 00:05:17.360 align:middle line:90% This is just a simple exercise. 00:05:17.360 --> 00:05:27.956 align:middle line:84% And it's the same as for V equals R. 00:05:27.956 --> 00:05:36.180 align:middle line:84% Now, notice I said something which is strictly 00:05:36.180 --> 00:05:39.570 align:middle line:84% weaker than what you encounter in either case V equals R. 00:05:39.570 --> 00:05:43.450 align:middle line:84% In the case V equals R, you have the theorem 00:05:43.450 --> 00:05:47.110 align:middle line:84% that if I have an absolutely summable series, 00:05:47.110 --> 00:05:48.910 align:middle line:90% then it's summable. 00:05:48.910 --> 00:05:52.810 align:middle line:84% Every absolutely convergent series is convergent. 00:05:52.810 --> 00:05:55.360 align:middle line:90% 00:05:55.360 --> 00:05:56.800 align:middle line:90% But I didn't say that here. 00:05:56.800 --> 00:06:01.240 align:middle line:84% I just said that the sequence of partial sums 00:06:01.240 --> 00:06:03.950 align:middle line:84% is Cauchy, not necessarily convergent. 00:06:03.950 --> 00:06:07.750 align:middle line:84% So when is the sequence of partial series convergent? 00:06:07.750 --> 00:06:11.530 align:middle line:84% When can I say that an absolutely summable series 00:06:11.530 --> 00:06:15.700 align:middle line:90% in this norm space is summable? 00:06:15.700 --> 00:06:19.660 align:middle line:84% And I can say that precisely when it's a Banach space. 00:06:19.660 --> 00:06:21.700 align:middle line:84% So the theorem that we're going to prove 00:06:21.700 --> 00:06:40.540 align:middle line:84% is that V is a Banach space if and only if every absolutely 00:06:40.540 --> 00:06:46.365 align:middle line:90% summable series is summable. 00:06:46.365 --> 00:06:55.886 align:middle line:90% 00:06:55.886 --> 00:06:59.280 align:middle line:84% OK, so this characterizes Banach spaces 00:06:59.280 --> 00:07:02.670 align:middle line:84% as those spaces for which this theorem 00:07:02.670 --> 00:07:05.370 align:middle line:84% you have from real analysis, that every absolutely 00:07:05.370 --> 00:07:08.460 align:middle line:84% convergent series converges, is precisely that. 00:07:08.460 --> 00:07:11.130 align:middle line:84% Every absolutely summable series is summable. 00:07:11.130 --> 00:07:16.380 align:middle line:84% And sometimes, that is an easier property 00:07:16.380 --> 00:07:21.210 align:middle line:84% to verify than going through the whole Cauchy business. 00:07:21.210 --> 00:07:26.780 align:middle line:84% And sometimes, it's exactly the same amount of work. 00:07:26.780 --> 00:07:29.600 align:middle line:84% We'll use this later when we deal with integration 00:07:29.600 --> 00:07:32.240 align:middle line:84% and measure theory to prove that the big Lp 00:07:32.240 --> 00:07:34.295 align:middle line:90% spaces are Banach spaces. 00:07:34.295 --> 00:07:37.040 align:middle line:90% 00:07:37.040 --> 00:07:40.520 align:middle line:84% All right, so we have two directions to prove-- 00:07:40.520 --> 00:07:43.370 align:middle line:84% that if V is a Banach space, then 00:07:43.370 --> 00:07:46.430 align:middle line:84% we get every absolutely summable series is summable. 00:07:46.430 --> 00:07:48.510 align:middle line:84% And this is pretty straightforward. 00:07:48.510 --> 00:07:53.800 align:middle line:90% So one direction-- if V-- 00:07:53.800 --> 00:08:00.600 align:middle line:90% suppose V is a Banach space. 00:08:00.600 --> 00:08:03.670 align:middle line:90% 00:08:03.670 --> 00:08:14.390 align:middle line:84% Then if vn, if this series, is absolutely 00:08:14.390 --> 00:08:16.880 align:middle line:84% summable by the previous theorem, which 00:08:16.880 --> 00:08:20.210 align:middle line:84% I didn't prove, but is very easy to prove, 00:08:20.210 --> 00:08:22.865 align:middle line:84% I get that the sequence of partial sums-- 00:08:22.865 --> 00:08:29.460 align:middle line:90% 00:08:29.460 --> 00:08:37.700 align:middle line:84% this is a Cauchy sequence in V. And because V 00:08:37.700 --> 00:08:41.210 align:middle line:84% is a Banach space, every Cauchy sequence converges. 00:08:41.210 --> 00:08:52.600 align:middle line:84% And therefore, it converges in V. 00:08:52.600 --> 00:08:55.840 align:middle line:84% And therefore, the series is summable. 00:08:55.840 --> 00:08:58.510 align:middle line:90% So that direction simple enough. 00:08:58.510 --> 00:09:01.170 align:middle line:90% 00:09:01.170 --> 00:09:05.300 align:middle line:84% Let's go the opposite direction and show that every-- 00:09:05.300 --> 00:09:08.660 align:middle line:84% that the condition that every absolutely summable series is 00:09:08.660 --> 00:09:10.965 align:middle line:84% summable implies that V is a Banach space. 00:09:10.965 --> 00:09:13.820 align:middle line:90% 00:09:13.820 --> 00:09:28.605 align:middle line:84% So every absolutely summable series is summable. 00:09:28.605 --> 00:09:32.760 align:middle line:90% 00:09:32.760 --> 00:09:36.570 align:middle line:84% Now, we want to show that every Cauchy sequence converges in V. 00:09:36.570 --> 00:09:53.080 align:middle line:84% So let vn be a Cauchy in V. So what we're going to do 00:09:53.080 --> 00:09:56.110 align:middle line:84% is we're, in fact, going to show that there's 00:09:56.110 --> 00:09:59.660 align:middle line:84% a subsequence of the sequence that converges. 00:09:59.660 --> 00:10:03.040 align:middle line:90% So what we're going to show-- 00:10:03.040 --> 00:10:16.030 align:middle line:90% 00:10:16.030 --> 00:10:21.120 align:middle line:84% so that this sequence has a convergent subsequence. 00:10:21.120 --> 00:10:30.720 align:middle line:90% 00:10:30.720 --> 00:10:33.450 align:middle line:84% And once we've done that, we're done because remember back 00:10:33.450 --> 00:10:36.780 align:middle line:90% to your real analysis days. 00:10:36.780 --> 00:10:39.450 align:middle line:84% If a Cauchy sequence has a convergent subsequence, 00:10:39.450 --> 00:10:41.280 align:middle line:84% then the entire sequence converges. 00:10:41.280 --> 00:10:49.474 align:middle line:90% 00:10:49.474 --> 00:11:08.130 align:middle line:84% And vn converges by metric space theory, all right? 00:11:08.130 --> 00:11:11.400 align:middle line:90% So real analysis stuff. 00:11:11.400 --> 00:11:15.720 align:middle line:84% OK, so let's find this subsequence. 00:11:15.720 --> 00:11:20.790 align:middle line:84% And basically, we're going to build this subsequence up 00:11:20.790 --> 00:11:24.420 align:middle line:84% by speeding up the convergence of vn or, if you like, 00:11:24.420 --> 00:11:27.830 align:middle line:84% speeding up the Cauchiness of vn. 00:11:27.830 --> 00:11:36.780 align:middle line:84% So the fact that the sequence is Cauchy 00:11:36.780 --> 00:11:43.290 align:middle line:84% implies that for all natural numbers, 00:11:43.290 --> 00:11:50.260 align:middle line:84% there exists a natural number k, also a natural number, 00:11:50.260 --> 00:11:57.930 align:middle line:84% such that for all nm bigger than or equal to N sub k, we have 00:11:57.930 --> 00:12:02.760 align:middle line:84% that the norm of v sub n minus the norm of v sub m 00:12:02.760 --> 00:12:06.120 align:middle line:90% is less than 2 to the minus k. 00:12:06.120 --> 00:12:08.040 align:middle line:84% All right, why did I choose to the minus k? 00:12:08.040 --> 00:12:09.630 align:middle line:84% Because that's summable, all right? 00:12:09.630 --> 00:12:10.630 align:middle line:90% And you'll see. 00:12:10.630 --> 00:12:16.840 align:middle line:84% And so what we're going to do is build up essentially 00:12:16.840 --> 00:12:22.436 align:middle line:90% a telescoping sum from these-- 00:12:22.436 --> 00:12:23.780 align:middle line:90% from Well-chosen guys. 00:12:23.780 --> 00:12:31.940 align:middle line:90% So define n sub k. 00:12:31.940 --> 00:12:32.990 align:middle line:90% What is this going to be? 00:12:32.990 --> 00:12:39.185 align:middle line:84% This is going to be equal to N sub 1 plus plus N sub k. 00:12:39.185 --> 00:12:46.520 align:middle line:90% 00:12:46.520 --> 00:12:52.280 align:middle line:84% So n sub 1 is less than n sub 2 is less than n sub 3 00:12:52.280 --> 00:12:54.660 align:middle line:84% because at each stage, I'm adding a natural number. 00:12:54.660 --> 00:12:57.470 align:middle line:84% So n sub 1 is equal to capital N sub 00:12:57.470 --> 00:13:00.927 align:middle line:84% 1. n sub 2 is equal to a capital N sub 1 plus n sub 2. 00:13:00.927 --> 00:13:02.010 align:middle line:90% These are natural numbers. 00:13:02.010 --> 00:13:05.330 align:middle line:84% So I'm always getting bigger at each stage. 00:13:05.330 --> 00:13:10.860 align:middle line:84% So this is an increasing sequence of integers. 00:13:10.860 --> 00:13:17.370 align:middle line:84% And for all k, n sub k is less than 00:13:17.370 --> 00:13:31.626 align:middle line:84% or equal to N sub k, capital N sub k 00:13:31.626 --> 00:13:33.770 align:middle line:84% because little n sub k is equal to sum 00:13:33.770 --> 00:13:35.900 align:middle line:90% integers plus capital N sub k. 00:13:35.900 --> 00:13:40.640 align:middle line:90% 00:13:40.640 --> 00:13:42.800 align:middle line:84% And so the v sub n sub k's are going 00:13:42.800 --> 00:13:47.225 align:middle line:84% to be essentially the guys which converge. 00:13:47.225 --> 00:13:51.580 align:middle line:90% 00:13:51.580 --> 00:13:56.200 align:middle line:84% Thus, for all k natural numbers, I 00:13:56.200 --> 00:14:01.720 align:middle line:84% get that v sub n sub k plus 1 minus v sub n sub k-- 00:14:01.720 --> 00:14:04.600 align:middle line:84% if I take the norm of that, so n sub k is bigger than 00:14:04.600 --> 00:14:06.280 align:middle line:90% or equal to capital N sub k. 00:14:06.280 --> 00:14:08.860 align:middle line:84% n sub k plus 1 is bigger than or equal to n sub 00:14:08.860 --> 00:14:11.320 align:middle line:84% k, which is bigger than or equal to capital N sub k. 00:14:11.320 --> 00:14:13.780 align:middle line:84% And therefore, by this condition-- 00:14:13.780 --> 00:14:17.431 align:middle line:90% how n sub k's are chosen-- 00:14:17.431 --> 00:14:20.380 align:middle line:90% 00:14:20.380 --> 00:14:24.800 align:middle line:84% so this, this, this, and what's in blue tells me 00:14:24.800 --> 00:14:27.890 align:middle line:84% that this is going to be less than 2 to the minus k. 00:14:27.890 --> 00:14:36.000 align:middle line:90% 00:14:36.000 --> 00:14:43.520 align:middle line:84% And therefore, thus, the sum v sub 00:14:43.520 --> 00:14:48.230 align:middle line:84% n sub k plus 1 minus v sub n sub k-- 00:14:48.230 --> 00:14:50.540 align:middle line:84% this is absolutely summable, right? 00:14:50.540 --> 00:14:52.850 align:middle line:84% Because the norm of this is less than 2 to the minus 00:14:52.850 --> 00:14:55.265 align:middle line:90% k, which you can sum. 00:14:55.265 --> 00:14:59.620 align:middle line:90% 00:14:59.620 --> 00:15:02.890 align:middle line:84% And by our assumption, that every absolutely summable 00:15:02.890 --> 00:15:05.890 align:middle line:84% series is summable, this implies that-- 00:15:05.890 --> 00:15:09.580 align:middle line:90% 00:15:09.580 --> 00:15:11.530 align:middle line:84% I didn't even say anything after that-- 00:15:11.530 --> 00:15:15.870 align:middle line:90% 00:15:15.870 --> 00:15:33.600 align:middle line:84% is absolutely summable, which implies that this is summable, 00:15:33.600 --> 00:15:39.290 align:middle line:90% which implies the limit as m-- 00:15:39.290 --> 00:15:43.105 align:middle line:84% or let me instead just finish with that-- 00:15:43.105 --> 00:15:49.980 align:middle line:90% 00:15:49.980 --> 00:15:51.900 align:middle line:90% i.e. 00:15:51.900 --> 00:15:57.120 align:middle line:84% the sequence of partial sums k equals 1 to m v sub n sub k 00:15:57.120 --> 00:15:59.760 align:middle line:90% plus 1 minus v sub n sub k-- 00:15:59.760 --> 00:16:04.740 align:middle line:90% 00:16:04.740 --> 00:16:13.898 align:middle line:84% this sequence of partial sums converges in V. 00:16:13.898 --> 00:16:17.650 align:middle line:84% OK, so to recap again, we started off 00:16:17.650 --> 00:16:19.240 align:middle line:90% with a Cauchy sequence. 00:16:19.240 --> 00:16:23.620 align:middle line:84% We go out in the Cauchy sequence far enough 00:16:23.620 --> 00:16:25.720 align:middle line:84% and pick certain guys so that they're 00:16:25.720 --> 00:16:26.920 align:middle line:90% pretty close to each other. 00:16:26.920 --> 00:16:27.970 align:middle line:90% And how close? 00:16:27.970 --> 00:16:33.670 align:middle line:84% So close that the sum of their norms is finite, 00:16:33.670 --> 00:16:35.230 align:middle line:84% so that it's absolutely summable. 00:16:35.230 --> 00:16:42.030 align:middle line:84% And therefore, the series is summable by our assumption. 00:16:42.030 --> 00:16:45.930 align:middle line:84% But this is essentially a telescoping sum. 00:16:45.930 --> 00:16:51.400 align:middle line:90% 00:16:51.400 --> 00:16:59.610 align:middle line:84% Thus, the series v sub n sub m, which equals-- 00:16:59.610 --> 00:17:04.619 align:middle line:84% so the sequence v sub n sub m, which is the sum from n 00:17:04.619 --> 00:17:11.130 align:middle line:84% equals 1 to m of v sub n sub k plus 1-- 00:17:11.130 --> 00:17:13.660 align:middle line:84% let's see, let me put a minus 1 here, 00:17:13.660 --> 00:17:21.930 align:middle line:84% minus v sub n sub k plus v sub n sub, m 00:17:21.930 --> 00:17:33.840 align:middle line:84% equals 1 to infinity converges in V. And I'm done. 00:17:33.840 --> 00:17:41.220 align:middle line:84% So v sub n sub m is equal to this telescoping sum plus-- 00:17:41.220 --> 00:17:44.310 align:middle line:84% so when I add this up, terms cancel. 00:17:44.310 --> 00:17:46.230 align:middle line:84% And I just pick up the last one, which 00:17:46.230 --> 00:17:51.630 align:middle line:84% is when I pick up m sub 1 here minus the first one, which 00:17:51.630 --> 00:17:53.110 align:middle line:90% is v sub n sub 1 here. 00:17:53.110 --> 00:17:57.345 align:middle line:84% So if I add on v sub n sub 1, I just pick up v sub n sub m. 00:17:57.345 --> 00:18:02.100 align:middle line:84% Now, as m goes to infinity, this converges to something 00:18:02.100 --> 00:18:04.560 align:middle line:90% because this sequence converges. 00:18:04.560 --> 00:18:08.340 align:middle line:90% And this is just fixed in m. 00:18:08.340 --> 00:18:10.230 align:middle line:90% So this sum converges. 00:18:10.230 --> 00:18:13.350 align:middle line:84% And therefore, v sub n sub m converges. 00:18:13.350 --> 00:18:16.230 align:middle line:84% And thus, this subsequence of our original Cauchy sequence 00:18:16.230 --> 00:18:19.560 align:middle line:84% converges, proving that the Cauchy sequence converges in V. 00:18:19.560 --> 00:18:20.280 align:middle line:90% And we're done. 00:18:20.280 --> 00:18:27.340 align:middle line:90% 00:18:27.340 --> 00:18:38.120 align:middle line:84% OK, so Banach spaces, these are a nice generalization 00:18:38.120 --> 00:18:41.720 align:middle line:84% of the spaces that you worked with in real analysis 00:18:41.720 --> 00:18:42.950 align:middle line:90% and linear algebra-- 00:18:42.950 --> 00:18:45.440 align:middle line:90% Rn, Cn, and so on. 00:18:45.440 --> 00:18:48.020 align:middle line:90% 00:18:48.020 --> 00:18:53.540 align:middle line:84% So what are the analogs of matrices, 00:18:53.540 --> 00:18:56.750 align:middle line:84% which you had to use in calculus and linear algebra? 00:18:56.750 --> 00:19:04.660 align:middle line:84% This is going to lead to our next topic, which 00:19:04.660 --> 00:19:07.630 align:middle line:90% is operators and functionals-- 00:19:07.630 --> 00:19:19.990 align:middle line:90% 00:19:19.990 --> 00:19:22.680 align:middle line:84% so operators being the analog of matrices 00:19:22.680 --> 00:19:27.090 align:middle line:84% that take one vector into another vector, 00:19:27.090 --> 00:19:33.900 align:middle line:84% functionals are the analog of taking a vector 00:19:33.900 --> 00:19:37.080 align:middle line:84% and taking its dot product with a fixed vector, 00:19:37.080 --> 00:19:39.370 align:middle line:90% spitting out a real number. 00:19:39.370 --> 00:19:43.740 align:middle line:84% So functionals will eat vectors and spit out 00:19:43.740 --> 00:19:47.580 align:middle line:84% real or complex numbers, depending on the field 00:19:47.580 --> 00:19:50.410 align:middle line:90% that you're working in. 00:19:50.410 --> 00:19:56.070 align:middle line:84% So let me write down just an example 00:19:56.070 --> 00:20:03.960 align:middle line:84% to keep in mind, as far as operators go. 00:20:03.960 --> 00:20:14.100 align:middle line:84% So I want you to keep this example 00:20:14.100 --> 00:20:19.220 align:middle line:84% in mind, which was the whole reason for really a lot 00:20:19.220 --> 00:20:21.050 align:middle line:90% of building all this machinery. 00:20:21.050 --> 00:20:26.110 align:middle line:84% I mean, this example came first, and then the machinery came-- 00:20:26.110 --> 00:20:29.210 align:middle line:84% was built later to be able to say all 00:20:29.210 --> 00:20:32.330 align:middle line:84% we can about these kind of operators-- 00:20:32.330 --> 00:20:34.250 align:middle line:84% or these kind of transformations. 00:20:34.250 --> 00:20:36.890 align:middle line:90% 00:20:36.890 --> 00:20:41.800 align:middle line:84% And depending on if you're in my class or not, 00:20:41.800 --> 00:20:47.880 align:middle line:84% maybe you saw a question about such a creature 00:20:47.880 --> 00:20:50.770 align:middle line:84% on an assignment or maybe on an exam. 00:20:50.770 --> 00:20:59.230 align:middle line:84% So let K be a function on 0, 1 cross 0, 1, 00:20:59.230 --> 00:21:01.450 align:middle line:84% let's say, into the complex numbers. 00:21:01.450 --> 00:21:05.690 align:middle line:90% 00:21:05.690 --> 00:21:16.320 align:middle line:84% And let's assume it's continuous for f, 00:21:16.320 --> 00:21:18.270 align:middle line:90% a continuous function on 0, 1. 00:21:18.270 --> 00:21:21.780 align:middle line:84% We can define a new function from f, 00:21:21.780 --> 00:21:28.740 align:middle line:84% Tf of x, to be the integral from 0 to 1 of K of xy times 00:21:28.740 --> 00:21:31.230 align:middle line:90% f of y dy. 00:21:31.230 --> 00:21:36.350 align:middle line:90% 00:21:36.350 --> 00:21:37.760 align:middle line:90% Now, these are things that-- 00:21:37.760 --> 00:21:39.802 align:middle line:84% I'm about to write a few things that you can just 00:21:39.802 --> 00:21:40.610 align:middle line:90% check by hand. 00:21:40.610 --> 00:21:50.435 align:middle line:84% But you can then check then Tf is also a continuous function. 00:21:50.435 --> 00:21:56.060 align:middle line:90% 00:21:56.060 --> 00:22:00.290 align:middle line:84% And it's linear in the argument f. 00:22:00.290 --> 00:22:08.390 align:middle line:84% And for all lambda 1, lambda 2 in C, f1, f2 00:22:08.390 --> 00:22:19.380 align:middle line:84% in 0, 1, T lambda 1, f1 plus lambda 2 f2 00:22:19.380 --> 00:22:25.216 align:middle line:84% equals lambda 1 Tf1 plus lambda 2 Tf2. 00:22:25.216 --> 00:22:29.373 align:middle line:90% 00:22:29.373 --> 00:22:30.790 align:middle line:84% And it has another property, which 00:22:30.790 --> 00:22:32.560 align:middle line:84% I'm going to say in a minute, namely 00:22:32.560 --> 00:22:36.410 align:middle line:84% that it's continuous on the space of continuous functions. 00:22:36.410 --> 00:22:38.980 align:middle line:84% So we've already proven that this space 00:22:38.980 --> 00:22:43.300 align:middle line:84% of continuous functions on 0, 1, this is a Banach space, right? 00:22:43.300 --> 00:22:46.660 align:middle line:84% This was a special example of the space 00:22:46.660 --> 00:22:48.980 align:middle line:84% of bounded continuous functions on a metric space 00:22:48.980 --> 00:22:54.700 align:middle line:84% we considered before because on the closed and bounded interval 00:22:54.700 --> 00:22:56.960 align:middle line:84% 0, 1, every continuous function is bounded. 00:22:56.960 --> 00:23:04.180 align:middle line:84% So this is just equal to C subscript infinity of 0, 1. 00:23:04.180 --> 00:23:08.760 align:middle line:84% So we know this is a Banach space. 00:23:08.760 --> 00:23:13.920 align:middle line:84% And so this is an example of what's 00:23:13.920 --> 00:23:16.930 align:middle line:90% called a linear operator. 00:23:16.930 --> 00:23:30.690 align:middle line:84% So definition-- let V and W be vector spaces. 00:23:30.690 --> 00:23:32.690 align:middle line:84% So you should have seen a linear transformation. 00:23:32.690 --> 00:23:34.315 align:middle line:84% I'm going to call it a linear operator. 00:23:34.315 --> 00:23:37.190 align:middle line:84% I'm just recalling what it means to be linear. 00:23:37.190 --> 00:23:39.410 align:middle line:90% Let V and W be vector spaces. 00:23:39.410 --> 00:23:43.240 align:middle line:90% 00:23:43.240 --> 00:23:49.755 align:middle line:84% You say a map T from V to W is linear-- 00:23:49.755 --> 00:23:54.390 align:middle line:90% 00:23:54.390 --> 00:24:02.020 align:middle line:84% so linear, if for all lambda 1, lambda 2 00:24:02.020 --> 00:24:04.720 align:middle line:90% in your field of scalars-- 00:24:04.720 --> 00:24:05.950 align:middle line:90% if they're either R or C-- 00:24:05.950 --> 00:24:08.800 align:middle line:90% 00:24:08.800 --> 00:24:19.150 align:middle line:84% and for all v1, v2 in V, T of lambda 1 v1 plus lambda 2 v2 00:24:19.150 --> 00:24:24.204 align:middle line:84% equals lambda 1 T v1 plus lambda 2 T v2. 00:24:24.204 --> 00:24:32.270 align:middle line:90% 00:24:32.270 --> 00:24:39.770 align:middle line:84% So in a map from one norm space-- 00:24:39.770 --> 00:24:45.740 align:middle line:90% 00:24:45.740 --> 00:24:47.752 align:middle line:84% so this is just a note off to the side. 00:24:47.752 --> 00:24:49.460 align:middle line:84% This is how I'm going to be using-- well, 00:24:49.460 --> 00:24:50.960 align:middle line:84% I'm not even going to write it down. 00:24:50.960 --> 00:24:53.380 align:middle line:90% 00:24:53.380 --> 00:24:57.480 align:middle line:84% So given two norm spaces, a linear map between them 00:24:57.480 --> 00:25:02.010 align:middle line:84% I will most often refer to as a linear operator. 00:25:02.010 --> 00:25:04.955 align:middle line:84% Rather than linear transformation, 00:25:04.955 --> 00:25:07.080 align:middle line:84% which is what you probably heard in linear algebra, 00:25:07.080 --> 00:25:10.410 align:middle line:84% I'll refer to these as operators. 00:25:10.410 --> 00:25:12.660 align:middle line:84% Something I meant to say as well was 00:25:12.660 --> 00:25:15.720 align:middle line:84% why do we care about such a guy like this other 00:25:15.720 --> 00:25:17.850 align:middle line:90% than it looks nice? 00:25:17.850 --> 00:25:27.730 align:middle line:84% You care about guys like these because operators of this form 00:25:27.730 --> 00:25:34.250 align:middle line:84% are essentially the inverse operators of differential 00:25:34.250 --> 00:25:35.750 align:middle line:90% operators. 00:25:35.750 --> 00:25:38.270 align:middle line:84% I mean, you know that from the fundamental theorem 00:25:38.270 --> 00:25:39.500 align:middle line:90% of calculus. 00:25:39.500 --> 00:25:44.450 align:middle line:84% The inverse operation of taking a derivative is taking 00:25:44.450 --> 00:25:46.910 align:middle line:90% is integrating. 00:25:46.910 --> 00:25:54.110 align:middle line:84% So it shouldn't be a surprise that the inverse operator, 00:25:54.110 --> 00:25:58.160 align:middle line:84% meaning if I take f as my data to some ODE, 00:25:58.160 --> 00:26:04.040 align:middle line:84% can be written as this, as this kind of linear operator. 00:26:04.040 --> 00:26:06.270 align:middle line:84% That shouldn't come as too much of a surprise. 00:26:06.270 --> 00:26:08.180 align:middle line:84% So that's why we care about them, 00:26:08.180 --> 00:26:13.520 align:middle line:84% is that operators of this form arise 00:26:13.520 --> 00:26:17.300 align:middle line:84% as the inverses of taking differentiable operators. 00:26:17.300 --> 00:26:22.670 align:middle line:90% 00:26:22.670 --> 00:26:30.660 align:middle line:84% OK, so now, in this class at least, 00:26:30.660 --> 00:26:35.390 align:middle line:84% we're not just interested in any old linear operator. 00:26:35.390 --> 00:26:38.450 align:middle line:84% We're going to be interested in a certain class 00:26:38.450 --> 00:26:40.220 align:middle line:90% of linear operators, those-- 00:26:40.220 --> 00:26:43.710 align:middle line:90% so those which are continuous. 00:26:43.710 --> 00:26:57.480 align:middle line:84% So let me recall for you an equivalent way of saying 00:26:57.480 --> 00:27:00.430 align:middle line:90% that a map is continuous. 00:27:00.430 --> 00:27:03.910 align:middle line:90% 00:27:03.910 --> 00:27:05.990 align:middle line:84% So this is just any map, any function, 00:27:05.990 --> 00:27:15.800 align:middle line:84% not necessarily a linear operator is continuous on V 00:27:15.800 --> 00:27:20.300 align:middle line:84% if-- and there's two ways to say this-- 00:27:20.300 --> 00:27:30.530 align:middle line:84% for every sequence vn converging to V-- 00:27:30.530 --> 00:27:31.940 align:middle line:90% so let me write it this way. 00:27:31.940 --> 00:27:37.540 align:middle line:90% 00:27:37.540 --> 00:27:49.290 align:middle line:84% For all v in E for all sequences converging to V, 00:27:49.290 --> 00:27:55.740 align:middle line:84% we obtain that T of v sub n convergence to T of v. 00:27:55.740 --> 00:28:05.360 align:middle line:84% And an equivalent way of stating this 00:28:05.360 --> 00:28:13.210 align:middle line:84% in terms of what one would call a topological notions 00:28:13.210 --> 00:28:17.240 align:middle line:84% is that the inverse image of open sets are open. 00:28:17.240 --> 00:28:26.570 align:middle line:84% So for all open u in W, the inverse image 00:28:26.570 --> 00:28:30.920 align:middle line:84% of u, which I will recall for you is the set of v in capital 00:28:30.920 --> 00:28:36.450 align:middle line:90% V such that T v is in u-- 00:28:36.450 --> 00:28:38.083 align:middle line:84% I'm not saying that T is invertible. 00:28:38.083 --> 00:28:39.125 align:middle line:90% That's the inverse image. 00:28:39.125 --> 00:28:43.130 align:middle line:90% 00:28:43.130 --> 00:28:52.060 align:middle line:84% The set is open in V. Remember, the notion of an open set 00:28:52.060 --> 00:28:55.360 align:middle line:84% is that for every point in that set, 00:28:55.360 --> 00:28:58.180 align:middle line:84% there's a small ball that's contained entirely 00:28:58.180 --> 00:29:00.400 align:middle line:84% in the set centered at that point. 00:29:00.400 --> 00:29:11.510 align:middle line:90% 00:29:11.510 --> 00:29:16.500 align:middle line:84% OK, now, for linear maps, there's 00:29:16.500 --> 00:29:18.930 align:middle line:84% a very simple way or equivalent way 00:29:18.930 --> 00:29:25.500 align:middle line:84% of finding when it's continuous on a norm space. 00:29:25.500 --> 00:29:27.510 align:middle line:84% Now, on finite dimensional spaces, 00:29:27.510 --> 00:29:30.840 align:middle line:84% any linear transformation-- every linear transformation 00:29:30.840 --> 00:29:31.680 align:middle line:90% is continuous. 00:29:31.680 --> 00:29:34.000 align:middle line:90% I should say that. 00:29:34.000 --> 00:29:37.830 align:middle line:84% So if you take any linear operator from Rn 00:29:37.830 --> 00:29:43.380 align:middle line:84% to Rm Cn to Cm or Rn to Cm, anything between two 00:29:43.380 --> 00:29:47.130 align:middle line:84% finite dimensional spaces, it will always 00:29:47.130 --> 00:29:49.590 align:middle line:90% be continuous if it's linear. 00:29:49.590 --> 00:29:53.610 align:middle line:84% That is not always the case between two Banach spaces. 00:29:53.610 --> 00:29:56.950 align:middle line:90% 00:29:56.950 --> 00:30:00.430 align:middle line:84% Now, again, is there a more efficient way 00:30:00.430 --> 00:30:02.500 align:middle line:84% of checking when something is continuous? 00:30:02.500 --> 00:30:06.340 align:middle line:84% Or what's an equivalent way of saying that? 00:30:06.340 --> 00:30:10.120 align:middle line:84% Or a more useful way is the following characterization 00:30:10.120 --> 00:30:11.110 align:middle line:90% that we have. 00:30:11.110 --> 00:30:24.250 align:middle line:84% So a linear operator T between two norm spaces now-- 00:30:24.250 --> 00:30:27.140 align:middle line:90% 00:30:27.140 --> 00:30:38.890 align:middle line:84% so between two norm spaces is continuous if there 00:30:38.890 --> 00:30:46.210 align:middle line:84% exists a C positive such that for all v in capital V, if I 00:30:46.210 --> 00:30:51.880 align:middle line:84% take T of v and take it's norm in W, this is less than 00:30:51.880 --> 00:31:19.000 align:middle line:84% or equal to a constant times the norm of v in capital V. 00:31:19.000 --> 00:31:25.730 align:middle line:90% OK, now, in this case, we say-- 00:31:25.730 --> 00:31:29.550 align:middle line:90% 00:31:29.550 --> 00:31:34.080 align:middle line:84% instead of saying that T is a continuous linear operator, 00:31:34.080 --> 00:31:46.350 align:middle line:84% we say T is a bounded linear operator. 00:31:46.350 --> 00:31:48.970 align:middle line:90% 00:31:48.970 --> 00:31:51.900 align:middle line:84% So we don't really say continuous linear operator. 00:31:51.900 --> 00:31:53.800 align:middle line:90% We say bounded linear operator. 00:31:53.800 --> 00:31:56.910 align:middle line:84% Now, this doesn't mean the image of V 00:31:56.910 --> 00:32:00.120 align:middle line:84% is a bounded set in W. That's not what this means. 00:32:00.120 --> 00:32:03.180 align:middle line:84% The only linear operator that takes the bounded set-- that 00:32:03.180 --> 00:32:08.130 align:middle line:84% takes a vector space into a bounded set 00:32:08.130 --> 00:32:09.730 align:middle line:90% is the zero operator. 00:32:09.730 --> 00:32:12.405 align:middle line:84% So we're not saying that it's taking all of V 00:32:12.405 --> 00:32:13.530 align:middle line:90% to a bounded set. 00:32:13.530 --> 00:32:16.080 align:middle line:84% But what this inequality does say 00:32:16.080 --> 00:32:35.220 align:middle line:84% is that it takes bounded subsets of V to bounded subsets of W. 00:32:35.220 --> 00:32:39.130 align:middle line:84% So let's prove-- let me put a star by this condition, 00:32:39.130 --> 00:32:41.460 align:middle line:84% so I don't have to write it out so much. 00:32:41.460 --> 00:32:59.142 align:middle line:90% 00:32:59.142 --> 00:33:04.240 align:middle line:84% OK, so let's go in this direction. 00:33:04.240 --> 00:33:14.720 align:middle line:84% So let's assume star and prove that this linear operator 00:33:14.720 --> 00:33:16.670 align:middle line:90% T is continuous. 00:33:16.670 --> 00:33:18.230 align:middle line:84% And this is not too difficult to do. 00:33:18.230 --> 00:33:20.645 align:middle line:90% We'll use this first-- 00:33:20.645 --> 00:33:25.660 align:middle line:90% 00:33:25.660 --> 00:33:29.080 align:middle line:84% we'll use this first characterization of continuity. 00:33:29.080 --> 00:33:34.184 align:middle line:90% 00:33:34.184 --> 00:33:43.160 align:middle line:84% Let v be in V. And suppose v sub n is a sequence in V converging 00:33:43.160 --> 00:33:55.380 align:middle line:84% to v. Then by star, if I look at T sub v sub n minus T sub v 00:33:55.380 --> 00:34:00.870 align:middle line:84% norm in W, this is less than or equal to a constant times 00:34:00.870 --> 00:34:02.730 align:middle line:90% the norm of v sub n minus-- 00:34:02.730 --> 00:34:04.290 align:middle line:84% well, first off, I'm kind of using-- 00:34:04.290 --> 00:34:07.710 align:middle line:90% 00:34:07.710 --> 00:34:08.850 align:middle line:90% let me add one little step. 00:34:08.850 --> 00:34:10.433 align:middle line:84% I'm using that it's a linear operator, 00:34:10.433 --> 00:34:14.100 align:middle line:84% meaning I can write this as T of v minus v sub n. 00:34:14.100 --> 00:34:18.670 align:middle line:90% 00:34:18.670 --> 00:34:22.550 align:middle line:84% And therefore, this is less than or equal to a constant times v 00:34:22.550 --> 00:34:28.250 align:middle line:84% sub n minus v in capital V. And so 00:34:28.250 --> 00:34:33.199 align:middle line:84% the norm of T of v sub n minus T of v 00:34:33.199 --> 00:34:36.800 align:middle line:84% is less than or equal to some fixed constant depending only 00:34:36.800 --> 00:34:39.469 align:middle line:84% on T times v sub n minus v. This goes 00:34:39.469 --> 00:34:41.420 align:middle line:90% to 0 as n goes to infinity. 00:34:41.420 --> 00:34:45.040 align:middle line:90% 00:34:45.040 --> 00:34:48.360 align:middle line:84% And therefore, by the squeeze theorem, this thing on the left 00:34:48.360 --> 00:34:49.290 align:middle line:90% must go to 0. 00:34:49.290 --> 00:34:51.810 align:middle line:84% It's always trivially bigger than 00:34:51.810 --> 00:35:04.060 align:middle line:84% or equal to 0, by the squeeze theorem. 00:35:04.060 --> 00:35:08.290 align:middle line:84% And therefore, T of v sub n converges to T of v. 00:35:08.290 --> 00:35:10.870 align:middle line:84% So that takes care of one direction. 00:35:10.870 --> 00:35:24.600 align:middle line:90% 00:35:24.600 --> 00:35:29.450 align:middle line:84% OK, so we've shown that this boundedness property of T 00:35:29.450 --> 00:35:33.470 align:middle line:84% implies that T is continuous, that the linear operator 00:35:33.470 --> 00:35:35.900 align:middle line:90% T is continuous. 00:35:35.900 --> 00:35:38.450 align:middle line:84% Now, let's show that continuity implies 00:35:38.450 --> 00:35:39.575 align:middle line:90% this boundedness property. 00:35:39.575 --> 00:35:50.670 align:middle line:90% 00:35:50.670 --> 00:35:53.470 align:middle line:90% And so for the continuity guy-- 00:35:53.470 --> 00:35:54.960 align:middle line:84% so I'm assuming T is continuous-- 00:35:54.960 --> 00:35:58.345 align:middle line:84% I'm going to use the second characterization of continuity 00:35:58.345 --> 00:35:58.845 align:middle line:90% here. 00:35:58.845 --> 00:36:05.730 align:middle line:90% 00:36:05.730 --> 00:36:15.440 align:middle line:84% So then the inverse image of every open set in W 00:36:15.440 --> 00:36:20.430 align:middle line:84% is an open set in V. So the inverse image 00:36:20.430 --> 00:36:31.040 align:middle line:84% of the ball centered at the zero vector in W of radius 1-- 00:36:31.040 --> 00:36:39.360 align:middle line:84% so let me just recall this is a set of v in V such 00:36:39.360 --> 00:36:42.045 align:middle line:90% that T v is in the ball. 00:36:42.045 --> 00:36:45.080 align:middle line:90% 00:36:45.080 --> 00:36:57.940 align:middle line:84% This is an open set in V because the ball of radius 1-- 00:36:57.940 --> 00:37:04.210 align:middle line:84% so this is the ball of elements in W such 00:37:04.210 --> 00:37:06.430 align:middle line:84% that their distance to 0 is less than 1. 00:37:06.430 --> 00:37:09.160 align:middle line:84% This must this is an open set in W. 00:37:09.160 --> 00:37:13.270 align:middle line:84% And therefore, its inverse image must be an open set and V. 00:37:13.270 --> 00:37:23.860 align:middle line:84% So we have here 0, 1, everything inside. 00:37:23.860 --> 00:37:27.240 align:middle line:90% Don't include the boundary. 00:37:27.240 --> 00:37:42.510 align:middle line:84% This must be an open set in V. And T takes one to the other. 00:37:42.510 --> 00:37:48.392 align:middle line:90% Now, what do I know? 00:37:48.392 --> 00:37:49.840 align:middle line:90% 0 is in here. 00:37:49.840 --> 00:37:54.530 align:middle line:84% And every linear transformation takes 0 to 0. 00:37:54.530 --> 00:37:59.500 align:middle line:84% So 0 has to be in the inverse image 00:37:59.500 --> 00:38:04.790 align:middle line:84% because any linear map takes 0 to 0. 00:38:04.790 --> 00:38:12.310 align:middle line:84% So 0 has to be in the inverse image. 00:38:12.310 --> 00:38:17.185 align:middle line:84% And therefore, since this set is open-- 00:38:17.185 --> 00:38:19.750 align:middle line:90% 00:38:19.750 --> 00:38:21.460 align:middle line:90% so this is 0. 00:38:21.460 --> 00:38:26.410 align:middle line:84% Since the set is open, I can find a small ball 00:38:26.410 --> 00:38:33.040 align:middle line:84% of radius r centered at 0, which remains inside the set V. 00:38:33.040 --> 00:38:40.740 align:middle line:84% And therefore, it gets sent to some other set, which 00:38:40.740 --> 00:38:44.850 align:middle line:84% is a subset of W. So this is the picture. 00:38:44.850 --> 00:38:47.250 align:middle line:84% Let me write down the math that goes with it. 00:38:47.250 --> 00:38:50.350 align:middle line:90% 00:38:50.350 --> 00:38:55.555 align:middle line:84% Since T of 0 equals 0, that implies 0 is in T inverse of-- 00:38:55.555 --> 00:38:58.380 align:middle line:90% 00:38:58.380 --> 00:39:04.040 align:middle line:84% which implies since this is open, 00:39:04.040 --> 00:39:08.240 align:middle line:84% there exists an r positive such that the ball 00:39:08.240 --> 00:39:19.550 align:middle line:84% in V centered at 0 of radius r is contained in T inverse of-- 00:39:19.550 --> 00:39:26.960 align:middle line:84% the inverse image of the ball of radius 1 centered at 0 in W. 00:39:26.960 --> 00:39:28.910 align:middle line:90% Just look at the picture here. 00:39:28.910 --> 00:39:38.635 align:middle line:90% 00:39:38.635 --> 00:39:44.770 align:middle line:84% Let v be in V. And let's not look at 0 00:39:44.770 --> 00:39:49.270 align:middle line:84% because v equals 0 will satisfy this inequality no matter 00:39:49.270 --> 00:39:50.020 align:middle line:90% what C you have. 00:39:50.020 --> 00:39:53.620 align:middle line:84% So we just need to look at v not equal to 0. 00:39:53.620 --> 00:39:58.690 align:middle line:84% I claim that I can take the constant to be 2 over R, 00:39:58.690 --> 00:39:59.290 align:middle line:90% all right? 00:39:59.290 --> 00:40:06.490 align:middle line:90% 00:40:06.490 --> 00:40:13.530 align:middle line:84% OK, then if I take v and I rescale it-- 00:40:13.530 --> 00:40:19.820 align:middle line:90% 00:40:19.820 --> 00:40:21.770 align:middle line:84% well, let me not write it as dividing 00:40:21.770 --> 00:40:28.250 align:middle line:84% v. Let me write it this way, r over 2 length v v v times v. So 00:40:28.250 --> 00:40:32.510 align:middle line:84% this is a vector in capital V. What is its length? 00:40:32.510 --> 00:40:33.650 align:middle line:90% Its length is r/2. 00:40:33.650 --> 00:40:37.200 align:middle line:90% 00:40:37.200 --> 00:40:39.420 align:middle line:84% So if I take it's length in v, this 00:40:39.420 --> 00:40:43.680 align:middle line:84% is equal to r/2 is less than r, which 00:40:43.680 --> 00:40:52.650 align:middle line:84% implies that r over 2 over v v is in the ball of radius r 00:40:52.650 --> 00:40:55.130 align:middle line:90% in V centered at 0. 00:40:55.130 --> 00:41:00.980 align:middle line:84% And therefore, so it has to be in this blue disk. 00:41:00.980 --> 00:41:04.220 align:middle line:84% And therefore, it gets mapped to something in this blue guy 00:41:04.220 --> 00:41:14.160 align:middle line:84% here, which is contained inside-- remember, 00:41:14.160 --> 00:41:18.720 align:middle line:84% this big yellow thing was the ball of radius 1 centered 00:41:18.720 --> 00:41:19.260 align:middle line:90% at 0-- 00:41:19.260 --> 00:41:25.280 align:middle line:90% 00:41:25.280 --> 00:41:29.240 align:middle line:90% of radius 1 in W. 00:41:29.240 --> 00:41:38.790 align:middle line:84% And therefore, the length of T r over 2 v v v W 00:41:38.790 --> 00:41:41.950 align:middle line:90% must be less than 1. 00:41:41.950 --> 00:41:48.900 align:middle line:84% And now, scalars pull out of linear transformation. 00:41:48.900 --> 00:41:50.030 align:middle line:90% So this comes out. 00:41:50.030 --> 00:41:52.130 align:middle line:84% And then it comes out of the norm 00:41:52.130 --> 00:41:54.690 align:middle line:90% by the homogeneity of the norm. 00:41:54.690 --> 00:41:57.410 align:middle line:90% And so I can divide through by-- 00:41:57.410 --> 00:42:01.310 align:middle line:84% multiply through by 2 norm v over r. 00:42:01.310 --> 00:42:18.190 align:middle line:84% And I get that T of v norm W is less than 2 over r norm V. 00:42:18.190 --> 00:42:26.300 align:middle line:84% And therefore, star holds with C given by 2 over r. 00:42:26.300 --> 00:42:31.290 align:middle line:90% 00:42:31.290 --> 00:42:39.610 align:middle line:84% OK, so continuous linear operators between norm spaces, 00:42:39.610 --> 00:42:41.380 align:middle line:84% we call them bounded linear operators 00:42:41.380 --> 00:42:44.350 align:middle line:84% because they satisfy this boundedness property, 00:42:44.350 --> 00:42:47.725 align:middle line:84% namely that they take bounded sets to bounded sets. 00:42:47.725 --> 00:42:50.760 align:middle line:90% 00:42:50.760 --> 00:42:55.740 align:middle line:84% Now, it's going to become quite tedious for me 00:42:55.740 --> 00:43:00.730 align:middle line:84% to keep writing norm sub W, norm sub V, and so on. 00:43:00.730 --> 00:43:04.050 align:middle line:84% So I'm going to stop using the subscripts. 00:43:04.050 --> 00:43:10.610 align:middle line:84% But it should be pretty clear from the context 00:43:10.610 --> 00:43:11.450 align:middle line:90% where the norm is. 00:43:11.450 --> 00:43:15.860 align:middle line:84% If I have a linear operator T from V to W, T of V-- 00:43:15.860 --> 00:43:33.420 align:middle line:90% 00:43:33.420 --> 00:43:38.070 align:middle line:84% so if I have a bounded linear operator or linear operator 00:43:38.070 --> 00:43:43.020 align:middle line:84% from V to W and you see me write norm of T v, 00:43:43.020 --> 00:43:49.170 align:middle line:84% you should equate this to the norm of T v in W. 00:43:49.170 --> 00:43:53.130 align:middle line:84% Or if you see v, you should equate this-- and v 00:43:53.130 --> 00:43:57.210 align:middle line:84% as an element of capital V, then you should interpret 00:43:57.210 --> 00:44:01.310 align:middle line:84% this as the norm of v in capital V. 00:44:01.310 --> 00:44:03.890 align:middle line:84% So I'm dropping subscripts just to save 00:44:03.890 --> 00:44:06.780 align:middle line:90% having to write too much. 00:44:06.780 --> 00:44:09.750 align:middle line:90% And then it'll soon get tedious. 00:44:09.750 --> 00:44:14.760 align:middle line:90% OK, so the definition-- 00:44:14.760 --> 00:44:19.170 align:middle line:90% 00:44:19.170 --> 00:44:23.220 align:middle line:84% in fact, before I do that, let's take 00:44:23.220 --> 00:44:29.800 align:middle line:84% a look at this linear operator I wrote up there a minute ago. 00:44:29.800 --> 00:44:35.810 align:middle line:84% Can we see that that's a bounded linear operator on the space 00:44:35.810 --> 00:44:37.800 align:middle line:90% of continuous functions? 00:44:37.800 --> 00:45:05.040 align:middle line:84% So given by-- where k is a continuous function-- 00:45:05.040 --> 00:45:13.040 align:middle line:90% 00:45:13.040 --> 00:45:20.400 align:middle line:84% this is a bounded linear operator. 00:45:20.400 --> 00:45:24.120 align:middle line:84% So it's clear-- pretty clear to see 00:45:24.120 --> 00:45:29.160 align:middle line:90% that it's linear in f, right? 00:45:29.160 --> 00:45:32.260 align:middle line:90% Scalars pull out and so on. 00:45:32.260 --> 00:45:33.700 align:middle line:90% Let's check that it's bounded. 00:45:33.700 --> 00:45:40.995 align:middle line:84% So recall that the norm on C 0, 1 is the infinity norm. 00:45:40.995 --> 00:45:45.882 align:middle line:90% 00:45:45.882 --> 00:45:48.485 align:middle line:84% So let f be a continuous function. 00:45:48.485 --> 00:45:54.540 align:middle line:90% 00:45:54.540 --> 00:45:57.350 align:middle line:84% Recall the norm on this space is the infinity norm 00:45:57.350 --> 00:46:07.480 align:middle line:84% given by sup of x in 0, 1, f of x, 00:46:07.480 --> 00:46:08.910 align:middle line:90% which is, in fact, a maximum. 00:46:08.910 --> 00:46:10.487 align:middle line:84% It's attained at a certain value. 00:46:10.487 --> 00:46:11.820 align:middle line:90% But I'll just write sup anyways. 00:46:11.820 --> 00:46:14.380 align:middle line:90% 00:46:14.380 --> 00:46:18.270 align:middle line:84% And so now, we want to estimate the norm of T of f 00:46:18.270 --> 00:46:21.465 align:middle line:90% of x in terms of the norm of f. 00:46:21.465 --> 00:46:23.460 align:middle line:84% That's what this boundedness property is. 00:46:23.460 --> 00:46:28.280 align:middle line:90% 00:46:28.280 --> 00:46:42.590 align:middle line:84% Then for all x in 0, 1, this is equal to the integral of k xy 00:46:42.590 --> 00:46:46.600 align:middle line:90% f of y dy. 00:46:46.600 --> 00:46:50.140 align:middle line:84% Now, the absolute value of the integral 00:46:50.140 --> 00:46:53.630 align:middle line:84% is less than or equal to the integral of the absolute value. 00:46:53.630 --> 00:47:03.990 align:middle line:84% So this is times the absolute value of k xy times f of y dy. 00:47:03.990 --> 00:47:07.890 align:middle line:84% Now, f of y for all y in 0, 1 is less than 00:47:07.890 --> 00:47:12.270 align:middle line:84% or equal to the infinity norm of f. 00:47:12.270 --> 00:47:15.225 align:middle line:84% So that only makes the integral bigger. 00:47:15.225 --> 00:47:24.670 align:middle line:90% 00:47:24.670 --> 00:47:29.600 align:middle line:90% And the same thing for k, right? 00:47:29.600 --> 00:47:32.420 align:middle line:84% k is a continuous function on 0, 1 cross 0, 1. 00:47:32.420 --> 00:47:36.131 align:middle line:84% And therefore, it's bounded by its-- 00:47:36.131 --> 00:47:43.690 align:middle line:84% it attains a max on this set, being a continuous function. 00:47:43.690 --> 00:47:48.600 align:middle line:84% So I can also replace it by its infinity norm. 00:47:48.600 --> 00:47:50.880 align:middle line:84% And here, you should interpret this 00:47:50.880 --> 00:47:56.330 align:middle line:84% as being the infinity norm on continuous functions 00:47:56.330 --> 00:47:57.990 align:middle line:90% on this set. 00:47:57.990 --> 00:48:03.650 align:middle line:84% So this infinity norm is the sup of k of xy for x and y in 0, 1 00:48:03.650 --> 00:48:04.505 align:middle line:90% cross 0, 1. 00:48:04.505 --> 00:48:22.500 align:middle line:90% 00:48:22.500 --> 00:48:25.170 align:middle line:90% And therefore, this equals-- 00:48:25.170 --> 00:48:26.340 align:middle line:90% these are just two numbers. 00:48:26.340 --> 00:48:31.180 align:middle line:90% 00:48:31.180 --> 00:48:36.600 align:middle line:90% And this held-- holds for all x. 00:48:36.600 --> 00:48:39.870 align:middle line:84% And therefore, its supremum is bounded by this number 00:48:39.870 --> 00:48:41.490 align:middle line:90% as well, so Tf-- 00:48:41.490 --> 00:48:49.400 align:middle line:90% 00:48:49.400 --> 00:48:55.040 align:middle line:84% so this is a bounded linear operator with constant given 00:48:55.040 --> 00:48:57.530 align:middle line:90% by the infinity norm of k. 00:48:57.530 --> 00:48:59.850 align:middle line:84% This k is usually referred to as a kernel. 00:48:59.850 --> 00:49:01.970 align:middle line:84% So if I've said that before and haven't explained 00:49:01.970 --> 00:49:05.180 align:middle line:84% where that comes from or why I use that word, 00:49:05.180 --> 00:49:07.220 align:middle line:84% this is usually referred to as the kernel 00:49:07.220 --> 00:49:11.390 align:middle line:90% of this linear operator. 00:49:11.390 --> 00:49:14.890 align:middle line:90% 00:49:14.890 --> 00:49:19.010 align:middle line:90% So there's an example for you. 00:49:19.010 --> 00:49:20.480 align:middle line:84% I've already used up both boards? 00:49:20.480 --> 00:49:24.500 align:middle line:84% Yeah, I missed the big room that had three of these. 00:49:24.500 --> 00:49:30.190 align:middle line:90% 00:49:30.190 --> 00:49:38.760 align:middle line:84% Now, given two norm spaces, V and W, 00:49:38.760 --> 00:49:44.460 align:middle line:84% we can consider the set of all bounded linear operators 00:49:44.460 --> 00:49:49.740 align:middle line:84% from V to W, which we denote by B V, W-- 00:49:49.740 --> 00:50:01.290 align:middle line:84% scripty looking B-- set of T. T is a bounded linear operator. 00:50:01.290 --> 00:50:05.490 align:middle line:90% 00:50:05.490 --> 00:50:10.190 align:middle line:84% So it's not difficult to see, again, that-- 00:50:10.190 --> 00:50:13.385 align:middle line:84% or to see that this is a vector space. 00:50:13.385 --> 00:50:22.920 align:middle line:90% 00:50:22.920 --> 00:50:26.370 align:middle line:84% So let me put this in a remark, that it's clear 00:50:26.370 --> 00:50:28.530 align:middle line:90% that this is a vector space. 00:50:28.530 --> 00:50:30.630 align:middle line:84% The sum of two linear transformations 00:50:30.630 --> 00:50:33.735 align:middle line:84% is a linear transformation-- or linear operator. 00:50:33.735 --> 00:50:37.050 align:middle line:84% The scalar multiple of a linear operator is a linear operator. 00:50:37.050 --> 00:50:42.580 align:middle line:84% And then those two operations preserve continuity. 00:50:42.580 --> 00:50:44.655 align:middle line:84% So this is clearly a vector space. 00:50:44.655 --> 00:50:49.040 align:middle line:90% 00:50:49.040 --> 00:50:52.915 align:middle line:84% Now, we can put a norm on this space. 00:50:52.915 --> 00:50:59.190 align:middle line:90% 00:50:59.190 --> 00:51:14.050 align:middle line:84% We define the operator norm of an element in here. 00:51:14.050 --> 00:51:19.550 align:middle line:90% 00:51:19.550 --> 00:51:26.210 align:middle line:84% This is defined to be the supremum over all unit length 00:51:26.210 --> 00:51:32.600 align:middle line:84% vectors v of T of v. So I want to recall that this norm is 00:51:32.600 --> 00:51:34.820 align:middle line:84% being taken in W because T of v is 00:51:34.820 --> 00:51:38.060 align:middle line:84% an element in W. This norm here is 00:51:38.060 --> 00:51:50.710 align:middle line:84% being taken in V because little v is an element in V. 00:51:50.710 --> 00:51:57.530 align:middle line:84% So maybe it's not clear at first. 00:51:57.530 --> 00:52:07.400 align:middle line:84% But let's go ahead and prove it that, in fact, the operator 00:52:07.400 --> 00:52:20.940 align:middle line:84% norm is, in fact, an actual norm so 00:52:20.940 --> 00:52:23.145 align:middle line:90% that this becomes a norm space. 00:52:23.145 --> 00:52:25.680 align:middle line:90% 00:52:25.680 --> 00:52:27.970 align:middle line:84% So the operator norm is an actual norm. 00:52:27.970 --> 00:52:32.820 align:middle line:84% It's not just a norm because I said it is. 00:52:32.820 --> 00:52:34.350 align:middle line:90% So let's prove this. 00:52:34.350 --> 00:52:43.720 align:middle line:90% 00:52:43.720 --> 00:52:51.390 align:middle line:84% It's not-- again, so it's not too difficult to see. 00:52:51.390 --> 00:52:55.190 align:middle line:84% So let's prove definiteness, namely the norm of T 00:52:55.190 --> 00:52:58.430 align:middle line:90% is 0 if and only if T is 0. 00:52:58.430 --> 00:53:04.350 align:middle line:84% So if T is a zero operator, then clearly its norm is 0. 00:53:04.350 --> 00:53:08.280 align:middle line:90% Suppose T of v equals 0. 00:53:08.280 --> 00:53:12.720 align:middle line:84% So this sup is 0 if and only if T of v is 0 for all unit length 00:53:12.720 --> 00:53:15.190 align:middle line:90% v. So suppose T is an operator. 00:53:15.190 --> 00:53:20.310 align:middle line:84% So that T of v equals 0 for all unit length v. Then that 00:53:20.310 --> 00:53:23.670 align:middle line:90% implies that T of-- 00:53:23.670 --> 00:53:29.250 align:middle line:90% 00:53:29.250 --> 00:53:40.720 align:middle line:90% that 0 equals T of v equals T. 00:53:40.720 --> 00:53:43.445 align:middle line:84% All I'm saying is you rescale now, OK? 00:53:43.445 --> 00:53:43.945 align:middle line:90% 0. 00:53:43.945 --> 00:53:47.650 align:middle line:90% 00:53:47.650 --> 00:54:03.530 align:middle line:84% So it implies for all v in V take away 0 that 0 equals T. 00:54:03.530 --> 00:54:10.940 align:middle line:84% So t is the zero operator if its norm is 0. 00:54:10.940 --> 00:54:11.480 align:middle line:90% 2. 00:54:11.480 --> 00:54:18.180 align:middle line:84% Homogeneity follows from the homogeneity of the norm 00:54:18.180 --> 00:54:26.420 align:middle line:84% on W. So lambda T, this is equal to-- 00:54:26.420 --> 00:54:32.930 align:middle line:84% so the norm of lambda times T, this is equal to the sup of v 00:54:32.930 --> 00:54:41.430 align:middle line:84% equals 1 of lambda T v. And this is equal to sup 00:54:41.430 --> 00:54:45.240 align:middle line:90% equals 1 times lambda times Tv. 00:54:45.240 --> 00:54:49.030 align:middle line:84% And if I take a set and multiply it by a non-negative number, 00:54:49.030 --> 00:54:51.510 align:middle line:84% then that non-negative number comes out of the sup. 00:54:51.510 --> 00:54:57.370 align:middle line:90% 00:54:57.370 --> 00:54:59.800 align:middle line:84% Hopefully, it was one of the first exercises 00:54:59.800 --> 00:55:02.113 align:middle line:90% you were ever given on supes. 00:55:02.113 --> 00:55:03.280 align:middle line:90% And therefore, this equals-- 00:55:03.280 --> 00:55:06.610 align:middle line:90% 00:55:06.610 --> 00:55:12.020 align:middle line:84% and now, so that proves homogeneity of the norm. 00:55:12.020 --> 00:55:14.780 align:middle line:84% And now, the triangle inequality follows 00:55:14.780 --> 00:55:19.760 align:middle line:84% from the triangle inequality for the norm on W again. 00:55:19.760 --> 00:55:32.300 align:middle line:84% So I take two operators and two bounded linear operators. 00:55:32.300 --> 00:55:43.920 align:middle line:84% And I take an element of V with norm equal to 1. 00:55:43.920 --> 00:55:54.210 align:middle line:84% Then S plus T applied to v norm, this is equal to Sv plus Tv. 00:55:54.210 --> 00:55:57.060 align:middle line:84% And by the triangle inequality for the norm on W, 00:55:57.060 --> 00:56:03.270 align:middle line:84% this is less than or equal to S times v plus T times v. 00:56:03.270 --> 00:56:09.470 align:middle line:84% And now, S times v-- so again, v's a unit length vector-- 00:56:09.470 --> 00:56:12.680 align:middle line:84% is less than or equal to the sup over all those norms, which 00:56:12.680 --> 00:56:15.500 align:middle line:84% is, again, just the operator norm of S 00:56:15.500 --> 00:56:19.820 align:middle line:84% and then the operator norm of T. So I've 00:56:19.820 --> 00:56:24.380 align:middle line:84% proven that for all unit length v, S plus T applied 00:56:24.380 --> 00:56:28.910 align:middle line:84% to v in norm is less than or equal to this number here. 00:56:28.910 --> 00:56:32.930 align:middle line:84% And therefore, the supremum over all such numbers 00:56:32.930 --> 00:56:36.050 align:middle line:84% as v ranges overall unit length vectors, which 00:56:36.050 --> 00:56:39.680 align:middle line:84% is the least upper bound of that set, must be-- 00:56:39.680 --> 00:56:45.375 align:middle line:84% sit below this number here, which implies that-- 00:56:45.375 --> 00:56:54.640 align:middle line:90% 00:56:54.640 --> 00:56:55.140 align:middle line:90% OK? 00:56:55.140 --> 00:56:58.020 align:middle line:90% 00:56:58.020 --> 00:57:03.170 align:middle line:84% And therefore, the operator norm is a norm 00:57:03.170 --> 00:57:11.220 align:middle line:90% both in name and in actuality. 00:57:11.220 --> 00:57:21.640 align:middle line:90% 00:57:21.640 --> 00:57:24.910 align:middle line:84% So if you like, what we did a minute ago here 00:57:24.910 --> 00:57:28.180 align:middle line:90% is we showed that-- 00:57:28.180 --> 00:57:32.620 align:middle line:84% so coming back to over here for this bounded linear operator 00:57:32.620 --> 00:57:34.870 align:middle line:84% from continuous functions to continuous functions, 00:57:34.870 --> 00:57:40.210 align:middle line:90% what this tells you-- 00:57:40.210 --> 00:57:45.570 align:middle line:84% so first off, if f has unit length, 00:57:45.570 --> 00:57:48.480 align:middle line:84% then I've shown that Tf in L infinity form is less than 00:57:48.480 --> 00:57:51.030 align:middle line:84% or equal to the L infinity norm of K. 00:57:51.030 --> 00:57:56.580 align:middle line:84% And therefore, I've shown that the operator norm of T, where 00:57:56.580 --> 00:58:04.120 align:middle line:84% T is defined over there, is less than or equal to the operator 00:58:04.120 --> 00:58:08.860 align:middle line:84% norm for K. We're actually a little wasteful there. 00:58:08.860 --> 00:58:13.170 align:middle line:90% This is not equality, actually. 00:58:13.170 --> 00:58:15.630 align:middle line:84% But I'll let you think about that when you have free time. 00:58:15.630 --> 00:58:19.150 align:middle line:90% 00:58:19.150 --> 00:58:24.910 align:middle line:84% OK, so we've talked about bounded linear operators 00:58:24.910 --> 00:58:28.300 align:middle line:90% from one norm space to another. 00:58:28.300 --> 00:58:32.290 align:middle line:84% What more can we say about this new space 00:58:32.290 --> 00:58:38.560 align:middle line:90% we formed from two norm spaces? 00:58:38.560 --> 00:58:41.330 align:middle line:84% When is this space complete, for example? 00:58:41.330 --> 00:58:42.980 align:middle line:90% What are sufficient conditions? 00:58:42.980 --> 00:58:56.150 align:middle line:84% And so theorem-- if W is a Banach space-- 00:58:56.150 --> 00:58:59.550 align:middle line:90% 00:58:59.550 --> 00:59:02.640 align:middle line:90% again, V and W are norm spaces. 00:59:02.640 --> 00:59:08.220 align:middle line:84% If I assume W is a Banach space, then the space 00:59:08.220 --> 00:59:10.800 align:middle line:84% of bounded linear operators from V to W 00:59:10.800 --> 00:59:19.380 align:middle line:84% is a Banach space no matter if v is a Banach space or not. 00:59:19.380 --> 00:59:26.410 align:middle line:90% 00:59:26.410 --> 00:59:31.620 align:middle line:84% OK, so what we're going to do is we're 00:59:31.620 --> 00:59:37.050 align:middle line:84% going to use that characterization we had earlier 00:59:37.050 --> 00:59:42.090 align:middle line:84% of when a norm space is a Banach space in terms 00:59:42.090 --> 00:59:45.630 align:middle line:84% of summability absolute summability. 00:59:45.630 --> 00:59:57.690 align:middle line:84% So suppose Tn is a sequence of elements 00:59:57.690 --> 01:00:08.400 align:middle line:84% of bounded linear operators such that this constant C, 01:00:08.400 --> 01:00:14.550 align:middle line:84% which is the sum of these norms, exists as a real number. 01:00:14.550 --> 01:00:19.440 align:middle line:84% So let's suppose that we have an absolutely summable series 01:00:19.440 --> 01:00:21.240 align:middle line:90% of linear operators. 01:00:21.240 --> 01:00:24.255 align:middle line:84% And to show that this is a Banach space, we want to show-- 01:00:24.255 --> 01:00:38.180 align:middle line:90% 01:00:38.180 --> 01:00:42.570 align:middle line:84% we want to show that this series is summable. 01:00:42.570 --> 01:00:45.510 align:middle line:84% Then by that theorem we have earlier, 01:00:45.510 --> 01:00:48.690 align:middle line:84% that if you have a norm space such that every absolutely 01:00:48.690 --> 01:00:51.750 align:middle line:84% summable series is summable, then it's a Banach space. 01:00:51.750 --> 01:00:55.560 align:middle line:84% We conclude the proof of the theorem. 01:00:55.560 --> 01:00:58.870 align:middle line:84% And how we're going to show this is summable is, again, 01:00:58.870 --> 01:01:01.050 align:middle line:84% kind of the same strategy we used 01:01:01.050 --> 01:01:04.890 align:middle line:84% to prove that the space of bounded continuous functions 01:01:04.890 --> 01:01:08.430 align:middle line:84% on a metric space is a Banach space. 01:01:08.430 --> 01:01:11.520 align:middle line:84% We're going to come up with a candidate for this, show 01:01:11.520 --> 01:01:13.610 align:middle line:84% that it's actually a bounded linear operator, 01:01:13.610 --> 01:01:15.360 align:middle line:84% and then that the convergence is uniform-- 01:01:15.360 --> 01:01:19.150 align:middle line:90% 01:01:19.150 --> 01:01:21.460 align:middle line:84% or not uniform, but the convergence 01:01:21.460 --> 01:01:24.520 align:middle line:84% is in the space in the operator norm. 01:01:24.520 --> 01:01:34.390 align:middle line:90% 01:01:34.390 --> 01:01:40.270 align:middle line:90% OK, so let me just make a-- 01:01:40.270 --> 01:01:47.850 align:middle line:90% 01:01:47.850 --> 01:01:50.820 align:middle line:84% let me just make a note of something real quick, which 01:01:50.820 --> 01:01:51.690 align:middle line:90% I meant to write. 01:01:51.690 --> 01:01:53.360 align:middle line:90% You know what? 01:01:53.360 --> 01:01:58.460 align:middle line:84% Let's write it up there because that's where it belongs. 01:01:58.460 --> 01:02:00.260 align:middle line:90% But I didn't write it down. 01:02:00.260 --> 01:02:02.780 align:middle line:84% So let me just make a remark here. 01:02:02.780 --> 01:02:05.900 align:middle line:90% 01:02:05.900 --> 01:02:09.590 align:middle line:84% The operator norm is defined for all unit length v. 01:02:09.590 --> 01:02:11.300 align:middle line:84% But it automatically gives us a bound. 01:02:11.300 --> 01:02:18.250 align:middle line:90% 01:02:18.250 --> 01:02:23.130 align:middle line:84% And I'm moving a little quickly because-- 01:02:23.130 --> 01:02:29.710 align:middle line:90% 01:02:29.710 --> 01:02:31.405 align:middle line:84% maybe quicker than I should have, but-- 01:02:31.405 --> 01:02:37.070 align:middle line:90% 01:02:37.070 --> 01:02:43.140 align:middle line:90% OK, what was I going to say? 01:02:43.140 --> 01:02:46.320 align:middle line:84% First off, one thing I should have said 01:02:46.320 --> 01:02:49.260 align:middle line:84% is that this is an actual finite number 01:02:49.260 --> 01:02:55.370 align:middle line:84% if I have a bounded linear operator because, remember, 01:02:55.370 --> 01:03:00.020 align:middle line:84% T, a bounded linear operator, implies 01:03:00.020 --> 01:03:07.780 align:middle line:84% there exists a constant such that for all v in V, 01:03:07.780 --> 01:03:11.890 align:middle line:84% the norm of Tv is less than or equal to a constant times 01:03:11.890 --> 01:03:15.910 align:middle line:84% the norm of V. So when v has unit length, 01:03:15.910 --> 01:03:18.070 align:middle line:90% this constant bounds-- 01:03:18.070 --> 01:03:22.810 align:middle line:84% that satisfies this inequality bounds these numbers 01:03:22.810 --> 01:03:24.250 align:middle line:90% for all v with unit length. 01:03:24.250 --> 01:03:27.700 align:middle line:84% And in fact, this supremum is the smallest such C 01:03:27.700 --> 01:03:29.940 align:middle line:90% that I can put here. 01:03:29.940 --> 01:03:31.690 align:middle line:84% So that's the one comment I wanted to say, 01:03:31.690 --> 01:03:35.030 align:middle line:84% that this is an actual, well-defined thing. 01:03:35.030 --> 01:03:38.620 align:middle line:90% OK, so I wanted to say that. 01:03:38.620 --> 01:03:42.660 align:middle line:84% And then also, now from rescaling, 01:03:42.660 --> 01:03:47.460 align:middle line:84% we get a bound for all v in terms of the operator norm. 01:03:47.460 --> 01:03:54.560 align:middle line:90% 01:03:54.560 --> 01:04:02.480 align:middle line:84% Then if I take v over its length and take its operator-- 01:04:02.480 --> 01:04:04.410 align:middle line:90% or I take its norm-- 01:04:04.410 --> 01:04:09.420 align:middle line:84% so this is a unit length vector in V. I applied T to it. 01:04:09.420 --> 01:04:11.540 align:middle line:84% So that's always less than or equal to the sup 01:04:11.540 --> 01:04:16.610 align:middle line:84% over all norm of T times something, 01:04:16.610 --> 01:04:18.720 align:middle line:84% where that something has unit length. 01:04:18.720 --> 01:04:22.140 align:middle line:84% But this is a linear operator so that this scalar 01:04:22.140 --> 01:04:24.750 align:middle line:84% comes out here, and then comes out of the norm 01:04:24.750 --> 01:04:28.010 align:middle line:90% again by homogeneity. 01:04:28.010 --> 01:04:36.330 align:middle line:84% And therefore, I get that Tv is that. 01:04:36.330 --> 01:04:41.120 align:middle line:84% So in short, the point of this remark was to say that-- 01:04:41.120 --> 01:04:43.700 align:middle line:90% 01:04:43.700 --> 01:04:46.040 align:middle line:84% this is not really maybe the best way to say it. 01:04:46.040 --> 01:04:49.820 align:middle line:84% But if I think of T acting on v as the product of T and v, 01:04:49.820 --> 01:04:53.060 align:middle line:84% then this says the norm of the product of T and v 01:04:53.060 --> 01:04:56.990 align:middle line:84% is less than or equal to the product of the norms of T 01:04:56.990 --> 01:05:11.148 align:middle line:90% and v. 01:05:11.148 --> 01:05:12.690 align:middle line:84% Sorry if I went a little quick there. 01:05:12.690 --> 01:05:14.232 align:middle line:84% Maybe you stopped and were wondering, 01:05:14.232 --> 01:05:15.840 align:middle line:90% why is that even a real-- 01:05:15.840 --> 01:05:17.160 align:middle line:90% why is that even a real thing? 01:05:17.160 --> 01:05:19.860 align:middle line:90% Why is that an actual number? 01:05:19.860 --> 01:05:21.390 align:middle line:90% What am I going on about? 01:05:21.390 --> 01:05:22.920 align:middle line:84% Maybe I'm just a little overexcited 01:05:22.920 --> 01:05:25.500 align:middle line:84% about teaching functional analysis because this 01:05:25.500 --> 01:05:40.060 align:middle line:84% is kind of your first adult analysis class, I will say. 01:05:40.060 --> 01:05:43.110 align:middle line:84% OK, so back to the proof at hand, 01:05:43.110 --> 01:05:46.820 align:middle line:84% we have this series of absolutely summable 01:05:46.820 --> 01:05:48.920 align:middle line:90% bounded linear operators. 01:05:48.920 --> 01:05:52.550 align:middle line:84% So the sum of the norms is convergent, 01:05:52.550 --> 01:05:56.240 align:middle line:84% meaning this series sum is finite. 01:05:56.240 --> 01:05:59.660 align:middle line:84% And we want to show that this series of the T 01:05:59.660 --> 01:06:02.720 align:middle line:84% sub n's is summable so that this has 01:06:02.720 --> 01:06:06.050 align:middle line:84% a limit in the space of bounded linear operators. 01:06:06.050 --> 01:06:09.840 align:middle line:90% 01:06:09.840 --> 01:06:14.488 align:middle line:84% So we're going to come up with a candidate. 01:06:14.488 --> 01:06:26.090 align:middle line:84% Let v be in V. If I look at the sum 01:06:26.090 --> 01:06:29.405 align:middle line:90% of the norms of T sub n of v-- 01:06:29.405 --> 01:06:36.475 align:middle line:84% so T sub n applied to v, this is bounded by-- 01:06:36.475 --> 01:06:37.850 align:middle line:84% I mean, I'm writing it like this. 01:06:37.850 --> 01:06:41.410 align:middle line:84% But let me be a little more careful. 01:06:41.410 --> 01:06:47.020 align:middle line:90% 01:06:47.020 --> 01:06:49.960 align:middle line:90% Let m be a natural number. 01:06:49.960 --> 01:06:59.340 align:middle line:84% Then sum from n equals 1 to m of T sub n of v norm, 01:06:59.340 --> 01:07:04.950 align:middle line:84% this is less than or equal to the sum from n equals 1 to m 01:07:04.950 --> 01:07:13.400 align:middle line:84% of T sub n times the norm of v. And so v, that's just a number. 01:07:13.400 --> 01:07:14.800 align:middle line:90% It comes outside the sum. 01:07:14.800 --> 01:07:19.180 align:middle line:84% And the sum from n equals 1 to m is bounded by the sum from n 01:07:19.180 --> 01:07:22.130 align:middle line:84% equals 1 to infinity of these non-negative numbers. 01:07:22.130 --> 01:07:29.860 align:middle line:84% So this is less than or equal to the norm of v times T sub n 01:07:29.860 --> 01:07:30.595 align:middle line:90% equals Cv. 01:07:30.595 --> 01:07:33.290 align:middle line:90% 01:07:33.290 --> 01:07:38.570 align:middle line:84% So for all m, I've shown that this thing is bounded 01:07:38.570 --> 01:07:47.300 align:middle line:84% by C times the norm of v. And therefore, the partial sums 01:07:47.300 --> 01:07:50.870 align:middle line:84% corresponding to this series of non-negative real numbers 01:07:50.870 --> 01:07:51.740 align:middle line:90% are bounded. 01:07:51.740 --> 01:07:54.040 align:middle line:84% And therefore, that series converges. 01:07:54.040 --> 01:07:58.640 align:middle line:90% 01:07:58.640 --> 01:08:06.660 align:middle line:84% So this sequence of partial sums of non-negative real numbers 01:08:06.660 --> 01:08:19.730 align:middle line:84% is bounded, which implies that it converges, 01:08:19.730 --> 01:08:30.979 align:middle line:90% which I'll write Tn v converges. 01:08:30.979 --> 01:08:34.370 align:middle line:90% Now, think of T sub n of v as-- 01:08:34.370 --> 01:08:38.810 align:middle line:84% so T sub n of v, this is an element in W for each n. 01:08:38.810 --> 01:08:51.660 align:middle line:84% So I've shown that the series T sub n of v, 01:08:51.660 --> 01:09:01.460 align:middle line:84% this is absolutely summable in W. These are elements of W. 01:09:01.460 --> 01:09:05.382 align:middle line:84% And their norms is an absolutely convergent series 01:09:05.382 --> 01:09:06.049 align:middle line:90% of real numbers. 01:09:06.049 --> 01:09:08.859 align:middle line:90% 01:09:08.859 --> 01:09:13.910 align:middle line:84% Now, since W is a Banach space, every absolutely summable 01:09:13.910 --> 01:09:24.739 align:middle line:84% series is summable, therefore summable. 01:09:24.739 --> 01:09:27.260 align:middle line:90% 01:09:27.260 --> 01:09:41.050 align:middle line:84% We therefore define a map from V to W via T sub v-- 01:09:41.050 --> 01:09:44.750 align:middle line:84% or T of v is defined to be the limit 01:09:44.750 --> 01:09:49.340 align:middle line:84% as m goes to infinity sum from n equals 1 to m 01:09:49.340 --> 01:09:53.760 align:middle line:84% of T sub n applied to V, which we've 01:09:53.760 --> 01:10:00.640 align:middle line:84% shown for every v is a convergent series in W. That 01:10:00.640 --> 01:10:02.560 align:middle line:84% was the point of everything that came before. 01:10:02.560 --> 01:10:08.910 align:middle line:84% For each v, this is a summable series. 01:10:08.910 --> 01:10:11.190 align:middle line:84% And we define its limit, which depends 01:10:11.190 --> 01:10:21.760 align:middle line:84% on v, as this map T going from V to W. 01:10:21.760 --> 01:10:23.510 align:middle line:84% So this is our candidate, which we'll show 01:10:23.510 --> 01:10:25.520 align:middle line:90% is a bounded linear operator. 01:10:25.520 --> 01:10:33.100 align:middle line:90% 01:10:33.100 --> 01:10:34.330 align:middle line:90% So let's show it's linear. 01:10:34.330 --> 01:10:40.130 align:middle line:90% 01:10:40.130 --> 01:10:41.480 align:middle line:90% So T is linear. 01:10:41.480 --> 01:10:42.710 align:middle line:90% Why? 01:10:42.710 --> 01:10:50.690 align:middle line:84% For all lambda 1, lambda 2 in the space of scalars, RC, v1, 01:10:50.690 --> 01:10:59.550 align:middle line:84% v2, in V, we have that T of lambda 1 v1 plus lambda 2 v2-- 01:10:59.550 --> 01:11:03.450 align:middle line:84% this is, by definition, equal to limit as m 01:11:03.450 --> 01:11:10.100 align:middle line:84% goes to infinity of the sum T sub n lambda 01:11:10.100 --> 01:11:13.760 align:middle line:90% 1 v1 plus lambda 2 v2. 01:11:13.760 --> 01:11:17.070 align:middle line:84% And now, each T sub n is a linear operator. 01:11:17.070 --> 01:11:21.900 align:middle line:84% So I can write this as limit as m 01:11:21.900 --> 01:11:38.585 align:middle line:84% goes to infinity of lambda 1 Tv sub 1 plus lambda 2 Tv sub 2. 01:11:38.585 --> 01:11:44.170 align:middle line:90% 01:11:44.170 --> 01:11:49.330 align:middle line:84% And now, this thing here converges to T of v 01:11:49.330 --> 01:11:50.930 align:middle line:90% as m goes to infinity. 01:11:50.930 --> 01:11:55.850 align:middle line:84% This thing here converges to T of v2 as m goes to infinity. 01:11:55.850 --> 01:12:00.625 align:middle line:84% And therefore, the limit of the sum is the sum of the limits. 01:12:00.625 --> 01:12:17.010 align:middle line:90% 01:12:17.010 --> 01:12:21.090 align:middle line:84% So technically, I did not prove that in a norm space 01:12:21.090 --> 01:12:24.930 align:middle line:84% if I have two sequences converging to v1 and v2, 01:12:24.930 --> 01:12:30.420 align:middle line:84% then the sum of that sequence converges to v1 plus v2. 01:12:30.420 --> 01:12:33.240 align:middle line:84% But it's the exact same proof as in R, all right? 01:12:33.240 --> 01:12:36.045 align:middle line:84% Just replace the absolute values with norms. 01:12:36.045 --> 01:12:39.190 align:middle line:90% 01:12:39.190 --> 01:12:42.240 align:middle line:90% So this should be believable. 01:12:42.240 --> 01:12:49.730 align:middle line:84% And therefore, T is a linear operator. 01:12:49.730 --> 01:12:52.270 align:middle line:90% 01:12:52.270 --> 01:12:58.105 align:middle line:84% Now, let's prove that it's a bounded linear operator. 01:12:58.105 --> 01:13:02.350 align:middle line:84% OK, so now, we'll show that T is a bounded linear operator. 01:13:02.350 --> 01:13:11.530 align:middle line:90% 01:13:11.530 --> 01:13:17.520 align:middle line:90% Let v be in V norm equal to 1. 01:13:17.520 --> 01:13:24.760 align:middle line:84% Then Tv, This is equal to the norm of the limit 01:13:24.760 --> 01:13:37.380 align:middle line:84% as m goes to infinity of n equals 1 to m T 01:13:37.380 --> 01:13:39.810 align:middle line:90% sub n of v norm. 01:13:39.810 --> 01:13:45.460 align:middle line:84% And norms of limits equal to limit of norms, 01:13:45.460 --> 01:13:52.210 align:middle line:84% just like in the case of R, so this is equal to limit as m 01:13:52.210 --> 01:13:53.485 align:middle line:90% goes to infinity of-- 01:13:53.485 --> 01:13:59.970 align:middle line:90% 01:13:59.970 --> 01:14:00.810 align:middle line:90% OK? 01:14:00.810 --> 01:14:02.640 align:middle line:84% Now, this is less than or equal to-- 01:14:02.640 --> 01:14:10.450 align:middle line:90% 01:14:10.450 --> 01:14:13.570 align:middle line:84% by the triangle inequality-- the triangle inequality for two 01:14:13.570 --> 01:14:17.980 align:middle line:84% things, this implies a triangle inequality for m things 01:14:17.980 --> 01:14:37.270 align:middle line:84% by induction, which is less than or equal to the norm of T sub n 01:14:37.270 --> 01:14:38.080 align:middle line:90% times-- 01:14:38.080 --> 01:14:39.910 align:middle line:84% I shouldn't even say v. It's just less 01:14:39.910 --> 01:14:43.890 align:middle line:84% than or equal to the norm of T sub n. 01:14:43.890 --> 01:14:50.100 align:middle line:84% And this is precisely equal to the sum 01:14:50.100 --> 01:14:55.310 align:middle line:84% of the norms, which I called C, this constant C, which 01:14:55.310 --> 01:14:56.600 align:middle line:90% we know is finite-- 01:14:56.600 --> 01:14:58.880 align:middle line:84% which we assumed is finite, right? 01:14:58.880 --> 01:15:01.940 align:middle line:84% So we assumed it was an absolutely summable series 01:15:01.940 --> 01:15:05.240 align:middle line:90% of bounded linear operators. 01:15:05.240 --> 01:15:10.250 align:middle line:84% So therefore, I've got that Tv is 01:15:10.250 --> 01:15:14.930 align:middle line:84% less than or equal to C for all unit length v. And therefore-- 01:15:14.930 --> 01:15:21.300 align:middle line:90% 01:15:21.300 --> 01:15:27.130 align:middle line:84% which again, by scaling arguments, 01:15:27.130 --> 01:15:32.420 align:middle line:84% this was for all v equals 1, which implies by scaling 01:15:32.420 --> 01:15:44.300 align:middle line:90% arguments T for all v. 01:15:44.300 --> 01:15:46.910 align:middle line:84% I guess I didn't have to start with norm of v 01:15:46.910 --> 01:15:50.570 align:middle line:84% equals 1 if that would have brought a norm of v here. 01:15:50.570 --> 01:15:53.150 align:middle line:84% And then that would just be C times the norm of v instead 01:15:53.150 --> 01:15:56.030 align:middle line:90% of doing this separate part. 01:15:56.030 --> 01:15:57.920 align:middle line:90% So change that in your notes. 01:15:57.920 --> 01:16:00.573 align:middle line:90% 01:16:00.573 --> 01:16:01.990 align:middle line:84% But I'm up against the clock here, 01:16:01.990 --> 01:16:04.750 align:middle line:84% so I'm not going to do this on the board. 01:16:04.750 --> 01:16:07.870 align:middle line:84% OK, so T is an actual bounded linear operator. 01:16:07.870 --> 01:16:14.760 align:middle line:90% 01:16:14.760 --> 01:16:20.060 align:middle line:84% Now, let's show that the sum of these operators 01:16:20.060 --> 01:16:24.530 align:middle line:84% converges to T in the operator norm. 01:16:24.530 --> 01:16:29.885 align:middle line:90% So now, we claim that T equals-- 01:16:29.885 --> 01:16:39.630 align:middle line:90% 01:16:39.630 --> 01:16:41.105 align:middle line:90% oh, that's awful. 01:16:41.105 --> 01:16:48.920 align:middle line:90% 01:16:48.920 --> 01:16:55.400 align:middle line:84% Tn converges to T as m goes to infinity 01:16:55.400 --> 01:17:00.470 align:middle line:84% in the space of bounded linear operators, 01:17:00.470 --> 01:17:06.830 align:middle line:84% meaning in the operator norm So I think, for this, this 01:17:06.830 --> 01:17:09.390 align:middle line:84% is the reason why I maybe accidentally wrote 01:17:09.390 --> 01:17:10.740 align:middle line:90% the norm of v equals 1 there. 01:17:10.740 --> 01:17:18.100 align:middle line:84% So let v be in V with norm of v equals 1. 01:17:18.100 --> 01:17:36.300 align:middle line:84% Then T of v minus T sum from n equals 1 to m 01:17:36.300 --> 01:17:39.900 align:middle line:90% of T sub n of v in norm. 01:17:39.900 --> 01:17:50.790 align:middle line:84% Now, T is equal to the whole sum n equals 1 to infinity. 01:17:50.790 --> 01:17:56.630 align:middle line:84% So I'll write this as maybe limit m-prime 01:17:56.630 --> 01:18:01.100 align:middle line:84% goes to infinity of sum from n equals 1 01:18:01.100 --> 01:18:07.970 align:middle line:84% to m-prime Tn V minus sum from n equals 01:18:07.970 --> 01:18:18.000 align:middle line:84% 1 to m Tn v. And this equals the limit m-prime goes 01:18:18.000 --> 01:18:27.190 align:middle line:84% to infinity of sum from n equals m plus 1 m-prime T sub n v 01:18:27.190 --> 01:18:27.690 align:middle line:90% norm. 01:18:27.690 --> 01:18:31.010 align:middle line:90% 01:18:31.010 --> 01:18:35.480 align:middle line:84% And now, this is less than or equal to the limit as m-prime 01:18:35.480 --> 01:18:37.160 align:middle line:90% goes to infinity of-- 01:18:37.160 --> 01:18:42.480 align:middle line:84% so the norm-- so this is, in fact, equal 01:18:42.480 --> 01:18:44.250 align:middle line:90% to the limit of the norm. 01:18:44.250 --> 01:18:45.840 align:middle line:84% And then I use the triangle inequality 01:18:45.840 --> 01:18:47.730 align:middle line:90% to bring the norm inside. 01:18:47.730 --> 01:18:53.040 align:middle line:84% This is less than or equal to sum equals m plus 1 m-prime T 01:18:53.040 --> 01:18:59.980 align:middle line:84% sub n of v. And now, this is less than or equal to limit 01:18:59.980 --> 01:19:05.660 align:middle line:84% m-prime goes to infinity of n equals m plus 1 01:19:05.660 --> 01:19:10.670 align:middle line:84% to m-prime T sub n norm because v has unit length. 01:19:10.670 --> 01:19:16.700 align:middle line:84% So T sub n applied to v in norm is bounded by the operator 01:19:16.700 --> 01:19:17.900 align:middle line:90% norm of T sub n. 01:19:17.900 --> 01:19:20.690 align:middle line:90% 01:19:20.690 --> 01:19:26.240 align:middle line:84% And this equals the sum from n equals m plus 1 01:19:26.240 --> 01:19:29.420 align:middle line:90% to infinity of T sub n. 01:19:29.420 --> 01:19:33.890 align:middle line:84% Now, what we know-- so this is just a series 01:19:33.890 --> 01:19:35.760 align:middle line:90% involving real numbers. 01:19:35.760 --> 01:19:37.920 align:middle line:84% And we know if the series converges, 01:19:37.920 --> 01:19:39.590 align:middle line:90% then the tails have to go to 0. 01:19:39.590 --> 01:19:45.100 align:middle line:84% So this goes to 0 as m goes to infinity, right? 01:19:45.100 --> 01:19:49.100 align:middle line:84% Or I shouldn't do that step just yet. 01:19:49.100 --> 01:19:50.290 align:middle line:90% Sorry, I'm making mistakes. 01:19:50.290 --> 01:19:52.290 align:middle line:84% I'm up against the clock that I see in the back. 01:19:52.290 --> 01:19:54.770 align:middle line:84% So I started off with this quantity 01:19:54.770 --> 01:19:59.510 align:middle line:84% here and ended up bounding it by this thing uniformly in V. So 01:19:59.510 --> 01:20:04.880 align:middle line:84% that implies that the operator norm of T minus sum from n 01:20:04.880 --> 01:20:07.970 align:middle line:84% equals 1 to m of T sub n is less than 01:20:07.970 --> 01:20:13.360 align:middle line:84% or equal to the sum from n equals 01:20:13.360 --> 01:20:17.470 align:middle line:84% m plus 1 operator norm of T sub n. 01:20:17.470 --> 01:20:21.730 align:middle line:84% And this last thing goes to 0 as m 01:20:21.730 --> 01:20:27.340 align:middle line:84% goes to infinity because it's the tail of a convergent series 01:20:27.340 --> 01:20:29.560 align:middle line:90% of non-negative terms. 01:20:29.560 --> 01:20:35.590 align:middle line:84% And therefore, the operator norm of T minus this partial sum 01:20:35.590 --> 01:20:50.310 align:middle line:84% goes to 0 as m goes to infinity, and therefore converges to T. 01:20:50.310 --> 01:20:53.280 align:middle line:84% So you see, I mean, it had the same basic format 01:20:53.280 --> 01:20:55.155 align:middle line:90% that the last argument did. 01:20:55.155 --> 01:21:00.635 align:middle line:90% 01:21:00.635 --> 01:21:02.510 align:middle line:90% You found a candidate. 01:21:02.510 --> 01:21:04.250 align:middle line:90% You showed it's in the space. 01:21:04.250 --> 01:21:08.160 align:middle line:84% And then you showed convergence of the space. 01:21:08.160 --> 01:21:10.190 align:middle line:84% So let me finish with a definition. 01:21:10.190 --> 01:21:13.690 align:middle line:90% 01:21:13.690 --> 01:21:21.480 align:middle line:84% If V is a norm space, then we denote 01:21:21.480 --> 01:21:27.420 align:middle line:84% v-prime as the space of bounded linear operators from V 01:21:27.420 --> 01:21:30.910 align:middle line:90% to the space of scalars. 01:21:30.910 --> 01:21:41.440 align:middle line:84% This is referred to as the dual space of V. 01:21:41.440 --> 01:21:45.460 align:middle line:84% And since the space of scholars is always R or C, both of which 01:21:45.460 --> 01:21:46.592 align:middle line:90% are complete-- 01:21:46.592 --> 01:21:48.925 align:middle line:84% I mean, they're the simplest examples of Banach spaces-- 01:21:48.925 --> 01:21:56.300 align:middle line:90% 01:21:56.300 --> 01:21:59.840 align:middle line:84% by the theorem we just proved, since the field of scalars 01:21:59.840 --> 01:22:07.020 align:middle line:84% is always complete, the dual space is always a Banach space. 01:22:07.020 --> 01:22:10.760 align:middle line:90% 01:22:10.760 --> 01:22:12.720 align:middle line:84% And let me just write here a simple example, 01:22:12.720 --> 01:22:18.360 align:middle line:84% which will be in the exercises, that, in a sense, 01:22:18.360 --> 01:22:24.590 align:middle line:84% for all p between 1 and strictly less than infinity, 01:22:24.590 --> 01:22:31.630 align:middle line:84% I can identify the dual space of little lp as little lp-prime, 01:22:31.630 --> 01:22:32.830 align:middle line:90% where p-prime-- 01:22:32.830 --> 01:22:39.910 align:middle line:84% this is now just a number bigger than 1, 01:22:39.910 --> 01:22:47.300 align:middle line:84% where p-prime and p satisfy this relation. 01:22:47.300 --> 01:22:57.310 align:middle line:84% So in particular, the dual of l1 is l infinity. 01:22:57.310 --> 01:23:00.790 align:middle line:90% The dual of l2 is l2. 01:23:00.790 --> 01:23:02.080 align:middle line:90% This is very special about l2. 01:23:02.080 --> 01:23:05.270 align:middle line:90% 01:23:05.270 --> 01:23:11.710 align:middle line:84% But if I take the dual of l infinity, p equals infinity. 01:23:11.710 --> 01:23:13.390 align:middle line:90% I would get p-prime equals 1. 01:23:13.390 --> 01:23:15.910 align:middle line:84% This, in fact, does not equal little l1. 01:23:15.910 --> 01:23:19.780 align:middle line:84% This is something of a headache that manifests itself 01:23:19.780 --> 01:23:23.620 align:middle line:90% for the big Lp spaces as well. 01:23:23.620 --> 01:23:27.850 align:middle line:84% And life would be a lot easier if this 01:23:27.850 --> 01:23:33.880 align:middle line:84% were the case, that the dual of l infinity was l1. 01:23:33.880 --> 01:23:35.380 align:middle line:90% But unfortunately, it's not. 01:23:35.380 --> 01:23:38.700 align:middle line:90% And that causes a headache. 01:23:38.700 --> 01:23:41.900 align:middle line:84% And l2, little l2, this is the only lp 01:23:41.900 --> 01:23:48.520 align:middle line:84% that has this property, that its dual is given by little l2. 01:23:48.520 --> 01:23:51.730 align:middle line:84% All right, and in the exercises, I'll 01:23:51.730 --> 01:23:54.400 align:middle line:84% discuss precisely in what way you 01:23:54.400 --> 01:24:01.030 align:middle line:84% can identify the duel with this little lp-prime in this way. 01:24:01.030 --> 01:24:03.330 align:middle line:90% All right, we'll stop there. 01:24:03.330 --> 01:24:11.000 align:middle line:90%