WEBVTT 00:00:00.000 --> 00:00:00.984 align:middle line:90% 00:00:00.984 --> 00:00:01.968 align:middle line:90% [SQUEAKING] 00:00:01.968 --> 00:00:03.936 align:middle line:90% [RUSTLING] 00:00:03.936 --> 00:00:07.380 align:middle line:90% [CLICKING] 00:00:07.380 --> 00:00:18.730 align:middle line:90% 00:00:18.730 --> 00:00:22.360 align:middle line:84% PROFESSOR: OK, so we're going to complete 00:00:22.360 --> 00:00:26.230 align:middle line:84% our discussion of Lebesgue measure and integration 00:00:26.230 --> 00:00:29.175 align:middle line:84% by discussing the big Lp spaces, which 00:00:29.175 --> 00:00:31.300 align:middle line:84% was kind of the whole point of this endeavor, which 00:00:31.300 --> 00:00:42.400 align:middle line:84% was to find, in some sense, the complete space 00:00:42.400 --> 00:00:45.490 align:middle line:84% of integratable functions, whatever integratable means-- 00:00:45.490 --> 00:00:48.520 align:middle line:84% I mean, we had to define an integral-- 00:00:48.520 --> 00:00:53.290 align:middle line:84% containing the space of continuous functions 00:00:53.290 --> 00:00:58.720 align:middle line:84% with norm given by, let's say, the integral of the p-th power 00:00:58.720 --> 00:01:01.480 align:middle line:90% of the continuous function. 00:01:01.480 --> 00:01:04.591 align:middle line:90% 00:01:04.591 --> 00:01:08.300 align:middle line:84% OK, so at the end of-- so at last lecture, 00:01:08.300 --> 00:01:11.230 align:middle line:84% we introduced the general Lebesgue integratable-- 00:01:11.230 --> 00:01:14.320 align:middle line:84% I mean the class of Lebesgue-integratable functions 00:01:14.320 --> 00:01:16.000 align:middle line:90% and the Lebesgue integral. 00:01:16.000 --> 00:01:20.110 align:middle line:90% And we proved that-- 00:01:20.110 --> 00:01:22.210 align:middle line:84% we proved the dominated convergence theorem, 00:01:22.210 --> 00:01:24.880 align:middle line:84% and one consequence of that was the fact 00:01:24.880 --> 00:01:27.970 align:middle line:84% that if I have a continuous function 00:01:27.970 --> 00:01:29.590 align:middle line:84% on a closed and bounded interval a, 00:01:29.590 --> 00:01:33.160 align:middle line:84% b, then the Lebesgue integral of that continuous function 00:01:33.160 --> 00:01:36.430 align:middle line:84% equals the Riemann integral of that continuous function. 00:01:36.430 --> 00:01:39.940 align:middle line:84% So you know how to compute integrals-- 00:01:39.940 --> 00:01:43.090 align:middle line:84% or the Lebesgue integral for every function 00:01:43.090 --> 00:01:47.080 align:middle line:84% that how to compute the Riemann integral for, 00:01:47.080 --> 00:01:49.562 align:middle line:84% which are mostly continuous functions. 00:01:49.562 --> 00:01:50.770 align:middle line:90% Now this can be strengthened. 00:01:50.770 --> 00:01:52.570 align:middle line:90% One can show that, in fact-- 00:01:52.570 --> 00:01:54.890 align:middle line:84% so we're not going to cover it in this class. 00:01:54.890 --> 00:01:59.740 align:middle line:84% You'll see it in another class that is devoted maybe just 00:01:59.740 --> 00:02:02.260 align:middle line:84% solely to measure theory for a longer bit, of time 00:02:02.260 --> 00:02:08.220 align:middle line:84% but one can show using the dominated convergence theorem 00:02:08.220 --> 00:02:10.770 align:middle line:84% that, in fact, every Riemann-integratable function-- 00:02:10.770 --> 00:02:13.500 align:middle line:84% not just continuous, but every Riemann-integratable function 00:02:13.500 --> 00:02:20.660 align:middle line:84% on a closed and bounded interval is Lebesgue-integratable, 00:02:20.660 --> 00:02:23.690 align:middle line:84% and that the Riemann integral equals the Lebesgue 00:02:23.690 --> 00:02:26.450 align:middle line:84% integral, even just for more general 00:02:26.450 --> 00:02:30.170 align:middle line:90% Riemann-integratable functions. 00:02:30.170 --> 00:02:35.605 align:middle line:84% And you can also, now using the machinery that we built up-- 00:02:35.605 --> 00:02:37.730 align:middle line:84% I don't know, maybe I'll put this in the assignment 00:02:37.730 --> 00:02:39.860 align:middle line:90% maybe not. 00:02:39.860 --> 00:02:45.080 align:middle line:84% That you can completely characterize those functions 00:02:45.080 --> 00:02:46.850 align:middle line:90% which are Riemann-integratable. 00:02:46.850 --> 00:02:49.820 align:middle line:84% And the statement is that a function-- a measurable 00:02:49.820 --> 00:02:53.750 align:middle line:84% function, say, is Riemann-integratable if 00:02:53.750 --> 00:02:57.728 align:middle line:84% and only if it is continuous almost everywhere. 00:02:57.728 --> 00:03:00.020 align:middle line:84% I'm not saying that it's equal to a continuous function 00:03:00.020 --> 00:03:01.010 align:middle line:90% almost everywhere. 00:03:01.010 --> 00:03:02.993 align:middle line:84% I'm saying that it's continuous at almost 00:03:02.993 --> 00:03:04.160 align:middle line:90% every point in the interval. 00:03:04.160 --> 00:03:07.792 align:middle line:90% 00:03:07.792 --> 00:03:08.750 align:middle line:90% OK. 00:03:08.750 --> 00:03:13.040 align:middle line:84% Now let's move on to analogs of the little Lp spaces 00:03:13.040 --> 00:03:16.040 align:middle line:84% that we saw earlier in the lectures 00:03:16.040 --> 00:03:17.990 align:middle line:90% and on the assignments. 00:03:17.990 --> 00:03:20.735 align:middle line:84% So these are usually referred to as the big Lp spaces. 00:03:20.735 --> 00:03:28.970 align:middle line:90% 00:03:28.970 --> 00:03:34.430 align:middle line:84% And so to define these, let me first 00:03:34.430 --> 00:03:39.300 align:middle line:84% define what will be, in the end, a norm. 00:03:39.300 --> 00:03:51.010 align:middle line:84% So if f from a measurable subset of R to the complex numbers 00:03:51.010 --> 00:03:59.250 align:middle line:84% is measurable, and 1 is less than 00:03:59.250 --> 00:04:02.200 align:middle line:84% or equal to p is less than infinity, 00:04:02.200 --> 00:04:09.600 align:middle line:84% then we define the following possibly-- 00:04:09.600 --> 00:04:16.019 align:middle line:84% or the following extended non-negative real number, 00:04:16.019 --> 00:04:23.910 align:middle line:84% norm Lp of E. This is defined to be the integral over E 00:04:23.910 --> 00:04:29.220 align:middle line:84% of f raised to the absolute value raised to the p to the 1 00:04:29.220 --> 00:04:29.880 align:middle line:90% over p. 00:04:29.880 --> 00:04:32.390 align:middle line:90% 00:04:32.390 --> 00:04:36.560 align:middle line:84% Now this is meaningful because no matter what f-- 00:04:36.560 --> 00:04:39.200 align:middle line:84% how f has behaved, because the absolute value 00:04:39.200 --> 00:04:42.610 align:middle line:84% of f raised to the p, this is a non-negative measurable 00:04:42.610 --> 00:04:43.110 align:middle line:90% function. 00:04:43.110 --> 00:04:45.360 align:middle line:84% So we can always define what the Lebesgue integral is. 00:04:45.360 --> 00:04:49.130 align:middle line:84% So this may be either infinite or finite, 00:04:49.130 --> 00:04:56.040 align:middle line:84% but it's a non-negative number, extended real number. 00:04:56.040 --> 00:04:58.730 align:middle line:84% And so maybe you ask, what's going on? 00:04:58.730 --> 00:05:02.550 align:middle line:84% Why did I leave out p equals infinity? 00:05:02.550 --> 00:05:06.020 align:middle line:84% We have a different definition for equals infinity, 00:05:06.020 --> 00:05:07.730 align:middle line:84% just like we had a different definition 00:05:07.730 --> 00:05:10.760 align:middle line:90% for the little l infinity. 00:05:10.760 --> 00:05:14.630 align:middle line:84% We define this quantity here, which 00:05:14.630 --> 00:05:17.600 align:middle line:84% I'm going to go ahead and start referring to as the Lp and L 00:05:17.600 --> 00:05:20.180 align:middle line:84% infinity norms even though I haven't proved their-- 00:05:20.180 --> 00:05:25.910 align:middle line:90% a norm yet on what space either. 00:05:25.910 --> 00:05:32.720 align:middle line:84% This is defined to be the infimum over M positive such 00:05:32.720 --> 00:05:41.280 align:middle line:84% that the measure of the set x and E such that f of x 00:05:41.280 --> 00:05:44.790 align:middle line:90% is bigger than m equals 0. 00:05:44.790 --> 00:05:47.320 align:middle line:90% 00:05:47.320 --> 00:05:49.700 align:middle line:84% So what does it mean for m to be in this set? 00:05:49.700 --> 00:05:53.260 align:middle line:84% This means that f of x is less than or equal 00:05:53.260 --> 00:05:54.610 align:middle line:90% to m almost everywhere. 00:05:54.610 --> 00:05:57.580 align:middle line:84% And then I take the minimum of all such-- 00:05:57.580 --> 00:06:01.580 align:middle line:90% almost everywhere upper bounds. 00:06:01.580 --> 00:06:05.460 align:middle line:84% And this is usually referred to as-- 00:06:05.460 --> 00:06:14.685 align:middle line:84% what is called the essential supremum of f of x. 00:06:14.685 --> 00:06:26.050 align:middle line:90% 00:06:26.050 --> 00:06:29.620 align:middle line:84% So just a little mini theorem about this L infinity norm 00:06:29.620 --> 00:06:32.065 align:middle line:90% here, what you'll see-- 00:06:32.065 --> 00:06:33.940 align:middle line:84% well, I guess you'll be seeing these lectures 00:06:33.940 --> 00:06:37.370 align:middle line:90% after the first exam. 00:06:37.370 --> 00:06:40.443 align:middle line:84% So you saw this guy actually on the exam, 00:06:40.443 --> 00:06:41.860 align:middle line:84% and you proved one of these facts. 00:06:41.860 --> 00:06:44.950 align:middle line:84% The other I will put on a future assignment. 00:06:44.950 --> 00:06:49.850 align:middle line:90% 00:06:49.850 --> 00:07:04.130 align:middle line:84% If f from E to C is measurable, then the absolute value 00:07:04.130 --> 00:07:12.340 align:middle line:84% of f of x is less than or equal to gL infinity 00:07:12.340 --> 00:07:15.520 align:middle line:90% almost everywhere on E. 00:07:15.520 --> 00:07:24.700 align:middle line:90% And another fact, if f-- 00:07:24.700 --> 00:07:27.580 align:middle line:90% 00:07:27.580 --> 00:07:38.590 align:middle line:84% if E is equal to a, b and f it is continuous on a, b, 00:07:38.590 --> 00:07:50.070 align:middle line:84% then this essential supremum is equal to, in fact, 00:07:50.070 --> 00:07:55.580 align:middle line:84% just the usual what we call the L infinity norm which, 00:07:55.580 --> 00:08:06.525 align:middle line:84% remember, was the sup over x and a, b absolute value of f of x. 00:08:06.525 --> 00:08:08.420 align:middle line:90% So why do I state this? 00:08:08.420 --> 00:08:15.650 align:middle line:84% Because the L infinity norm bounds f from above 00:08:15.650 --> 00:08:17.270 align:middle line:84% for almost every x in the same way 00:08:17.270 --> 00:08:20.600 align:middle line:84% that the little l infinity norm for sequences bounded, 00:08:20.600 --> 00:08:25.490 align:middle line:84% the sequence-- every entry in the sequence-- for every entry 00:08:25.490 --> 00:08:26.900 align:middle line:90% in the sequence. 00:08:26.900 --> 00:08:29.000 align:middle line:84% But now for the essential supremum, 00:08:29.000 --> 00:08:31.160 align:middle line:84% we have just an almost everywhere statement. 00:08:31.160 --> 00:08:36.110 align:middle line:84% But this norm is the same as the L infinity norm 00:08:36.110 --> 00:08:39.690 align:middle line:84% or the infinity norm for continuous functions. 00:08:39.690 --> 00:08:43.055 align:middle line:84% So it shouldn't be something that's too crazy. 00:08:43.055 --> 00:08:47.030 align:middle line:90% 00:08:47.030 --> 00:08:47.530 align:middle line:90% OK. 00:08:47.530 --> 00:08:53.590 align:middle line:84% So now I'm just going to state a couple of theorems 00:08:53.590 --> 00:08:57.400 align:middle line:84% because you already gave the proof-- 00:08:57.400 --> 00:09:02.357 align:middle line:84% the proofs, I should say, when you did-- 00:09:02.357 --> 00:09:04.440 align:middle line:84% I think it was probably the first assignment, when 00:09:04.440 --> 00:09:06.870 align:middle line:84% you did the corresponding statements for little lp 00:09:06.870 --> 00:09:09.630 align:middle line:84% spaces, except now you replace the-- 00:09:09.630 --> 00:09:13.050 align:middle line:84% for those, you replace an integral with a sum. 00:09:13.050 --> 00:09:17.140 align:middle line:84% I mean, you should always think of an integral as a sum. 00:09:17.140 --> 00:09:23.970 align:middle line:84% So we have the following two theorems, two inequalities. 00:09:23.970 --> 00:09:26.250 align:middle line:90% We have Holder's inequality. 00:09:26.250 --> 00:09:29.160 align:middle line:90% If one is between p-- 00:09:29.160 --> 00:09:36.840 align:middle line:84% if p is between 1 and infinity and q is the dual exponent 00:09:36.840 --> 00:09:47.420 align:middle line:84% to p, meaning 1 over p plus 1 over q equals 1, 00:09:47.420 --> 00:09:57.140 align:middle line:84% and you have f as an Lp of E and g is an Lp of E, 00:09:57.140 --> 00:10:05.080 align:middle line:84% then f times g integrated over E-- 00:10:05.080 --> 00:10:07.845 align:middle line:90% 00:10:07.845 --> 00:10:10.273 align:middle line:84% well I haven't even said-- sorry. 00:10:10.273 --> 00:10:11.440 align:middle line:90% Getting way ahead of myself. 00:10:11.440 --> 00:10:14.010 align:middle line:90% 00:10:14.010 --> 00:10:27.240 align:middle line:84% So-- and if f and g are two measurable functions, then 00:10:27.240 --> 00:10:32.770 align:middle line:84% the integral over E of f times g absolute value, 00:10:32.770 --> 00:10:39.750 align:middle line:84% this is less than or equal to the Lp norm of f 00:10:39.750 --> 00:10:45.660 align:middle line:90% times the Lp norm of g. 00:10:45.660 --> 00:10:47.940 align:middle line:84% Now of course, this inequality is only interesting 00:10:47.940 --> 00:10:50.160 align:middle line:84% if the right hand-side is finite. 00:10:50.160 --> 00:10:54.105 align:middle line:84% If this is infinite, then this is all vacuously true. 00:10:54.105 --> 00:10:58.120 align:middle line:90% 00:10:58.120 --> 00:11:00.810 align:middle line:84% So this is the analog of the Holder's inequality 00:11:00.810 --> 00:11:05.255 align:middle line:84% which you proved for sequences where we had a sum here 00:11:05.255 --> 00:11:07.380 align:middle line:84% instead of the integral and where we had a sum here 00:11:07.380 --> 00:11:09.270 align:middle line:90% instead of the integral. 00:11:09.270 --> 00:11:11.850 align:middle line:84% And it's proven essentially the same way. 00:11:11.850 --> 00:11:16.749 align:middle line:84% You just replace a sigma with a little s. 00:11:16.749 --> 00:11:20.421 align:middle line:90% 00:11:20.421 --> 00:11:23.250 align:middle line:84% And so from Holder's inequality, you 00:11:23.250 --> 00:11:25.080 align:middle line:90% obtained Minkowski's inequality. 00:11:25.080 --> 00:11:35.240 align:middle line:90% 00:11:35.240 --> 00:11:42.210 align:middle line:84% If p is between 1 and infinity, and f, 00:11:42.210 --> 00:11:53.375 align:middle line:84% g are two measurable functions, then by-- 00:11:53.375 --> 00:11:57.760 align:middle line:90% 00:11:57.760 --> 00:12:00.040 align:middle line:84% take the Lp norm of f plus g, this 00:12:00.040 --> 00:12:08.320 align:middle line:84% is less than or equal to the Lp norm of f plus the Lp 00:12:08.320 --> 00:12:11.950 align:middle line:84% norm of E. And again, you proved this exactly 00:12:11.950 --> 00:12:16.510 align:middle line:84% the same way as you did for the little Lp spaces 00:12:16.510 --> 00:12:18.130 align:middle line:90% using Holder's inequality. 00:12:18.130 --> 00:12:24.730 align:middle line:90% 00:12:24.730 --> 00:12:27.370 align:middle line:84% Of course, that requires a slightly different argument 00:12:27.370 --> 00:12:30.400 align:middle line:84% for p equals infinity you know for this essential supremum, 00:12:30.400 --> 00:12:33.220 align:middle line:84% but in fact, that's what you did in the exam 00:12:33.220 --> 00:12:35.890 align:middle line:90% you took a few days ago. 00:12:35.890 --> 00:12:40.270 align:middle line:90% 00:12:40.270 --> 00:12:41.890 align:middle line:84% So I've been calling these things 00:12:41.890 --> 00:12:48.230 align:middle line:84% a norm even though I haven't proved their norm yet 00:12:48.230 --> 00:12:50.188 align:middle line:84% and on what space are they a norm, 00:12:50.188 --> 00:12:51.355 align:middle line:90% so now I'm going to do that. 00:12:51.355 --> 00:12:53.910 align:middle line:90% 00:12:53.910 --> 00:12:57.620 align:middle line:84% So when it's clear-- also let me make a small remark. 00:12:57.620 --> 00:13:10.740 align:middle line:90% 00:13:10.740 --> 00:13:20.160 align:middle line:84% I'll denote this thing just by shorthand with just a p. 00:13:20.160 --> 00:13:23.610 align:middle line:84% And it should be clear from the context what 00:13:23.610 --> 00:13:27.810 align:middle line:90% set I'm taking this norm over. 00:13:27.810 --> 00:13:30.240 align:middle line:84% Or what set am I taking this integral 00:13:30.240 --> 00:13:33.720 align:middle line:90% that defines this norm over. 00:13:33.720 --> 00:13:35.670 align:middle line:84% OK, so now let me define the actual space 00:13:35.670 --> 00:13:40.430 align:middle line:90% that this will be a norm on. 00:13:40.430 --> 00:13:44.840 align:middle line:84% And it involves a slight abuse of terminology and notation 00:13:44.840 --> 00:13:48.890 align:middle line:84% in the end, which is just tradition, 00:13:48.890 --> 00:13:50.720 align:middle line:90% not just in this subject, but-- 00:13:50.720 --> 00:13:57.720 align:middle line:84% I mean, abuse of notation is tradition in all of math. 00:13:57.720 --> 00:14:07.520 align:middle line:84% So for 1-- for p between 1 and infinity, 00:14:07.520 --> 00:14:16.310 align:middle line:84% we define the space Lp of E-- so E here is know-- 00:14:16.310 --> 00:14:18.560 align:middle line:84% if I don't say it, it's always a measurable subset 00:14:18.560 --> 00:14:24.490 align:middle line:84% of R. This is the set of all functions from E to C, 00:14:24.490 --> 00:14:34.400 align:middle line:84% which are measurable, that have finite Lp norm. 00:14:34.400 --> 00:14:40.430 align:middle line:90% 00:14:40.430 --> 00:14:51.270 align:middle line:84% Now let me make a second caveat to this space. 00:14:51.270 --> 00:14:55.767 align:middle line:84% So as I've written it down now, it's a space of functions. 00:14:55.767 --> 00:14:58.350 align:middle line:84% And I'm going to keep referring to it as a space of functions. 00:14:58.350 --> 00:15:02.500 align:middle line:84% I'm going to keep referring to elements of it as functions. 00:15:02.500 --> 00:15:07.010 align:middle line:84% But the actual space itself is not a space of functions, 00:15:07.010 --> 00:15:08.920 align:middle line:84% it's a space of equivalence classes 00:15:08.920 --> 00:15:12.610 align:middle line:84% in order for this quantity, which I keep calling a norm, 00:15:12.610 --> 00:15:15.260 align:middle line:84% to actually be a norm on this space. 00:15:15.260 --> 00:15:16.330 align:middle line:90% So let me add here-- 00:15:16.330 --> 00:15:27.076 align:middle line:90% 00:15:27.076 --> 00:15:32.695 align:middle line:84% if we consider two elements of this set, the-- 00:15:32.695 --> 00:15:37.940 align:middle line:90% 00:15:37.940 --> 00:15:43.045 align:middle line:84% I shouldn't say equal, but to be the same element-- 00:15:43.045 --> 00:15:47.270 align:middle line:90% 00:15:47.270 --> 00:15:54.360 align:middle line:84% and let me give two elements, say f 00:15:54.360 --> 00:16:05.160 align:middle line:84% and g in Lp, to be the same element if f equals g 00:16:05.160 --> 00:16:07.480 align:middle line:90% almost everywhere. 00:16:07.480 --> 00:16:10.920 align:middle line:84% So as I've written it down, Lp of E 00:16:10.920 --> 00:16:15.360 align:middle line:84% consists of all measurable functions with finite Lp norm, 00:16:15.360 --> 00:16:22.250 align:middle line:84% and now I'm saying that I will consider two elements 00:16:22.250 --> 00:16:23.840 align:middle line:90% in this space to be the same-- 00:16:23.840 --> 00:16:27.410 align:middle line:84% to be the same element if they equal each other almost 00:16:27.410 --> 00:16:28.460 align:middle line:90% everywhere. 00:16:28.460 --> 00:16:34.572 align:middle line:90% 00:16:34.572 --> 00:16:36.905 align:middle line:84% So strictly speaking, let me just make this as a remark. 00:16:36.905 --> 00:16:47.070 align:middle line:90% 00:16:47.070 --> 00:17:12.819 align:middle line:84% This means an element of Lp of E is an equivalence class 00:17:12.819 --> 00:17:15.160 align:middle line:90% of the form-- 00:17:15.160 --> 00:17:18.190 align:middle line:84% so little brackets f to indicate the equivalence class. 00:17:18.190 --> 00:17:20.800 align:middle line:84% I haven't told you what the equivalence relation is, 00:17:20.800 --> 00:17:24.440 align:middle line:84% so I'll explain that as I describe the equivalence class. 00:17:24.440 --> 00:17:27.220 align:middle line:84% So the equivalence class of f is equal to the set 00:17:27.220 --> 00:17:31.540 align:middle line:84% of all function g, which are measurable. 00:17:31.540 --> 00:17:43.860 align:middle line:90% 00:17:43.860 --> 00:17:48.990 align:middle line:84% So Lp of E is really a set of all equivalents-- 00:17:48.990 --> 00:17:52.620 align:middle line:84% is a set of equivalence classes of measurable functions 00:17:52.620 --> 00:17:57.600 align:middle line:84% with finite Lp norm where two equivalence classes are 00:17:57.600 --> 00:18:03.300 align:middle line:84% equal if and only if the representative 00:18:03.300 --> 00:18:05.040 align:middle line:84% from the first equivalent class equals 00:18:05.040 --> 00:18:07.500 align:middle line:84% the representative of the second equivalent class 00:18:07.500 --> 00:18:09.310 align:middle line:90% almost everywhere. 00:18:09.310 --> 00:18:15.800 align:middle line:90% So this is what Lp of E is. 00:18:15.800 --> 00:18:19.400 align:middle line:90% Now why am I making this-- 00:18:19.400 --> 00:18:22.910 align:middle line:84% why am I adding this caveat that you 00:18:22.910 --> 00:18:27.800 align:middle line:84% have to consider two elements to be equal almost everywhere? 00:18:27.800 --> 00:18:32.570 align:middle line:84% Because this is what allows me to put-- 00:18:32.570 --> 00:18:35.540 align:middle line:84% or say this Lp norm is an actual norm. 00:18:35.540 --> 00:18:39.770 align:middle line:84% Otherwise it is just a semi-norm if I really consider Lp of E 00:18:39.770 --> 00:18:42.990 align:middle line:84% to be this space of actual functions. 00:18:42.990 --> 00:18:49.760 align:middle line:84% So again, this is a small point, that Lp of E 00:18:49.760 --> 00:18:54.200 align:middle line:84% is, in fact, a set of equivalence classes 00:18:54.200 --> 00:18:58.190 align:middle line:84% where two functions are two equivalence classes or equal if 00:18:58.190 --> 00:19:00.890 align:middle line:84% and only if the representative functions equal-- 00:19:00.890 --> 00:19:04.970 align:middle line:90% are equal almost everywhere. 00:19:04.970 --> 00:19:07.700 align:middle line:84% But now that I've explained all of that 00:19:07.700 --> 00:19:10.610 align:middle line:84% and carefully told you what Lp of E is, 00:19:10.610 --> 00:19:14.420 align:middle line:84% it is customary to never refer to them as equivalence classes 00:19:14.420 --> 00:19:17.540 align:middle line:84% or elements of Lp of E as an equivalence class. 00:19:17.540 --> 00:19:20.570 align:middle line:84% We usually still just refer to them as functions. 00:19:20.570 --> 00:19:22.085 align:middle line:90% Let me make that point clear. 00:19:22.085 --> 00:19:48.840 align:middle line:90% 00:19:48.840 --> 00:19:52.470 align:middle line:84% So rather than speaking of an equivalence class or elements 00:19:52.470 --> 00:19:54.450 align:middle line:84% of Lp of E as an equivalence class, 00:19:54.450 --> 00:19:56.430 align:middle line:84% I will just refer to them as functions 00:19:56.430 --> 00:20:03.660 align:middle line:84% with the understanding that two functions in Lp of E 00:20:03.660 --> 00:20:07.320 align:middle line:84% are considered to be the same function, the same element 00:20:07.320 --> 00:20:11.310 align:middle line:84% if they equal each other almost everywhere. 00:20:11.310 --> 00:20:18.450 align:middle line:84% Now maybe you think this is a little weird, 00:20:18.450 --> 00:20:20.200 align:middle line:84% but you've been doing this your whole life 00:20:20.200 --> 00:20:25.780 align:middle line:84% already, because the rational numbers themselves are defined, 00:20:25.780 --> 00:20:29.587 align:middle line:84% if you look back into algebra or if you're taking algebra now, 00:20:29.587 --> 00:20:32.170 align:middle line:84% when you actually sit down and construct the rational numbers, 00:20:32.170 --> 00:20:35.170 align:middle line:84% they're constructed as equivalence classes 00:20:35.170 --> 00:20:37.450 align:middle line:90% of pairs of integers. 00:20:37.450 --> 00:20:40.270 align:middle line:84% But you don't think of them as that, 00:20:40.270 --> 00:20:44.260 align:middle line:90% you think of them as 3 over 2. 00:20:44.260 --> 00:20:48.520 align:middle line:84% Not the equivalence class of 3, 2. 00:20:48.520 --> 00:20:53.600 align:middle line:84% So again, two elements in Lp of E. I mean, 00:20:53.600 --> 00:20:58.820 align:middle line:84% we think of Lp of E as a set of functions with finite Lp norm, 00:20:58.820 --> 00:21:01.370 align:middle line:84% and with the caveat that two functions-- 00:21:01.370 --> 00:21:04.610 align:middle line:90% or two elements of Lp and E-- 00:21:04.610 --> 00:21:06.800 align:middle line:90% Lp of E are equal-- 00:21:06.800 --> 00:21:09.650 align:middle line:84% are the same element if the two functions 00:21:09.650 --> 00:21:13.350 align:middle line:90% agree almost everywhere. 00:21:13.350 --> 00:21:18.640 align:middle line:84% OK, now with that minor detail out of the way, 00:21:18.640 --> 00:21:19.950 align:middle line:90% let's move on to-- 00:21:19.950 --> 00:21:24.710 align:middle line:90% let me state the theorem that-- 00:21:24.710 --> 00:21:33.480 align:middle line:90% 00:21:33.480 --> 00:21:41.120 align:middle line:84% so Lp with the natural scalar multiplication 00:21:41.120 --> 00:21:44.870 align:middle line:84% where you just multiply-- the scalar multiple of an element 00:21:44.870 --> 00:21:48.520 align:middle line:84% as just multiplying the function by a scalar. 00:21:48.520 --> 00:21:50.800 align:middle line:84% And the sum of two elements is just the pointwise 00:21:50.800 --> 00:21:54.800 align:middle line:90% why some of the two functions. 00:21:54.800 --> 00:21:59.080 align:middle line:84% So with the obvious definitions of scalar multiplication 00:21:59.080 --> 00:21:59.815 align:middle line:90% and addition. 00:21:59.815 --> 00:22:03.810 align:middle line:90% 00:22:03.810 --> 00:22:08.955 align:middle line:90% Operations is a vector space. 00:22:08.955 --> 00:22:14.380 align:middle line:90% 00:22:14.380 --> 00:22:17.830 align:middle line:84% Moreover, this function which I've 00:22:17.830 --> 00:22:27.350 align:middle line:84% been referring to as a norm is actually a norm on Lp. 00:22:27.350 --> 00:22:30.950 align:middle line:90% 00:22:30.950 --> 00:22:32.210 align:middle line:90% Now I'm not going to-- 00:22:32.210 --> 00:22:38.640 align:middle line:84% I'm only going to prove part of this 00:22:38.640 --> 00:22:41.850 align:middle line:84% because to verify something as a vector space, 00:22:41.850 --> 00:22:44.460 align:middle line:84% you have to verify the operations are-- 00:22:44.460 --> 00:22:49.800 align:middle line:90% satisfy certain properties. 00:22:49.800 --> 00:22:53.070 align:middle line:84% And when I state the-- or give the proof of what 00:22:53.070 --> 00:22:55.290 align:middle line:84% I'm going to prove out of this theorem, 00:22:55.290 --> 00:22:56.880 align:middle line:84% this will be the last time I actually 00:22:56.880 --> 00:23:00.030 align:middle line:84% refer to the fact that these are equivalence glasses and not 00:23:00.030 --> 00:23:04.200 align:middle line:84% functions, but just to make a point. 00:23:04.200 --> 00:23:10.390 align:middle line:90% 00:23:10.390 --> 00:23:12.630 align:middle line:84% Again, this space, strictly speaking, 00:23:12.630 --> 00:23:15.310 align:middle line:90% is a set of equivalence classes. 00:23:15.310 --> 00:23:18.450 align:middle line:84% So one would have to check that addition and scalar 00:23:18.450 --> 00:23:21.960 align:middle line:84% multiplication are well-defined on this set, 00:23:21.960 --> 00:23:25.320 align:middle line:84% meaning if I have two representatives 00:23:25.320 --> 00:23:28.020 align:middle line:84% of the same equivalence class and I multiply one 00:23:28.020 --> 00:23:31.020 align:middle line:84% by a scalar and the other by a scalar, 00:23:31.020 --> 00:23:35.527 align:middle line:84% then I get the same equivalence class in the end, which 00:23:35.527 --> 00:23:37.360 align:middle line:84% is easy to check, and the same for addition. 00:23:37.360 --> 00:23:50.530 align:middle line:84% So again, in a strictly speaking sense, 00:23:50.530 --> 00:23:53.440 align:middle line:84% that Lp is a set of equivalence classes, 00:23:53.440 --> 00:23:57.070 align:middle line:84% these things are well-defined, scalar multiplication 00:23:57.070 --> 00:23:59.800 align:middle line:90% and addition. 00:23:59.800 --> 00:24:05.350 align:middle line:90% So let's check that Lp-- 00:24:05.350 --> 00:24:07.750 align:middle line:84% or that this quantity here is a norm. 00:24:07.750 --> 00:24:11.080 align:middle line:90% And I'm just going to-- 00:24:11.080 --> 00:24:16.540 align:middle line:84% So first off, note that taking the Lp norm of an element of Lp 00:24:16.540 --> 00:24:17.890 align:middle line:90% is actually well-defined. 00:24:17.890 --> 00:24:21.400 align:middle line:84% Remember, so this is going to be the only theorem 00:24:21.400 --> 00:24:24.160 align:middle line:84% and proof where I actually refer to the fact 00:24:24.160 --> 00:24:27.430 align:middle line:84% that Lp is actually a set of equivalence classes, 00:24:27.430 --> 00:24:30.760 align:middle line:84% and then after that we just won't do that anymore 00:24:30.760 --> 00:24:33.190 align:middle line:84% and we'll just think of Lp as a set of functions 00:24:33.190 --> 00:24:35.770 align:middle line:84% where two functions are the same element if they equal 00:24:35.770 --> 00:24:37.060 align:middle line:90% each other almost everywhere. 00:24:37.060 --> 00:24:41.890 align:middle line:84% But first thing to note is that this is actually 00:24:41.890 --> 00:24:46.670 align:middle line:84% well-defined on this set of equivalence classes. 00:24:46.670 --> 00:24:55.780 align:middle line:84% So note, if I have two representatives 00:24:55.780 --> 00:25:06.310 align:middle line:84% of an equivalence class, then by what we developed 00:25:06.310 --> 00:25:08.240 align:middle line:90% for the Lebesgue integral-- 00:25:08.240 --> 00:25:10.900 align:middle line:84% So f, take the absolute value raised to the p 00:25:10.900 --> 00:25:13.743 align:middle line:84% is also going to be equal to g absolute value rates of the p 00:25:13.743 --> 00:25:15.910 align:middle line:84% almost everywhere, and therefore these two integrals 00:25:15.910 --> 00:25:18.550 align:middle line:90% are going to equal each other. 00:25:18.550 --> 00:25:20.120 align:middle line:84% These integrals are real numbers. 00:25:20.120 --> 00:25:22.120 align:middle line:84% They're not equivalence classes of real numbers, 00:25:22.120 --> 00:25:26.230 align:middle line:84% this is a real number, a non-negative real number. 00:25:26.230 --> 00:25:38.220 align:middle line:84% And therefore, if in the notation I had before, 00:25:38.220 --> 00:25:41.370 align:middle line:84% if I have an equivalence class and I 00:25:41.370 --> 00:25:43.140 align:middle line:84% have two different representatives of this 00:25:43.140 --> 00:25:48.840 align:middle line:84% and I define the Lp norm of this equivalence class 00:25:48.840 --> 00:25:51.390 align:middle line:84% to be the integral of the representative, 00:25:51.390 --> 00:25:55.590 align:middle line:84% I get the same number regardless of the representative 00:25:55.590 --> 00:25:56.715 align:middle line:90% for that equivalence class. 00:25:56.715 --> 00:26:25.420 align:middle line:90% 00:26:25.420 --> 00:26:35.785 align:middle line:90% I.e.-- OK? 00:26:35.785 --> 00:26:40.450 align:middle line:90% So this function is defined by-- 00:26:40.450 --> 00:26:43.030 align:middle line:84% if I take that Lp norm of an equivalence class 00:26:43.030 --> 00:26:45.370 align:middle line:84% given by just the integral of the representative, 00:26:45.370 --> 00:26:46.643 align:middle line:90% this is well-defined. 00:26:46.643 --> 00:26:48.310 align:middle line:84% If I take two different representatives, 00:26:48.310 --> 00:26:49.570 align:middle line:90% I get the same number out. 00:26:49.570 --> 00:26:54.620 align:middle line:90% 00:26:54.620 --> 00:27:05.240 align:middle line:84% Now this quantity here equals 0 if and only if-- 00:27:05.240 --> 00:27:09.320 align:middle line:84% by what we've developed in our theory of integration, 00:27:09.320 --> 00:27:13.610 align:middle line:90% f equals 0 almost everywhere. 00:27:13.610 --> 00:27:16.820 align:middle line:90% So the Lp norm or the-- 00:27:16.820 --> 00:27:22.730 align:middle line:84% so I should say this equals 0 almost everywhere-- i.e., 00:27:22.730 --> 00:27:24.620 align:middle line:90% f equals 0 almost everywhere. 00:27:24.620 --> 00:27:27.130 align:middle line:90% 00:27:27.130 --> 00:27:28.990 align:middle line:90% But possibly not everywhere. 00:27:28.990 --> 00:27:33.990 align:middle line:90% 00:27:33.990 --> 00:27:35.610 align:middle line:84% The equivalence class of this function 00:27:35.610 --> 00:27:38.350 align:middle line:90% is equal to the 0 element. 00:27:38.350 --> 00:27:41.190 align:middle line:84% So this was the whole reason why I 00:27:41.190 --> 00:27:46.315 align:middle line:84% went through this trouble of actually explaining to you why, 00:27:46.315 --> 00:27:50.850 align:middle line:84% strictly speaking, the most rigorous way of defining Lp 00:27:50.850 --> 00:27:52.680 align:middle line:84% is as a space of equivalence classes, 00:27:52.680 --> 00:27:55.140 align:middle line:84% even though we won't think about them that way 00:27:55.140 --> 00:27:57.690 align:middle line:84% or talk about them that way in the future, that's, 00:27:57.690 --> 00:27:59.700 align:middle line:90% in fact, what they are. 00:27:59.700 --> 00:28:03.510 align:middle line:84% Because then, this quantity here, this Lp norm 00:28:03.510 --> 00:28:04.740 align:middle line:90% is, in fact, a norm. 00:28:04.740 --> 00:28:11.775 align:middle line:84% If the Lp norm is 0, then that element is the 0 element. 00:28:11.775 --> 00:28:18.550 align:middle line:90% 00:28:18.550 --> 00:28:19.090 align:middle line:90% OK. 00:28:19.090 --> 00:28:23.380 align:middle line:84% So that's about the last time I'm ever 00:28:23.380 --> 00:28:28.480 align:middle line:84% going to refer to the elements of Lp as equivalence classes. 00:28:28.480 --> 00:28:32.470 align:middle line:84% I will now be referring to them as functions. 00:28:32.470 --> 00:28:44.990 align:middle line:84% But homogeneity and the triangle inequality for the Lp norm, 00:28:44.990 --> 00:28:51.480 align:middle line:84% then follow, simply from the definition, 00:28:51.480 --> 00:28:55.560 align:middle line:84% for homogeneity, the fact that a scalar-- 00:28:55.560 --> 00:28:57.720 align:middle line:84% that the Lp norm of a scalar multiple of f 00:28:57.720 --> 00:29:01.770 align:middle line:84% is equal to the absolute value of the scalar times 00:29:01.770 --> 00:29:03.990 align:middle line:84% the Lp norm of f, and the triangle inequality 00:29:03.990 --> 00:29:15.830 align:middle line:84% follow from the definition and Minkowski's inequality. 00:29:15.830 --> 00:29:26.460 align:middle line:90% 00:29:26.460 --> 00:29:27.270 align:middle line:90% OK. 00:29:27.270 --> 00:29:35.020 align:middle line:84% So Lp is the space of measurable functions with finite Lp norms. 00:29:35.020 --> 00:29:35.520 align:middle line:90% See? 00:29:35.520 --> 00:29:38.020 align:middle line:84% I'm Already not going to refer to it ever again as the space 00:29:38.020 --> 00:29:39.210 align:middle line:90% of equivalence classes. 00:29:39.210 --> 00:29:41.460 align:middle line:84% You should think of them as space functions, 00:29:41.460 --> 00:29:45.600 align:middle line:84% just with two elements equal if these functions equal 00:29:45.600 --> 00:29:50.010 align:middle line:84% each other almost everywhere pointwise on E. 00:29:50.010 --> 00:29:54.140 align:middle line:90% Now we come to the big question. 00:29:54.140 --> 00:29:58.640 align:middle line:84% We now have this norm space Lp of E corresponding 00:29:58.640 --> 00:30:03.095 align:middle line:84% to all these functions that have finite Lp norm. 00:30:03.095 --> 00:30:06.080 align:middle line:90% 00:30:06.080 --> 00:30:09.260 align:middle line:90% First off, is this non-empty? 00:30:09.260 --> 00:30:14.470 align:middle line:84% Let's maybe give you the simplest example. 00:30:14.470 --> 00:30:17.730 align:middle line:84% In fact, let me prove the following simple theorem. 00:30:17.730 --> 00:30:21.830 align:middle line:90% 00:30:21.830 --> 00:30:24.290 align:middle line:90% So-- which is the following. 00:30:24.290 --> 00:30:27.530 align:middle line:90% 00:30:27.530 --> 00:30:39.220 align:middle line:84% Let E be measurable, then f is an Lp of E 00:30:39.220 --> 00:30:50.760 align:middle line:84% if and only if the limit as R goes to infinity over-- 00:30:50.760 --> 00:30:54.990 align:middle line:84% of the integral of minus R to R intersect E 00:30:54.990 --> 00:31:00.080 align:middle line:90% of f raised to the p is finite. 00:31:00.080 --> 00:31:02.960 align:middle line:84% Notice, as R going to plus infinity-- 00:31:02.960 --> 00:31:05.580 align:middle line:90% 00:31:05.580 --> 00:31:09.140 align:middle line:84% so maybe let's not make it R. We can 00:31:09.140 --> 00:31:10.895 align:middle line:84% make it n we're in as a natural number. 00:31:10.895 --> 00:31:17.720 align:middle line:90% 00:31:17.720 --> 00:31:20.550 align:middle line:84% Then this is an increasing sequence of numbers. 00:31:20.550 --> 00:31:25.190 align:middle line:84% So let's give the proof real quick. 00:31:25.190 --> 00:31:40.580 align:middle line:84% Let's assume f is in Lp implies that this quantity here 00:31:40.580 --> 00:31:41.090 align:middle line:90% is finite. 00:31:41.090 --> 00:31:44.130 align:middle line:90% 00:31:44.130 --> 00:31:53.208 align:middle line:84% Since the sequence of integrals over minus n to n intersect E 00:31:53.208 --> 00:31:59.370 align:middle line:84% f raised to the p, this is an increasing sequence. 00:31:59.370 --> 00:32:03.030 align:middle line:84% Because at each step, at each entry, 00:32:03.030 --> 00:32:05.310 align:middle line:84% I'm taking the integral of this non-negative quantity 00:32:05.310 --> 00:32:09.030 align:middle line:90% over a bigger set. 00:32:09.030 --> 00:32:10.770 align:middle line:84% So this is an increasing sequence. 00:32:10.770 --> 00:32:23.560 align:middle line:90% 00:32:23.560 --> 00:32:26.710 align:middle line:84% So, in fact, this limit always exists, 00:32:26.710 --> 00:32:29.070 align:middle line:84% so it was meaningful to actually refer to this limit. 00:32:29.070 --> 00:32:36.180 align:middle line:90% 00:32:36.180 --> 00:32:43.020 align:middle line:84% Since this is an increasing sequence, limit as n 00:32:43.020 --> 00:32:52.330 align:middle line:84% goes to infinity minus n intersect E actually exists 00:32:52.330 --> 00:32:56.515 align:middle line:84% as a possibly extended real number. 00:32:56.515 --> 00:33:03.270 align:middle line:90% 00:33:03.270 --> 00:33:06.500 align:middle line:84% Now since for all n, we have that the integral 00:33:06.500 --> 00:33:13.850 align:middle line:84% of minus n to n intersect E raised to the p 00:33:13.850 --> 00:33:15.770 align:middle line:90% is less than or equal to-- 00:33:15.770 --> 00:33:24.680 align:middle line:84% well, in fact, I'm being a little inefficient here. 00:33:24.680 --> 00:33:30.755 align:middle line:90% Let's just erase one bit here. 00:33:30.755 --> 00:33:40.650 align:middle line:90% 00:33:40.650 --> 00:33:41.670 align:middle line:90% OK. 00:33:41.670 --> 00:33:43.700 align:middle line:90% Yeah, let's do it this way. 00:33:43.700 --> 00:33:44.700 align:middle line:90% We'll do it much faster. 00:33:44.700 --> 00:33:48.600 align:middle line:90% 00:33:48.600 --> 00:34:00.910 align:middle line:84% Now note that intersect E f raised to the p, 00:34:00.910 --> 00:34:05.460 align:middle line:84% this is equal to integral over E chi minus n 00:34:05.460 --> 00:34:08.290 align:middle line:90% to n f raised to the p. 00:34:08.290 --> 00:34:11.889 align:middle line:90% So I can think of this as-- 00:34:11.889 --> 00:34:14.380 align:middle line:84% I can think of what's here in the integrand 00:34:14.380 --> 00:34:16.480 align:middle line:90% as a function for each n. 00:34:16.480 --> 00:34:41.060 align:middle line:84% So since this is a sequence that is pointwise increasing, 00:34:41.060 --> 00:34:47.449 align:middle line:84% and for all x in E, we have limit 00:34:47.449 --> 00:34:55.196 align:middle line:84% as n goes to infinity of chi minus n to n, 00:34:55.196 --> 00:35:03.810 align:middle line:84% f of x raised to the p equals f of x raised of the p. 00:35:03.810 --> 00:35:12.630 align:middle line:84% This-- by the monotone convergence theorem, 00:35:12.630 --> 00:35:15.280 align:middle line:84% the integral of the limit as n goes to infinity, 00:35:15.280 --> 00:35:22.050 align:middle line:84% which is just f raised to the p, is equal to the limit 00:35:22.050 --> 00:35:30.030 align:middle line:84% as n goes to infinity of the integral of this quantity 00:35:30.030 --> 00:35:32.910 align:middle line:84% here over E, which is, again, the interval over minus n 00:35:32.910 --> 00:35:36.890 align:middle line:84% to n intersect E raised to the p. 00:35:36.890 --> 00:35:39.230 align:middle line:84% And therefore, this quantity here 00:35:39.230 --> 00:35:43.790 align:middle line:84% is finite if and only if this quantity here is finite. 00:35:43.790 --> 00:35:46.310 align:middle line:84% And therefore, f is an Lp if and only 00:35:46.310 --> 00:35:51.100 align:middle line:84% if this quantity here is finite, and in fact, 00:35:51.100 --> 00:35:52.150 align:middle line:90% they equal each other. 00:35:52.150 --> 00:36:01.180 align:middle line:90% 00:36:01.180 --> 00:36:10.230 align:middle line:84% So using that theorem, if you like, you can prove-- 00:36:10.230 --> 00:36:20.310 align:middle line:84% and I'll leave it to you-- that if f from, let's say, R to C 00:36:20.310 --> 00:36:35.000 align:middle line:84% is measurable and there exists a constant non negative and q 00:36:35.000 --> 00:36:45.700 align:middle line:84% bigger than 1 such that for almost every x in R, 00:36:45.700 --> 00:36:49.610 align:middle line:84% f of x is less than or equal to a constant times 1 00:36:49.610 --> 00:37:07.490 align:middle line:84% plus x raised to the minus q, then f is an Lp of R 00:37:07.490 --> 00:37:09.350 align:middle line:84% for all p bigger than or equal to 1. 00:37:09.350 --> 00:37:13.610 align:middle line:90% 00:37:13.610 --> 00:37:16.530 align:middle line:90% So how do you do that? 00:37:16.530 --> 00:37:23.840 align:middle line:84% So OK, maybe I'll just indicate why we use the previous theorem 00:37:23.840 --> 00:37:26.980 align:middle line:84% and look at the integral from minus n 00:37:26.980 --> 00:37:35.520 align:middle line:84% to n intersect R. So this here, by this estimate, 00:37:35.520 --> 00:37:37.410 align:middle line:84% is less than or equal to the integral 00:37:37.410 --> 00:37:49.325 align:middle line:84% from minus n to n times constant 1 plus x to the minus q. 00:37:49.325 --> 00:37:53.020 align:middle line:84% Now this is a continuous function over a compact-- 00:37:53.020 --> 00:37:56.380 align:middle line:84% or closed and bounded interval, so this is 00:37:56.380 --> 00:37:57.655 align:middle line:90% equal to its Riemann integral. 00:37:57.655 --> 00:38:05.490 align:middle line:90% 00:38:05.490 --> 00:38:10.350 align:middle line:84% I will often, on exams and so on, right the Lebesgue 00:38:10.350 --> 00:38:12.540 align:middle line:84% integral also in this form, though. 00:38:12.540 --> 00:38:15.362 align:middle line:84% So I don't want to say that I'm just-- 00:38:15.362 --> 00:38:17.070 align:middle line:84% you're going to use this kind of notation 00:38:17.070 --> 00:38:20.270 align:middle line:90% for the Riemann integral. 00:38:20.270 --> 00:38:22.580 align:middle line:90% But anyways. 00:38:22.580 --> 00:38:25.760 align:middle line:84% And I leave it to you to show that as long 00:38:25.760 --> 00:38:30.470 align:middle line:84% as q is bigger than 1, this is less than or equal to-- so I 00:38:30.470 --> 00:38:34.020 align:middle line:90% should put p times q. 00:38:34.020 --> 00:38:35.640 align:middle line:90% p times q. 00:38:35.640 --> 00:38:46.900 align:middle line:84% That as long as q is bigger than 1, 00:38:46.900 --> 00:38:53.620 align:middle line:84% this integral here is less than or equal to some constant 00:38:53.620 --> 00:38:54.640 align:middle line:90% depending on p. 00:38:54.640 --> 00:39:06.990 align:middle line:90% 00:39:06.990 --> 00:39:08.700 align:middle line:90% OK. 00:39:08.700 --> 00:39:18.420 align:middle line:84% So there's many functions that are in Lp, so it's not exactly 00:39:18.420 --> 00:39:19.680 align:middle line:90% a trivial space. 00:39:19.680 --> 00:39:22.770 align:middle line:90% 00:39:22.770 --> 00:39:24.690 align:middle line:90% But what kind of space is it? 00:39:24.690 --> 00:39:28.730 align:middle line:90% Now let me state the following. 00:39:28.730 --> 00:39:34.370 align:middle line:84% That, in fact, this is what you proved in the assignment right 00:39:34.370 --> 00:39:38.970 align:middle line:84% before -- or the assignment I at least assigned before the exam, 00:39:38.970 --> 00:39:40.410 align:middle line:90% which was following. 00:39:40.410 --> 00:39:44.645 align:middle line:90% Let a less than b-- less than b. 00:39:44.645 --> 00:39:48.420 align:middle line:90% 00:39:48.420 --> 00:39:51.610 align:middle line:90% And p between 1 and infinity. 00:39:51.610 --> 00:39:55.170 align:middle line:84% So in fact, you did it for L1, but the same proof 00:39:55.170 --> 00:39:56.220 align:middle line:90% carries over for Lp. 00:39:56.220 --> 00:39:59.040 align:middle line:90% 00:39:59.040 --> 00:40:02.775 align:middle line:90% fE in Lp of a, b. 00:40:02.775 --> 00:40:07.310 align:middle line:90% 00:40:07.310 --> 00:40:11.600 align:middle line:84% So I keep adding stuff and epsilon would be positive. 00:40:11.600 --> 00:40:15.610 align:middle line:90% 00:40:15.610 --> 00:40:20.670 align:middle line:84% Then there exists a continuous function 00:40:20.670 --> 00:40:31.410 align:middle line:84% in a, b, which I can also impose vanishes at the end points 00:40:31.410 --> 00:40:36.300 align:middle line:90% and is close to f and Lp norm. 00:40:36.300 --> 00:40:42.310 align:middle line:90% 00:40:42.310 --> 00:40:50.710 align:middle line:84% So what this states is that the space of continuous functions 00:40:50.710 --> 00:40:59.065 align:middle line:90% is dense an Lp of a, b. 00:40:59.065 --> 00:41:02.460 align:middle line:90% 00:41:02.460 --> 00:41:06.870 align:middle line:84% And it's a proper subset of Lp of a, b. 00:41:06.870 --> 00:41:14.560 align:middle line:84% I can find elements in Lp that are not continuous or even not 00:41:14.560 --> 00:41:17.480 align:middle line:84% equal to a continuous function almost everywhere. 00:41:17.480 --> 00:41:20.845 align:middle line:90% So it is dense and also proper. 00:41:20.845 --> 00:41:25.010 align:middle line:90% 00:41:25.010 --> 00:41:29.030 align:middle line:84% And now the final theorem will prove about integration 00:41:29.030 --> 00:41:33.540 align:middle line:84% in Lp spaces is that Lp is complete. 00:41:33.540 --> 00:41:38.920 align:middle line:90% So this is due to Reece and-- 00:41:38.920 --> 00:41:39.420 align:middle line:90% let's see. 00:41:39.420 --> 00:41:42.000 align:middle line:84% Does this have a C or is it just-- 00:41:42.000 --> 00:41:48.550 align:middle line:84% yeah, it does have a C. This is a Banach space. 00:41:48.550 --> 00:41:52.130 align:middle line:90% 00:41:52.130 --> 00:41:54.422 align:middle line:90% For p between 1 and infinity. 00:41:54.422 --> 00:41:57.380 align:middle line:90% 00:41:57.380 --> 00:41:58.970 align:middle line:90% Including 1 and infinity. 00:41:58.970 --> 00:42:05.050 align:middle line:84% Now I'll give the proof for p between strictly 00:42:05.050 --> 00:42:06.020 align:middle line:90% less than infinity. 00:42:06.020 --> 00:42:10.990 align:middle line:90% So p equals infinity. 00:42:10.990 --> 00:42:19.030 align:middle line:84% This will appear in an assignment. 00:42:19.030 --> 00:42:26.280 align:middle line:84% So we'll do the case of p between 1 and infinity. 00:42:26.280 --> 00:42:32.920 align:middle line:90% 00:42:32.920 --> 00:42:34.520 align:middle line:90% So how are we going to do this? 00:42:34.520 --> 00:42:38.470 align:middle line:84% We are going to, in fact, use that criterion 00:42:38.470 --> 00:42:43.390 align:middle line:84% from several weeks ago about when is a norm space 00:42:43.390 --> 00:42:44.350 align:middle line:90% a Banach space? 00:42:44.350 --> 00:42:47.440 align:middle line:84% So we proved this equivalent criterion-- so remember, 00:42:47.440 --> 00:42:50.390 align:middle line:84% a Banach space is a norm space that's complete with respect 00:42:50.390 --> 00:42:50.890 align:middle line:90% to the norm. 00:42:50.890 --> 00:42:53.410 align:middle line:84% So you would have to check that all Cauchy sequences 00:42:53.410 --> 00:42:56.410 align:middle line:84% in the space converge to something in the space. 00:42:56.410 --> 00:43:01.450 align:middle line:90% 00:43:01.450 --> 00:43:08.990 align:middle line:84% But now we came up with this other criterion-- 00:43:08.990 --> 00:43:10.040 align:middle line:90% I shouldn't say we did. 00:43:10.040 --> 00:43:12.200 align:middle line:84% Somebody did, and then I showed it to you, 00:43:12.200 --> 00:43:15.620 align:middle line:84% that an equivalent way to prove that is 00:43:15.620 --> 00:43:20.150 align:middle line:84% to prove that all absolutely summable series in the norm 00:43:20.150 --> 00:43:23.180 align:middle line:90% space are summable. 00:43:23.180 --> 00:43:25.850 align:middle line:84% Remember, absolutely summable means that some of the norms 00:43:25.850 --> 00:43:26.660 align:middle line:90% is finite. 00:43:26.660 --> 00:43:28.340 align:middle line:84% So that's what we're going to use-- 00:43:28.340 --> 00:43:29.715 align:middle line:84% or that's what we're going to do. 00:43:29.715 --> 00:43:34.480 align:middle line:90% 00:43:34.480 --> 00:43:54.750 align:middle line:84% We will show that every absolutely summable series 00:43:54.750 --> 00:43:55.250 align:middle line:90% is summable. 00:43:55.250 --> 00:43:59.120 align:middle line:90% 00:43:59.120 --> 00:44:11.910 align:middle line:84% So, suppose I have a sequence in Lk, it's a sequence in-- 00:44:11.910 --> 00:44:14.550 align:middle line:90% f is an Lp, not Lk. 00:44:14.550 --> 00:44:19.700 align:middle line:90% 00:44:19.700 --> 00:44:23.310 align:middle line:84% That form an absolutely seminal series. 00:44:23.310 --> 00:44:29.360 align:middle line:84% So such that sum over k of-- if I take the LP norm of fk. 00:44:29.360 --> 00:44:33.500 align:middle line:84% So this is now a series of non-negative numbers, 00:44:33.500 --> 00:44:35.420 align:middle line:84% let's assume this converges, meaning 00:44:35.420 --> 00:44:41.425 align:middle line:84% this I'm writing it as this is finite. 00:44:41.425 --> 00:44:45.740 align:middle line:90% 00:44:45.740 --> 00:44:48.230 align:middle line:90% So, in fact, this equals a-- 00:44:48.230 --> 00:44:52.070 align:middle line:84% let me call this a convergence series something. 00:44:52.070 --> 00:44:54.005 align:middle line:84% Call it m, which is a finite number. 00:44:54.005 --> 00:44:59.056 align:middle line:90% 00:44:59.056 --> 00:45:06.580 align:middle line:84% All right, so we have this absolutely summable series, 00:45:06.580 --> 00:45:15.850 align:middle line:84% and we want to prove that now the series, sum of fk's, 00:45:15.850 --> 00:45:20.350 align:middle line:90% converge to something in Lp. 00:45:20.350 --> 00:45:30.640 align:middle line:90% 00:45:30.640 --> 00:45:32.260 align:middle line:90% And what do we want to show now? 00:45:32.260 --> 00:45:35.440 align:middle line:84% So let me just-- so that that's clear ahead of time, 00:45:35.440 --> 00:45:43.910 align:middle line:84% we want to show there exists a function in Lp 00:45:43.910 --> 00:45:53.320 align:middle line:90% such that k equals 1 to n. 00:45:53.320 --> 00:45:56.380 align:middle line:84% The partial sums converts to f as n 00:45:56.380 --> 00:46:03.168 align:middle line:90% goes to infinity and Lp of eie. 00:46:03.168 --> 00:46:07.980 align:middle line:84% Equivalent way of writing that is that the limit as n 00:46:07.980 --> 00:46:12.730 align:middle line:84% goes to infinity of the norm, some k equals 1 to n 00:46:12.730 --> 00:46:17.670 align:middle line:90% of fk minus fp equals 0. 00:46:17.670 --> 00:46:21.290 align:middle line:90% 00:46:21.290 --> 00:46:24.680 align:middle line:84% That will show that every summable series is-- 00:46:24.680 --> 00:46:26.720 align:middle line:84% absolutely summable series is summable. 00:46:26.720 --> 00:46:29.820 align:middle line:90% 00:46:29.820 --> 00:46:33.960 align:middle line:84% So we have to identify a candidate f and then 00:46:33.960 --> 00:46:37.530 align:middle line:90% show that the norm of-- 00:46:37.530 --> 00:46:43.901 align:middle line:84% that this norm here goes to 0 as n goes to infinity. 00:46:43.901 --> 00:46:52.030 align:middle line:90% OK, so define gn from E to-- 00:46:52.030 --> 00:46:55.960 align:middle line:90% 00:46:55.960 --> 00:46:58.240 align:middle line:84% so it's a non-negative number-- or it's a non-negative 00:46:58.240 --> 00:47:00.770 align:middle line:90% function-- 00:47:00.770 --> 00:47:09.320 align:middle line:84% by gn of x equals sum from k equals 1 to n 00:47:09.320 --> 00:47:11.540 align:middle line:84% of the absolute value of fk of x. 00:47:11.540 --> 00:47:16.140 align:middle line:90% 00:47:16.140 --> 00:47:17.990 align:middle line:84% This is, again, a measurable function 00:47:17.990 --> 00:47:20.390 align:middle line:84% because it's the sum of measurable functions. 00:47:20.390 --> 00:47:22.873 align:middle line:90% 00:47:22.873 --> 00:47:23.665 align:middle line:90% So what do we know? 00:47:23.665 --> 00:47:27.170 align:middle line:90% 00:47:27.170 --> 00:47:29.210 align:middle line:84% By the triangle inequality, we know 00:47:29.210 --> 00:47:40.340 align:middle line:84% that if I take the Lp norm of gn, which is this finite sum-- 00:47:40.340 --> 00:47:48.030 align:middle line:90% 00:47:48.030 --> 00:47:50.360 align:middle line:84% so by the triangle inequality for the Lp norm, 00:47:50.360 --> 00:47:54.800 align:middle line:84% this is less than or equal to sum from k equals 1 to n 00:47:54.800 --> 00:47:58.700 align:middle line:90% of the Lp norms of the fk's. 00:47:58.700 --> 00:48:02.450 align:middle line:84% And this is a partial sum corresponding 00:48:02.450 --> 00:48:07.160 align:middle line:84% to the series of the norms of fk's which sum to M. 00:48:07.160 --> 00:48:12.340 align:middle line:84% So this is always less than or equal to M, 00:48:12.340 --> 00:48:14.635 align:middle line:84% which we're assuming, again, is a finite number. 00:48:14.635 --> 00:48:17.810 align:middle line:90% 00:48:17.810 --> 00:48:34.400 align:middle line:84% And therefore, which implies by Fatou's lemma 00:48:34.400 --> 00:48:41.360 align:middle line:84% that the liminf as n goes to infinity of g sub 00:48:41.360 --> 00:48:52.170 align:middle line:90% n over E, which is equal to-- 00:48:52.170 --> 00:48:56.596 align:middle line:90% Now for each x, this-- 00:48:56.596 --> 00:48:59.140 align:middle line:90% 00:48:59.140 --> 00:49:03.110 align:middle line:84% so as n goes to infinity-- for each x as n goes to infinity, 00:49:03.110 --> 00:49:04.570 align:middle line:90% this converges to something. 00:49:04.570 --> 00:49:07.490 align:middle line:84% It's either finite or equal to infinity, 00:49:07.490 --> 00:49:08.480 align:middle line:90% so it always converges. 00:49:08.480 --> 00:49:15.000 align:middle line:84% So this is equal to the infinite sum. 00:49:15.000 --> 00:49:18.440 align:middle line:84% I should say, let's raise this to the p. 00:49:18.440 --> 00:49:21.860 align:middle line:90% 00:49:21.860 --> 00:49:22.865 align:middle line:90% Which is equal to-- 00:49:22.865 --> 00:49:38.470 align:middle line:90% 00:49:38.470 --> 00:49:42.640 align:middle line:84% so-- OK, I kind of got this backwards. 00:49:42.640 --> 00:49:44.750 align:middle line:84% I should have said this is equal to this. 00:49:44.750 --> 00:49:47.950 align:middle line:90% But anyways, by Fatou's lemma-- 00:49:47.950 --> 00:49:50.650 align:middle line:90% so let me reverse these things. 00:49:50.650 --> 00:50:02.590 align:middle line:90% 00:50:02.590 --> 00:50:06.000 align:middle line:84% So by Fatou's lemma, so I have this is equal to-- 00:50:06.000 --> 00:50:08.700 align:middle line:90% so this does not used Fatou. 00:50:08.700 --> 00:50:12.515 align:middle line:84% It just uses the definition of gn. 00:50:12.515 --> 00:50:15.950 align:middle line:90% 00:50:15.950 --> 00:50:19.260 align:middle line:90% This is less than or equal to-- 00:50:19.260 --> 00:50:22.550 align:middle line:90% now I'm using Fatou's lemma. 00:50:22.550 --> 00:50:27.920 align:middle line:84% Integral over E gn raised to the p. 00:50:27.920 --> 00:50:32.420 align:middle line:84% And now I use that bound which I have right here-- 00:50:32.420 --> 00:50:35.480 align:middle line:84% this is always less than or equal to-- 00:50:35.480 --> 00:50:38.390 align:middle line:84% remember, the Lp norm of g sub n was 00:50:38.390 --> 00:50:41.840 align:middle line:84% less than or equal to M. So raising that to the p power, 00:50:41.840 --> 00:50:44.210 align:middle line:84% I have this is less than or equal to M to the p. 00:50:44.210 --> 00:50:50.780 align:middle line:84% So I started off with the integral 00:50:50.780 --> 00:50:53.270 align:middle line:84% of this non-negative measurable function, which 00:50:53.270 --> 00:50:57.410 align:middle line:84% is the series of f sub k raised to the p, 00:50:57.410 --> 00:51:00.005 align:middle line:84% and showed that that integral is finite. 00:51:00.005 --> 00:51:02.560 align:middle line:90% 00:51:02.560 --> 00:51:06.530 align:middle line:84% Thus-- so by another theorem that we 00:51:06.530 --> 00:51:09.110 align:middle line:84% proved from integration, if I have a non-negative measurable 00:51:09.110 --> 00:51:11.360 align:middle line:84% function that has finite integral, 00:51:11.360 --> 00:51:13.760 align:middle line:84% then that measurable function has 00:51:13.760 --> 00:51:17.900 align:middle line:90% to be finite almost everywhere. 00:51:17.900 --> 00:51:25.190 align:middle line:84% So thus, this quantity has to converge 00:51:25.190 --> 00:51:38.280 align:middle line:84% or is finite for almost every x in E. 00:51:38.280 --> 00:51:42.195 align:middle line:84% So, what I get is that for almost-- 00:51:42.195 --> 00:51:44.220 align:middle line:90% let me include an x in here. 00:51:44.220 --> 00:51:53.190 align:middle line:90% 00:51:53.190 --> 00:51:58.818 align:middle line:84% So I've proven for almost every x in E, 00:51:58.818 --> 00:52:02.920 align:middle line:84% fk of x is absolutely convergent. 00:52:02.920 --> 00:52:04.860 align:middle line:84% So this series is absolutely convergent. 00:52:04.860 --> 00:52:07.740 align:middle line:84% And therefore, it converges for almost every x in E. 00:52:07.740 --> 00:52:10.500 align:middle line:84% And I'll define my function away, right? 00:52:10.500 --> 00:52:12.150 align:middle line:84% Because in the end, remember, we're 00:52:12.150 --> 00:52:16.020 align:middle line:84% trying to find a function f so that These partial sums 00:52:16.020 --> 00:52:17.640 align:middle line:90% converge to f and Lp. 00:52:17.640 --> 00:52:22.560 align:middle line:84% So if we can define f at somehow as the almost everywhere 00:52:22.560 --> 00:52:24.600 align:middle line:84% pointwise limit of s sub k, maybe we 00:52:24.600 --> 00:52:26.340 align:middle line:84% can use the dominated convergence theorem 00:52:26.340 --> 00:52:30.860 align:middle line:84% in some way, and that'll be what we do. 00:52:30.860 --> 00:52:45.000 align:middle line:84% So define f of x to be exactly what I 00:52:45.000 --> 00:52:49.390 align:middle line:84% get from this convergent series when it converges, absolutely. 00:52:49.390 --> 00:52:57.820 align:middle line:90% So if-- 0 or otherwise. 00:52:57.820 --> 00:53:02.780 align:middle line:90% 00:53:02.780 --> 00:53:10.030 align:middle line:84% And I'm going to define g of x to be, again, if you like, 00:53:10.030 --> 00:53:13.460 align:middle line:84% it's when the g sub n's converge. 00:53:13.460 --> 00:53:23.240 align:middle line:84% So if this is finite, and 0 otherwise. 00:53:23.240 --> 00:53:27.520 align:middle line:90% 00:53:27.520 --> 00:53:31.350 align:middle line:84% So now I have these two measurable real value 00:53:31.350 --> 00:53:33.920 align:middle line:90% functions. 00:53:33.920 --> 00:53:38.930 align:middle line:84% And this guy will end up being what the fk's converged to. 00:53:38.930 --> 00:53:44.130 align:middle line:90% 00:53:44.130 --> 00:53:47.070 align:middle line:84% So then we have a couple of things. 00:53:47.070 --> 00:53:54.350 align:middle line:84% Then limit as n goes to infinity of sum from k equals 1 to n, 00:53:54.350 --> 00:53:55.800 align:middle line:90% fk of x-- 00:53:55.800 --> 00:53:58.550 align:middle line:90% 00:53:58.550 --> 00:54:02.920 align:middle line:90% you can put a minus f of x-- 00:54:02.920 --> 00:54:07.450 align:middle line:84% equals 0 almost everywhere on E. Why is that? 00:54:07.450 --> 00:54:10.060 align:middle line:84% This is just because f of x is simply 00:54:10.060 --> 00:54:13.750 align:middle line:84% defined to be the infinite sum when the sum is absolutely 00:54:13.750 --> 00:54:16.540 align:middle line:84% convergent, which is almost everywhere by what we've done. 00:54:16.540 --> 00:54:20.710 align:middle line:90% 00:54:20.710 --> 00:54:21.535 align:middle line:90% What else? 00:54:21.535 --> 00:54:26.520 align:middle line:90% 00:54:26.520 --> 00:54:31.020 align:middle line:84% If I-- so this is one crucial thing. 00:54:31.020 --> 00:54:36.250 align:middle line:90% 00:54:36.250 --> 00:54:51.570 align:middle line:84% And minus f of x, this is less than or equal to g of x-- 00:54:51.570 --> 00:55:00.430 align:middle line:84% so it's the p raised the p almost everywhere on E. OK, 00:55:00.430 --> 00:55:02.800 align:middle line:90% why is this? 00:55:02.800 --> 00:55:05.380 align:middle line:84% Holds, for example, when the series 00:55:05.380 --> 00:55:07.270 align:middle line:90% is absolutely convergent. 00:55:07.270 --> 00:55:11.435 align:middle line:84% Because then f of x is equal to the infinite sum of the fk's 00:55:11.435 --> 00:55:11.935 align:middle line:90% of x. 00:55:11.935 --> 00:55:14.605 align:middle line:90% 00:55:14.605 --> 00:55:16.448 align:middle line:84% And so by the triangle inequality, 00:55:16.448 --> 00:55:17.990 align:middle line:84% the absolute value of that difference 00:55:17.990 --> 00:55:20.480 align:middle line:84% is going to be less than or equal to the sum 00:55:20.480 --> 00:55:22.250 align:middle line:90% of the absolute values. 00:55:22.250 --> 00:55:27.590 align:middle line:84% That's by definition equal to g sub n or g sub n plus 1. 00:55:27.590 --> 00:55:31.460 align:middle line:84% g is the limit of the g sub n, so the g sub n's 00:55:31.460 --> 00:55:32.390 align:middle line:90% are increasing. 00:55:32.390 --> 00:55:34.685 align:middle line:84% So that always sits below g of x. 00:55:34.685 --> 00:55:49.930 align:middle line:90% 00:55:49.930 --> 00:55:59.500 align:middle line:84% Now, by-- I should say, by this, I don't want to pull it up. 00:55:59.500 --> 00:56:17.620 align:middle line:84% So let's just say, since we proved that the Lp norm of this 00:56:17.620 --> 00:56:23.280 align:middle line:84% is less than or equal to M, which this here 00:56:23.280 --> 00:56:27.870 align:middle line:84% is equal to g almost everywhere, this 00:56:27.870 --> 00:56:36.940 align:middle line:84% implies that the LP norm of g is equal to this Lp norm, 00:56:36.940 --> 00:56:38.800 align:middle line:84% and therefore is less than or equal to M 00:56:38.800 --> 00:56:49.985 align:middle line:84% And therefore, that means the Lp norm raised the PE is finite. 00:56:49.985 --> 00:56:57.600 align:middle line:90% 00:56:57.600 --> 00:57:00.465 align:middle line:84% Moreover, what else do we deduce? 00:57:00.465 --> 00:57:02.970 align:middle line:90% 00:57:02.970 --> 00:57:09.180 align:middle line:84% We also deduce from this that f, which 00:57:09.180 --> 00:57:13.560 align:middle line:84% is bounded above by almost everywhere by this quantity, 00:57:13.560 --> 00:57:17.140 align:middle line:90% this is less than or equal to-- 00:57:17.140 --> 00:57:19.330 align:middle line:84% which is equal to g almost everywhere, 00:57:19.330 --> 00:57:21.900 align:middle line:84% the Lp form of f is less than or equal to the Lp 00:57:21.900 --> 00:57:25.710 align:middle line:84% norm of g, which, as we said, is less than or equal to M-- 00:57:25.710 --> 00:57:31.770 align:middle line:90% i.e., f is an Lp of E. 00:57:31.770 --> 00:57:33.840 align:middle line:84% So again, almost everywhere we have 00:57:33.840 --> 00:57:37.680 align:middle line:84% that f which is equal to the sum without the absolute values. 00:57:37.680 --> 00:57:40.980 align:middle line:84% That's always less than or equal to an absolute value. 00:57:40.980 --> 00:57:42.780 align:middle line:90% The sum from k equals-- 00:57:42.780 --> 00:57:45.742 align:middle line:84% the sum over k with absolute values inside. 00:57:45.742 --> 00:57:47.700 align:middle line:84% And we know that the Lp form of this is finite, 00:57:47.700 --> 00:57:50.850 align:middle line:84% we proved that earlier, which tells me that the Lp norm of g 00:57:50.850 --> 00:57:53.700 align:middle line:84% is finite, because g is equal to this quantity 00:57:53.700 --> 00:57:54.450 align:middle line:90% almost everywhere. 00:57:54.450 --> 00:57:57.630 align:middle line:84% It's 0 on a set of measure 0, which we just threw 00:57:57.630 --> 00:58:00.610 align:middle line:90% in there to make it finite. 00:58:00.610 --> 00:58:08.700 align:middle line:84% So we have f is an Lp, we have g is an Lp. 00:58:08.700 --> 00:58:11.950 align:middle line:84% So at least this can be a possible candidate. 00:58:11.950 --> 00:58:15.330 align:middle line:84% And we have these two facts here. 00:58:15.330 --> 00:58:22.670 align:middle line:84% Now we apply the dominated convergence theorem 00:58:22.670 --> 00:58:24.680 align:middle line:90% using this fact, this-- 00:58:24.680 --> 00:58:29.240 align:middle line:84% we have converging to 0 almost everywhere on E. 00:58:29.240 --> 00:58:32.750 align:middle line:84% And since that's true, I can put, if you like, 00:58:32.750 --> 00:58:34.700 align:middle line:90% raised to the p. 00:58:34.700 --> 00:58:36.680 align:middle line:84% So this quantity here is converging to 0 00:58:36.680 --> 00:58:38.060 align:middle line:90% almost everywhere. 00:58:38.060 --> 00:58:42.950 align:middle line:84% This quantity is also bounded above by this quantity here 00:58:42.950 --> 00:58:44.810 align:middle line:84% on the right, which is integrable. 00:58:44.810 --> 00:58:47.780 align:middle line:84% It has the integral of g to the p is finite. 00:58:47.780 --> 00:58:50.900 align:middle line:84% So by the dominated convergence theorem, 00:58:50.900 --> 00:58:55.220 align:middle line:84% I can conclude that the limit as n 00:58:55.220 --> 00:59:02.930 align:middle line:84% goes to infinity of sum from k equals 1 to n of the fk's minus 00:59:02.930 --> 00:59:08.750 align:middle line:84% f raised to the p over E, this equals the integral 00:59:08.750 --> 00:59:10.790 align:middle line:84% of the limit, which, remember, is 0. 00:59:10.790 --> 00:59:15.350 align:middle line:90% 00:59:15.350 --> 00:59:16.805 align:middle line:90% I.e., we've shown that. 00:59:16.805 --> 00:59:32.275 align:middle line:90% 00:59:32.275 --> 00:59:33.650 align:middle line:84% Which is what we wanted to prove. 00:59:33.650 --> 00:59:37.807 align:middle line:90% 00:59:37.807 --> 00:59:39.890 align:middle line:84% Now of course, you need a different argument for p 00:59:39.890 --> 00:59:42.905 align:middle line:84% equals infinity, and it's a little bit simpler. 00:59:42.905 --> 00:59:47.310 align:middle line:90% 00:59:47.310 --> 00:59:52.420 align:middle line:84% So we've proved that Lp is a Banach space. 00:59:52.420 --> 00:59:57.300 align:middle line:84% And notice, we used a lot of different tools and things 00:59:57.300 --> 01:00:02.670 align:middle line:84% that we had developed over the course of this-- 01:00:02.670 --> 01:00:05.475 align:middle line:84% over the course of this course so far. 01:00:05.475 --> 01:00:10.330 align:middle line:90% 01:00:10.330 --> 01:00:12.370 align:middle line:84% I keep using this word completion even though we 01:00:12.370 --> 01:00:13.900 align:middle line:90% haven't really talked about it. 01:00:13.900 --> 01:00:15.940 align:middle line:84% I kind of left it out of the first chapter 01:00:15.940 --> 01:00:18.340 align:middle line:84% because I wasn't going to use the second chapter 01:00:18.340 --> 01:00:24.980 align:middle line:84% from the lecture notes that are usually used for this course. 01:00:24.980 --> 01:00:29.268 align:middle line:90% But from this fact-- 01:00:29.268 --> 01:00:30.810 align:middle line:84% so you should think of the completion 01:00:30.810 --> 01:00:36.470 align:middle line:84% as the smallest Banach space containing a certain norm 01:00:36.470 --> 01:00:36.970 align:middle line:90% space. 01:00:36.970 --> 01:00:40.900 align:middle line:84% But what this statement here along with the theorem 01:00:40.900 --> 01:00:43.960 align:middle line:84% due to Riesz and Fischer is that the completion 01:00:43.960 --> 01:00:46.930 align:middle line:84% of continuous functions over a, b with norm 01:00:46.930 --> 01:00:49.640 align:middle line:84% given by the Lp norm, which is, in fact, 01:00:49.640 --> 01:00:51.550 align:middle line:84% a norm on continuous functions because if 01:00:51.550 --> 01:00:54.040 align:middle line:84% a continuous function is equal to 0 almost everywhere, 01:00:54.040 --> 01:00:56.380 align:middle line:90% it has to be 0. 01:00:56.380 --> 01:00:58.750 align:middle line:84% So what this proves is that the completion 01:00:58.750 --> 01:01:01.900 align:middle line:84% of the continuous functions with this norm 01:01:01.900 --> 01:01:03.810 align:middle line:90% is equal to the space Lp. 01:01:03.810 --> 01:01:14.970 align:middle line:90% 01:01:14.970 --> 01:01:20.230 align:middle line:84% So now we're going to move on back 01:01:20.230 --> 01:01:23.650 align:middle line:84% to some more general theory of functional analysis. 01:01:23.650 --> 01:01:29.727 align:middle line:84% This was specific to measure and integration 01:01:29.727 --> 01:01:30.560 align:middle line:90% on the real numbers. 01:01:30.560 --> 01:01:41.500 align:middle line:84% Now we're going to go to more general topics and probably 01:01:41.500 --> 01:01:54.350 align:middle line:84% more intuitive topics because a lot of it has analogs. 01:01:54.350 --> 01:01:58.130 align:middle line:84% I mean, a lot of functional analysis has-- 01:01:58.130 --> 01:02:01.820 align:middle line:84% it's supposed to be in some way analogous to stuff 01:02:01.820 --> 01:02:05.270 align:middle line:90% you've seen in linear algebra. 01:02:05.270 --> 01:02:06.635 align:middle line:90% Some of it definitely is not. 01:02:06.635 --> 01:02:12.490 align:middle line:90% 01:02:12.490 --> 01:02:14.640 align:middle line:84% For example, in one of the assignments, 01:02:14.640 --> 01:02:19.770 align:middle line:84% you proved that the unit ball and little Lp 01:02:19.770 --> 01:02:21.030 align:middle line:90% is not compact for-- 01:02:21.030 --> 01:02:23.640 align:middle line:90% 01:02:23.640 --> 01:02:25.980 align:middle line:84% over the natural numbers, the set 01:02:25.980 --> 01:02:29.160 align:middle line:84% of sequences that have little-- finite little Lp norm. 01:02:29.160 --> 01:02:30.390 align:middle line:90% That that's not compact. 01:02:30.390 --> 01:02:41.660 align:middle line:84% While from calculus, that in Rn, the unit ball is compact. 01:02:41.660 --> 01:02:46.040 align:middle line:84% That's the Burrill theorem or Bolzano-Weierstrass depending 01:02:46.040 --> 01:02:52.730 align:middle line:84% on if you take as your starting definition of compactness 01:02:52.730 --> 01:02:55.400 align:middle line:84% in terms of open sets or in terms of sequences having 01:02:55.400 --> 01:03:00.620 align:middle line:84% a convergent subsequence, which are equivalent for metric-- 01:03:00.620 --> 01:03:01.790 align:middle line:90% at least in a metric space. 01:03:01.790 --> 01:03:05.190 align:middle line:90% 01:03:05.190 --> 01:03:05.760 align:middle line:90% OK. 01:03:05.760 --> 01:03:11.750 align:middle line:84% Now here we're going to move on to the topic of Hilbert spaces. 01:03:11.750 --> 01:03:25.450 align:middle line:84% So these are special in the sense 01:03:25.450 --> 01:03:28.653 align:middle line:84% that the norm comes from an inner product-- maybe 01:03:28.653 --> 01:03:30.445 align:middle line:84% you saw an inner product in linear algebra. 01:03:30.445 --> 01:03:33.460 align:middle line:90% 01:03:33.460 --> 01:03:35.830 align:middle line:90% Or you should have at least. 01:03:35.830 --> 01:03:38.050 align:middle line:84% And therefore, you have notions of being orthogonal, 01:03:38.050 --> 01:03:40.210 align:middle line:84% you have notions of projections, and these-- 01:03:40.210 --> 01:03:45.727 align:middle line:84% we saw things that have that flavor when 01:03:45.727 --> 01:03:47.560 align:middle line:84% we're talking about Banach spaces in general 01:03:47.560 --> 01:03:52.940 align:middle line:84% and we were talking about modding out by a subspace, 01:03:52.940 --> 01:03:55.960 align:middle line:84% and you can think of the equivalence class corresponding 01:03:55.960 --> 01:04:00.640 align:middle line:84% to an element as the projection onto the complement 01:04:00.640 --> 01:04:07.100 align:middle line:84% of that subspace, but it wasn't an exact analogy. 01:04:07.100 --> 01:04:09.020 align:middle line:84% But a lot of exact analogies will now 01:04:09.020 --> 01:04:10.550 align:middle line:84% occur for Hilbert spaces, and then 01:04:10.550 --> 01:04:13.430 align:middle line:84% certain operators on Hilbert spaces 01:04:13.430 --> 01:04:22.460 align:middle line:84% will be very analogous to self-adjoint matrices 01:04:22.460 --> 01:04:28.490 align:middle line:84% or symmetric matrices which you saw in linear algebra. 01:04:28.490 --> 01:04:31.520 align:middle line:84% And of course, from a applied standpoint-- 01:04:31.520 --> 01:04:33.410 align:middle line:90% I should say applied-- 01:04:33.410 --> 01:04:36.980 align:middle line:84% Hilbert spaces, this is where the action-- 01:04:36.980 --> 01:04:39.978 align:middle line:90% 01:04:39.978 --> 01:04:42.020 align:middle line:84% this is where-- the setting of quantum mechanics. 01:04:42.020 --> 01:04:47.470 align:middle line:84% Quantum mechanics takes place in a Hilbert space. 01:04:47.470 --> 01:04:50.570 align:middle line:84% The elements are square integrable, 01:04:50.570 --> 01:04:54.633 align:middle line:84% now that we've dealt with that, functions over R3, 01:04:54.633 --> 01:04:56.300 align:middle line:84% if we're in three dimensions or if we're 01:04:56.300 --> 01:05:03.230 align:middle line:84% on the line R, that have L2 norm equal to 1, 01:05:03.230 --> 01:05:05.150 align:middle line:84% along with the Schrodinger equation. 01:05:05.150 --> 01:05:08.592 align:middle line:84% So Hilbert spaces are very important. 01:05:08.592 --> 01:05:10.175 align:middle line:84% They arise naturally in many problems. 01:05:10.175 --> 01:05:12.770 align:middle line:90% 01:05:12.770 --> 01:05:16.640 align:middle line:84% And because of this additional structure of them-- 01:05:16.640 --> 01:05:20.550 align:middle line:84% of the norm coming from an inner product, 01:05:20.550 --> 01:05:25.260 align:middle line:84% you can say a lot more things about them. 01:05:25.260 --> 01:05:29.000 align:middle line:90% 01:05:29.000 --> 01:05:34.060 align:middle line:84% So before we get to Hilbert spaces, 01:05:34.060 --> 01:05:39.620 align:middle line:90% let me add a "pre" before that. 01:05:39.620 --> 01:05:41.400 align:middle line:90% So pre-Hilbert spaces. 01:05:41.400 --> 01:05:43.220 align:middle line:84% So I said that Hilbert spaces are 01:05:43.220 --> 01:05:46.310 align:middle line:84% going to be Banach spaces that come from a norm. 01:05:46.310 --> 01:05:47.750 align:middle line:84% Pre-Hilbert spaces, these are just 01:05:47.750 --> 01:05:50.400 align:middle line:84% norm spaces that come from an inner product. 01:05:50.400 --> 01:05:53.180 align:middle line:84% So make the following definition. 01:05:53.180 --> 01:05:58.860 align:middle line:90% 01:05:58.860 --> 01:06:07.850 align:middle line:84% A pre-Hilbert space H, this is a vector space. 01:06:07.850 --> 01:06:11.010 align:middle line:90% 01:06:11.010 --> 01:06:13.920 align:middle line:84% Typically over C, but you can also just take it 01:06:13.920 --> 01:06:17.290 align:middle line:90% over R, that's fine as well. 01:06:17.290 --> 01:06:19.357 align:middle line:84% But I'll just for definiteness say over 01:06:19.357 --> 01:06:32.765 align:middle line:84% C. With a Hermitian inner product. 01:06:32.765 --> 01:06:35.730 align:middle line:90% 01:06:35.730 --> 01:06:38.250 align:middle line:84% So this is new terminology, maybe this 01:06:38.250 --> 01:06:44.460 align:middle line:84% is new terminology, too, so let me write out what it means-- 01:06:44.460 --> 01:06:46.330 align:middle line:84% what is a Hermitian inner product. 01:06:46.330 --> 01:06:55.310 align:middle line:84% So this is, i.e., a map, usually denoted 01:06:55.310 --> 01:07:01.010 align:middle line:84% using brackets from H-cross-H into the complex numbers. 01:07:01.010 --> 01:07:06.120 align:middle line:84% Satisfying certain properties-- so such that-- 01:07:06.120 --> 01:07:25.700 align:middle line:84% so one, it is linear and the first variable-- 01:07:25.700 --> 01:07:29.940 align:middle line:84% so for all lambda 1, lambda 2, v1, v2, 01:07:29.940 --> 01:07:35.370 align:middle line:84% if I look at the inner product of v1, lambda 1 times v1 01:07:35.370 --> 01:07:41.040 align:middle line:84% plus lambda 2 times v2 with w, this is equal to lambda 1 times 01:07:41.040 --> 01:07:47.230 align:middle line:84% v1 times the inner product with w plus lambda 2 v2 01:07:47.230 --> 01:07:48.190 align:middle line:90% inner product with w. 01:07:48.190 --> 01:07:56.580 align:middle line:90% 01:07:56.580 --> 01:07:59.340 align:middle line:90% For all V, w, and-- 01:07:59.340 --> 01:08:01.290 align:middle line:84% I wrote capital V, I should have-- 01:08:01.290 --> 01:08:02.790 align:middle line:90% written H a minute ago. 01:08:02.790 --> 01:08:07.020 align:middle line:90% 01:08:07.020 --> 01:08:09.690 align:middle line:84% Now we're in Hilbert spaces, pre-Hilbert spaces. 01:08:09.690 --> 01:08:13.200 align:middle line:84% For all v and w, the inner product of v and w 01:08:13.200 --> 01:08:18.750 align:middle line:84% is equal to the complex conjugate of the inner product 01:08:18.750 --> 01:08:28.200 align:middle line:90% of w with v. 01:08:28.200 --> 01:08:31.830 align:middle line:84% And the following, which is positive definiteness 01:08:31.830 --> 01:08:37.335 align:middle line:84% of the inner product, if to call it that, for all v and H, 01:08:37.335 --> 01:08:42.640 align:middle line:84% m the inner product would be with itself, 01:08:42.640 --> 01:08:45.670 align:middle line:84% this is bigger than or equal to 0, so it's a real number 01:08:45.670 --> 01:08:48.250 align:middle line:90% and it's non-negative. 01:08:48.250 --> 01:08:56.050 align:middle line:84% And this one, it equals 0 if and only if v is a zero element. 01:08:56.050 --> 01:09:06.229 align:middle line:90% 01:09:06.229 --> 01:09:07.970 align:middle line:90% OK. 01:09:07.970 --> 01:09:14.729 align:middle line:84% So let me make a couple of remarks. 01:09:14.729 --> 01:09:19.474 align:middle line:84% First off, let's say this is remark 1. 01:09:19.474 --> 01:09:23.180 align:middle line:90% 01:09:23.180 --> 01:09:25.390 align:middle line:84% The third quantity does imply that the only thing 01:09:25.390 --> 01:09:28.040 align:middle line:84% orthogonal to everything in the space is a zero element. 01:09:28.040 --> 01:09:35.609 align:middle line:84% So v is in H and v is orthogonal to everything-- 01:09:35.609 --> 01:09:37.800 align:middle line:84% I keep using the word orthogonal. 01:09:37.800 --> 01:09:40.500 align:middle line:84% I should just the inner product with zero. 01:09:40.500 --> 01:09:53.069 align:middle line:84% It gives me 0 for every element, this implies that v equals 0, 01:09:53.069 --> 01:09:54.990 align:middle line:84% and of course, the converse applies, too, 01:09:54.990 --> 01:09:58.500 align:middle line:84% that v is equal to 0, then the inner product with 0 01:09:58.500 --> 01:10:02.430 align:middle line:84% and every element is 0 just by linearity. 01:10:02.430 --> 01:10:09.930 align:middle line:84% Two is that if I have two elements v and w and a scalar 01:10:09.930 --> 01:10:16.890 align:middle line:84% lambda, then this is equal to the complex conjugate of lambda 01:10:16.890 --> 01:10:27.080 align:middle line:84% times w with v, and by the first property, the lambda pops out, 01:10:27.080 --> 01:10:29.930 align:middle line:90% and therefore, if I-- 01:10:29.930 --> 01:10:31.910 align:middle line:84% complex conjugate of a product is a product 01:10:31.910 --> 01:10:33.260 align:middle line:90% of the complex conjugates. 01:10:33.260 --> 01:10:38.000 align:middle line:84% And then if I undo this, I can get this. 01:10:38.000 --> 01:10:42.650 align:middle line:84% So it's linear in the first entry, 01:10:42.650 --> 01:10:44.570 align:middle line:84% meaning the constants just come out. 01:10:44.570 --> 01:10:46.610 align:middle line:84% But if I have a constant in the second-- 01:10:46.610 --> 01:10:50.600 align:middle line:84% or a scalar multiple in the second entry, 01:10:50.600 --> 01:10:53.840 align:middle line:84% then that comes out as well, but with a complex conjugate 01:10:53.840 --> 01:10:54.930 align:middle line:90% over it. 01:10:54.930 --> 01:10:56.270 align:middle line:90% That's all I wanted to say. 01:10:56.270 --> 01:11:00.150 align:middle line:90% 01:11:00.150 --> 01:11:04.830 align:middle line:84% So I said that pre Hilbert spaces are-- 01:11:04.830 --> 01:11:08.540 align:middle line:84% you naturally think of them as norm spaces where the norm 01:11:08.540 --> 01:11:10.400 align:middle line:90% comes from the inner product. 01:11:10.400 --> 01:11:12.170 align:middle line:90% Here, we have in a product. 01:11:12.170 --> 01:11:13.010 align:middle line:90% Where's the norm? 01:11:13.010 --> 01:11:16.910 align:middle line:90% 01:11:16.910 --> 01:11:18.200 align:middle line:90% So definition. 01:11:18.200 --> 01:11:29.420 align:middle line:84% If H is a pre-Hilbert space with inner product denoted 01:11:29.420 --> 01:11:42.810 align:middle line:84% as before, we define using this-- 01:11:42.810 --> 01:11:44.350 align:middle line:84% So I'm not calling it a norm yet. 01:11:44.350 --> 01:11:48.510 align:middle line:84% I'm just saying we define this function on H 01:11:48.510 --> 01:11:54.990 align:middle line:84% to be in a product v with itself raised to the 1/2 power. 01:11:54.990 --> 01:11:56.670 align:middle line:84% In the end or in a minute we'll show 01:11:56.670 --> 01:11:58.140 align:middle line:90% that this is, in fact, a norm. 01:11:58.140 --> 01:12:02.900 align:middle line:90% 01:12:02.900 --> 01:12:09.020 align:middle line:84% But for now I'm just going to call it this function on H 01:12:09.020 --> 01:12:18.650 align:middle line:84% or possible norm on H. All right, so we 01:12:18.650 --> 01:12:21.140 align:middle line:90% have the following theorem. 01:12:21.140 --> 01:12:23.450 align:middle line:84% So this is valid in any pre-Hilbert space. 01:12:23.450 --> 01:12:27.230 align:middle line:90% 01:12:27.230 --> 01:12:30.590 align:middle line:84% For all u, v, and H of pre-Hilbert space, 01:12:30.590 --> 01:12:36.770 align:middle line:84% if I take the absolute value of the inner product of u and v, 01:12:36.770 --> 01:12:46.070 align:middle line:84% this is less than or equal to this norm-looking thing 01:12:46.070 --> 01:12:51.850 align:middle line:84% of u times the norm-looking thing of v. 01:12:51.850 --> 01:13:01.230 align:middle line:84% So this shouldn't come as a complete surprise. 01:13:01.230 --> 01:13:05.350 align:middle line:84% Right down-- if we took H to be Rn, 01:13:05.350 --> 01:13:10.860 align:middle line:84% and so then now this is a vector space over R, 01:13:10.860 --> 01:13:13.980 align:middle line:84% then and the inner product would just be the dot product. 01:13:13.980 --> 01:13:17.610 align:middle line:84% This is stating the Cauchy-Schwarz inequality 01:13:17.610 --> 01:13:20.080 align:middle line:84% that you know and love from before. 01:13:20.080 --> 01:13:23.760 align:middle line:90% So what's the proof? 01:13:23.760 --> 01:13:37.800 align:middle line:84% Let's let f of t be the norm of u plus t times v. 01:13:37.800 --> 01:13:40.510 align:middle line:84% So I said norm, but I haven't proved it's a norm yet, 01:13:40.510 --> 01:13:42.802 align:middle line:84% so you're going to have to forgive if I keep calling it 01:13:42.802 --> 01:13:45.570 align:middle line:90% a norm, but in the end it is. 01:13:45.570 --> 01:13:50.700 align:middle line:84% Let f of t be this thing squared, which, we note, 01:13:50.700 --> 01:13:52.380 align:middle line:90% it's a non-negative number. 01:13:52.380 --> 01:13:56.040 align:middle line:84% Because the product of v with v is a non-negative number taking 01:13:56.040 --> 01:14:04.310 align:middle line:84% that to the 1/2 power, so this is u plus tv inner product u 01:14:04.310 --> 01:14:10.570 align:middle line:90% plus tv which is non-negative. 01:14:10.570 --> 01:14:20.910 align:middle line:84% Now if we compute all this out using how the linearity works 01:14:20.910 --> 01:14:25.090 align:middle line:90% for inner products, this-- 01:14:25.090 --> 01:14:30.060 align:middle line:84% I get u inner product u plus t squared 01:14:30.060 --> 01:14:35.100 align:middle line:84% v inner product v plus t times u inner product 01:14:35.100 --> 01:14:40.820 align:middle line:84% v plus t times inner product of v with u. 01:14:40.820 --> 01:14:43.690 align:middle line:90% 01:14:43.690 --> 01:14:46.120 align:middle line:84% And this is the complex conjugate 01:14:46.120 --> 01:14:48.410 align:middle line:90% of this complex number. 01:14:48.410 --> 01:14:54.250 align:middle line:84% So let me just rewrite this-- u squared plus t squared or v 01:14:54.250 --> 01:14:56.530 align:middle line:90% squared plus-- 01:14:56.530 --> 01:14:59.530 align:middle line:84% so again, this number is the complex conjugate 01:14:59.530 --> 01:15:00.430 align:middle line:90% of this number. 01:15:00.430 --> 01:15:03.910 align:middle line:84% And when I add a complex number to its complex conjugate, 01:15:03.910 --> 01:15:06.920 align:middle line:90% I get twice times the real part. 01:15:06.920 --> 01:15:20.390 align:middle line:90% 2t real part of u and v. 01:15:20.390 --> 01:15:25.670 align:middle line:84% Now, this is just a polynomial with a non-negative thing 01:15:25.670 --> 01:15:26.940 align:middle line:90% out in front of t squared. 01:15:26.940 --> 01:15:28.340 align:middle line:90% So it has a minimum somewhere. 01:15:28.340 --> 01:15:30.470 align:middle line:84% And this minimum has to be non-negative 01:15:30.470 --> 01:15:32.765 align:middle line:84% since this function is greater than or equal to 0. 01:15:32.765 --> 01:15:42.390 align:middle line:90% 01:15:42.390 --> 01:15:47.890 align:middle line:84% Is greater than or equal to 0, I should say, just as a sentence. 01:15:47.890 --> 01:15:59.980 align:middle line:90% 01:15:59.980 --> 01:16:01.810 align:middle line:84% Now, where does this minimum occur? 01:16:01.810 --> 01:16:05.380 align:middle line:84% It occurs where the derivative is equal to 0. 01:16:05.380 --> 01:16:11.920 align:middle line:84% Now f prime of t then equals 0 implies-- 01:16:11.920 --> 01:16:18.040 align:middle line:84% or if and only if t min is equal to minus the real part. 01:16:18.040 --> 01:16:21.310 align:middle line:84% So I will leave this calculus to you. 01:16:21.310 --> 01:16:26.420 align:middle line:90% 01:16:26.420 --> 01:16:27.860 align:middle line:84% I mean, this is just a polynomial. 01:16:27.860 --> 01:16:29.402 align:middle line:84% Take the derivative with respect to t 01:16:29.402 --> 01:16:32.150 align:middle line:84% and solve for when that's equal to 0. 01:16:32.150 --> 01:16:36.590 align:middle line:84% And therefore, I get that this minimum-- 01:16:36.590 --> 01:16:39.420 align:middle line:84% so f evaluated at this point is non-negative, 01:16:39.420 --> 01:16:45.530 align:middle line:84% so 0 is less than or equal to f of t min, which, 01:16:45.530 --> 01:16:48.950 align:middle line:84% when you stick that into this here, 01:16:48.950 --> 01:16:58.580 align:middle line:84% you get norm of u squared minus real part of u v squared 01:16:58.580 --> 01:17:03.540 align:middle line:90% over norm of v squared. 01:17:03.540 --> 01:17:12.310 align:middle line:84% And therefore, the real part of u, v absolute value-- 01:17:12.310 --> 01:17:15.750 align:middle line:84% so this-- all of this is bigger than or equal to 0. 01:17:15.750 --> 01:17:17.685 align:middle line:84% So if I move this over to that side 01:17:17.685 --> 01:17:20.790 align:middle line:84% and multiplied by norm squared, take the square root, 01:17:20.790 --> 01:17:24.780 align:middle line:84% I get that this is less than or equal to-- 01:17:24.780 --> 01:17:25.410 align:middle line:90% le me put a-- 01:17:25.410 --> 01:17:29.290 align:middle line:90% 01:17:29.290 --> 01:17:31.690 align:middle line:90% u times norm v. 01:17:31.690 --> 01:17:34.490 align:middle line:90% So this is almost what I want. 01:17:34.490 --> 01:17:40.570 align:middle line:84% I want the absolute value of the inner product with u and v. But 01:17:40.570 --> 01:17:47.030 align:middle line:84% all I have is the real part of inner product with u and v. 01:17:47.030 --> 01:17:49.970 align:middle line:84% So of course, if the inner product of u and v is 0, 01:17:49.970 --> 01:17:54.380 align:middle line:84% this inequality I want to prove is automatic. 01:17:54.380 --> 01:17:59.420 align:middle line:84% Also, if u or v is 0, then this inequality I wanted to prove 01:17:59.420 --> 01:18:00.180 align:middle line:90% is automatic. 01:18:00.180 --> 01:18:02.180 align:middle line:84% So that's why I'm actually just dealing 01:18:02.180 --> 01:18:03.780 align:middle line:90% with the non-trivial case. 01:18:03.780 --> 01:18:11.450 align:middle line:84% So if this inner product is 0, then we're 01:18:11.450 --> 01:18:13.250 align:middle line:84% already done because the right-hand side 01:18:13.250 --> 01:18:14.210 align:middle line:90% is non-negative. 01:18:14.210 --> 01:18:25.250 align:middle line:84% So suppose this is non-zero, but lambda be this complex number 01:18:25.250 --> 01:18:29.210 align:middle line:84% u, v inner product complex conjugate 01:18:29.210 --> 01:18:39.480 align:middle line:84% over the absolute value, then the absolute value 01:18:39.480 --> 01:18:41.880 align:middle line:84% of this complex number is equal to 1 01:18:41.880 --> 01:18:44.820 align:middle line:84% because the absolute value of the conjugate 01:18:44.820 --> 01:18:48.045 align:middle line:84% is equal to the absolute value of the original complex number. 01:18:48.045 --> 01:18:51.590 align:middle line:90% 01:18:51.590 --> 01:19:03.560 align:middle line:84% And lambda times u, v, which is equal to lambda. 01:19:03.560 --> 01:19:18.000 align:middle line:84% So I should say u, v, this is equal to lambda times u, v, 01:19:18.000 --> 01:19:23.010 align:middle line:84% which is equal to lambda u times v. 01:19:23.010 --> 01:19:26.910 align:middle line:84% Now this is equal to this, and therefore, it's 01:19:26.910 --> 01:19:29.010 align:middle line:90% equal to the real part of it. 01:19:29.010 --> 01:19:32.490 align:middle line:84% So this is equal to a real number, 01:19:32.490 --> 01:19:36.055 align:middle line:84% and therefore it's equal to its real part. 01:19:36.055 --> 01:19:37.430 align:middle line:84% We pulled a similar trick when we 01:19:37.430 --> 01:19:39.180 align:middle line:84% were talking about the triangle inequality 01:19:39.180 --> 01:19:43.520 align:middle line:84% for integral functions that now take 01:19:43.520 --> 01:19:45.050 align:middle line:90% values and the complex number-- 01:19:45.050 --> 01:19:46.020 align:middle line:90% complex numbers. 01:19:46.020 --> 01:19:48.320 align:middle line:84% So this is equal to the real part of this, which 01:19:48.320 --> 01:19:49.460 align:middle line:90% is less than or equal to-- 01:19:49.460 --> 01:19:52.960 align:middle line:90% 01:19:52.960 --> 01:19:55.120 align:middle line:84% by what we've done here, we've already 01:19:55.120 --> 01:19:57.940 align:middle line:84% proven this inequality holds for every u and v, 01:19:57.940 --> 01:20:06.350 align:middle line:84% so it also holds for lambda times u times v. And now-- 01:20:06.350 --> 01:20:11.780 align:middle line:84% so simple off to the side, that if I take lambda u, 01:20:11.780 --> 01:20:13.460 align:middle line:84% an inner product with lambda u, if I 01:20:13.460 --> 01:20:16.010 align:middle line:84% take the 1/2 power of that, that gives me this quantity. 01:20:16.010 --> 01:20:17.420 align:middle line:90% But let me just compute this. 01:20:17.420 --> 01:20:18.710 align:middle line:90% This is equal to-- 01:20:18.710 --> 01:20:20.780 align:middle line:90% lambda comes out from here. 01:20:20.780 --> 01:20:26.040 align:middle line:84% And then complex conjugate of lambda comes out from there, 01:20:26.040 --> 01:20:27.320 align:middle line:90% and this is equal to-- 01:20:27.320 --> 01:20:32.410 align:middle line:90% 01:20:32.410 --> 01:20:35.550 align:middle line:90% but this is equal to 1. 01:20:35.550 --> 01:20:43.670 align:middle line:84% So we see that this is equal to this, and therefore, 01:20:43.670 --> 01:20:47.030 align:middle line:84% raising it to the 1/2 power is equal to-- 01:20:47.030 --> 01:20:49.910 align:middle line:90% 01:20:49.910 --> 01:20:52.220 align:middle line:84% this quantity here is equal to this quantity. 01:20:52.220 --> 01:20:53.990 align:middle line:84% And we proven what we wanted to do. 01:20:53.990 --> 01:20:57.490 align:middle line:90% 01:20:57.490 --> 01:20:59.440 align:middle line:90% OK. 01:20:59.440 --> 01:21:04.220 align:middle line:84% And so this was a Cauchy-Schwarz inequality 01:21:04.220 --> 01:21:07.450 align:middle line:84% and a general pre-Hilbert space with this quantity 01:21:07.450 --> 01:21:09.340 align:middle line:84% that I referred to as a norm but I 01:21:09.340 --> 01:21:11.200 align:middle line:90% haven't proved it's a norm yet. 01:21:11.200 --> 01:21:14.680 align:middle line:84% Next time we will use the Cauchy-Schwarz inequality 01:21:14.680 --> 01:21:19.300 align:middle line:84% to prove that, in fact, this thing that I'm masquerading 01:21:19.300 --> 01:21:22.070 align:middle line:84% around in a norm notation is, in fact, 01:21:22.070 --> 01:21:25.210 align:middle line:90% a norm on a pre-Hilbert space. 01:21:25.210 --> 01:21:28.450 align:middle line:90% 01:21:28.450 --> 01:21:34.140 align:middle line:84% And from there, we'll introduce Hilbert spaces 01:21:34.140 --> 01:21:39.090 align:middle line:84% which are those pre-Hilbert spaces with this norm that 01:21:39.090 --> 01:21:42.400 align:middle line:90% are actually complete. 01:21:42.400 --> 01:21:46.170 align:middle line:90% And really, for any kind of-- 01:21:46.170 --> 01:21:53.730 align:middle line:84% so for-- and we'll prove this at some point, 01:21:53.730 --> 01:21:58.890 align:middle line:84% there's really only two types of reasonable Hilbert spaces. 01:21:58.890 --> 01:22:01.630 align:middle line:90% 01:22:01.630 --> 01:22:04.220 align:middle line:84% And I mean this in a very strong sense, 01:22:04.220 --> 01:22:10.180 align:middle line:84% not in a loose sense, that the first type is 01:22:10.180 --> 01:22:13.150 align:middle line:84% just finite-dimensional-- so think of C raised to the n. 01:22:13.150 --> 01:22:17.350 align:middle line:84% So n tuples of complex numbers where the inner product 01:22:17.350 --> 01:22:21.650 align:middle line:90% is definable in a natural way. 01:22:21.650 --> 01:22:28.970 align:middle line:90% Or a little L2 which is these-- 01:22:28.970 --> 01:22:36.400 align:middle line:84% this set of sequences that have finite little L2 norm. 01:22:36.400 --> 01:22:39.520 align:middle line:84% This is basically the only other type of Hilbert space 01:22:39.520 --> 01:22:43.780 align:middle line:84% that there is, and I'll say what I mean by that. 01:22:43.780 --> 01:22:46.590 align:middle line:84% How it'll work is we'll basically 01:22:46.590 --> 01:22:50.910 align:middle line:84% show that every separable Hilbert space, which 01:22:50.910 --> 01:22:57.330 align:middle line:84% is what I mean by reasonable, that it's 01:22:57.330 --> 01:23:02.370 align:middle line:84% the most reasonable spaces we work with are separable, 01:23:02.370 --> 01:23:04.770 align:middle line:84% meaning they have countable dense subset, 01:23:04.770 --> 01:23:09.420 align:middle line:84% has a countable orthogonal basis, which is what-- 01:23:09.420 --> 01:23:11.730 align:middle line:84% Orthonormal basis I haven't defined. 01:23:11.730 --> 01:23:17.250 align:middle line:84% It's not a Hamel basis, but it serves the same purpose, 01:23:17.250 --> 01:23:19.590 align:middle line:84% meaning you can't write every element 01:23:19.590 --> 01:23:22.470 align:middle line:84% as a finite linear combination of orthonormal basis, 01:23:22.470 --> 01:23:25.230 align:middle line:84% but you can write it as an infinite expansion 01:23:25.230 --> 01:23:26.620 align:middle line:90% in orthonormal basis. 01:23:26.620 --> 01:23:31.600 align:middle line:90% Think Fourier series. 01:23:31.600 --> 01:23:34.230 align:middle line:84% And this is what provides this identification 01:23:34.230 --> 01:23:41.100 align:middle line:84% of a separable Hilbert space with either a C to the n 01:23:41.100 --> 01:23:44.670 align:middle line:84% if this or orthonormal basis is finite, or little L2 01:23:44.670 --> 01:23:48.120 align:middle line:84% if this or orthonormal basis is countably infinite. 01:23:48.120 --> 01:23:52.850 align:middle line:84% And we'll get to that possibly by the end of next lecture. 01:23:52.850 --> 01:23:58.000 align:middle line:90%