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Title: Closed Operators
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Series: Unbounded Operators
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YouTube-Title: Unbounded Operators 3 | Closed Operators
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Bright video: https://youtu.be/nSrQBWM-Tsc
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Subtitle on GitHub: uo03_sub_eng.srt
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Timestamps (n/a)
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Subtitle in English
1 00:00:00,400 –> 00:00:04,243 Hello and welcome back to unbounded operators.
2 00:00:04,300 –> 00:00:07,670 The video series in the field of functional analysis
3 00:00:07,870 –> 00:00:12,646 and in today’s part 3 we will define so called closed operators.
4 00:00:13,100 –> 00:00:20,971 Indeed it will turn out that these closed operators are related to the continuity property that bounded operators have.
5 00:00:21,300 –> 00:00:25,332 So in some sense this is the generalization we want now.
6 00:00:25,532 –> 00:00:28,029 However, you already know before we start
7 00:00:28,100 –> 00:00:33,916 I first want to thank all the nice people who support this channel on Steady, here on YouTube or via Patreon
8 00:00:34,116 –> 00:00:41,419 and please don’t forget, you can also download quizzes and PDF versions for all the videos with the link in the description.
9 00:00:41,619 –> 00:00:46,370 Ok, then let’s start again by telling what we mean by an operator.
10 00:00:46,570 –> 00:00:54,217 So an operator T is just a linear map between two normed spaces X and Y, but we can have a domain
11 00:00:54,417 –> 00:00:58,830 and usually what we do is that we call this domain D(T).
12 00:00:59,030 –> 00:01:03,775 So in general T is not defined on the whole normed space X,
13 00:01:03,975 –> 00:01:08,392 but still, it has to be a linear mapping between the two spaces.
14 00:01:09,086 –> 00:01:15,474 Ok and now since this is a well defined map, we can identify it with its graph
15 00:01:15,674 –> 00:01:21,787 and this you might know. It’s just a special subset in the Cartesian product X times Y.
16 00:01:22,200 –> 00:01:26,872 This means we can simply sketch it into a coordinate system.
17 00:01:27,457 –> 00:01:31,510 Hence in general you would sketch a graph like that.
18 00:01:31,710 –> 00:01:39,365 However since we have a linear mapping here, we should also make it clear that we have a linear relation as a graph.
19 00:01:39,565 –> 00:01:42,460 Moreover since we work in infinite dimensions,
20 00:01:42,660 –> 00:01:48,225 in reality we would expect that the picture looks much more complicated than this line,
21 00:01:48,629 –> 00:01:55,457 but still the general idea here of this set given by this graph of T still holds.
22 00:01:55,657 –> 00:02:01,101 Ok, then let’s give this set a short name. Let’s call it G index T
23 00:02:01,301 –> 00:02:05,842 and now we can define it as a subset of X times Y.
24 00:02:06,429 –> 00:02:12,366 This means we just have to write down what one point (x,y) has to fulfill.
25 00:02:12,566 –> 00:02:20,956 First that x lies in the domain of T and second we need that Tx is equal to y.
26 00:02:21,156 –> 00:02:25,177 So the image of our x is exactly the point y.
27 00:02:25,614 –> 00:02:31,464 So not so surprising this is how in general we would define the graph of a map.
28 00:02:31,664 –> 00:02:38,429 However here we have a little bit more, because this Cartesian product here is a normed space again.
29 00:02:38,629 –> 00:02:44,813 Hence it’s not just a set. It’s a whole vector space, where we can also measure lengths
30 00:02:45,271 –> 00:02:49,750 and at this point you should ask what is the corresponding norm here.
31 00:02:49,950 –> 00:02:54,515 In other words what is the correct length of the pair (x,y)?
32 00:02:54,715 –> 00:03:01,591 Indeed there are different possibilities to do that, but the common one is just to add two norms we already know.
33 00:03:01,791 –> 00:03:06,309 So first take the norm of x and then add the norm of y
34 00:03:06,509 –> 00:03:11,429 and now it’s not hard to check at all, that this here defines a norm again.
35 00:03:12,057 –> 00:03:19,768 Hence we always have a normed space with the Cartesian product, with X and Y are normed spaces from the beginning
36 00:03:19,968 –> 00:03:26,337 and therefore you can say now in the graph of T we can also measure distances
37 00:03:26,537 –> 00:03:31,974 or more precisely we can use all the topological terms we have for the graph of T.
38 00:03:32,174 –> 00:03:38,178 In particular we can also check if the graph of T is a closed set
39 00:03:38,378 –> 00:03:44,593 and that’s exactly what we do now in order to define so called closed operators.
40 00:03:45,357 –> 00:03:54,358 So you see we call T a closed operator if its graph is a closed set in the normed space X times Y.
41 00:03:54,558 –> 00:04:01,984 So not a complicated definition at all, we just have to know what closed sets in normed spaces are
42 00:04:02,184 –> 00:04:06,924 and the good thing is, we can assume that we already know a lot.
43 00:04:07,124 –> 00:04:13,882 In particular we know that we can also work with sequences to define closed sets
44 00:04:14,082 –> 00:04:19,400 and this means we immediately get an equivalent definition for a closed operator.
45 00:04:19,771 –> 00:04:26,066 More precisely instead of working with a closed set, we can also work with sequences.
46 00:04:26,471 –> 00:04:32,946 So, what we have to do is to take arbitrary sequence x_n inside the domain
47 00:04:33,146 –> 00:04:36,271 and this one has to fulfill 2 properties.
48 00:04:36,471 –> 00:04:41,714 So first it should be a convergent sequence in the normed space X.
49 00:04:42,414 –> 00:04:47,013 This means it has a limit that lies in the space X.
50 00:04:47,213 –> 00:04:51,357 So we don’t have to assume that the limit lies in the domain
51 00:04:51,557 –> 00:04:57,559 and second we assume that the sequence of the images is also convergent.
52 00:04:57,759 –> 00:05:03,185 So Tx_n converges inside the space Y to a limit y
53 00:05:03,543 –> 00:05:10,486 and now the claim is that for each sequence with these two properties we can conclude 2 other properties.
54 00:05:10,871 –> 00:05:16,241 The first one is indeed this limit x lies in the domain of T
55 00:05:16,441 –> 00:05:22,758 and this immediately implies that we are allowed to apply T to this x as well
56 00:05:22,958 –> 00:05:28,640 and then we get the second property. This image is indeed equal to y.
57 00:05:28,929 –> 00:05:37,427 So this sounds not so strange, but maybe at this point you can immediately compare this definition to the continuity definition.
58 00:05:37,886 –> 00:05:44,747 Indeed, this is what I mentioned before. We want to use this definition as a substitute for the continuity.
59 00:05:45,229 –> 00:05:50,542 This is because we already know, unbounded operators are not continuous.
60 00:05:50,886 –> 00:05:56,030 Ok, now I thing we can also write down the proof for this statement from before.
61 00:05:56,357 –> 00:05:59,621 So we want to improve the equivalence stated here
62 00:06:00,057 –> 00:06:06,824 and in fact I already gave you the hint. Closed sets can be equivalently described by sequences.
63 00:06:07,600 –> 00:06:14,634 So you should know, this works in general, in metric spaces. So in particular here in normed spaces.
64 00:06:15,300 –> 00:06:22,336 So what you have to do is to take a sequence from the set and you have to assume that this one is convergent.
65 00:06:22,536 –> 00:06:29,191 So please don’t forget this means the sequence is convergent in our normed space X times Y.
66 00:06:29,471 –> 00:06:36,671 So maybe here let’s call the limit also z and the important thing is, it lies in X times Y,
67 00:06:36,871 –> 00:06:43,609 because now the important conclusion here is that this z also lies in G_T.
68 00:06:43,809 –> 00:06:51,760 So you should see, this is the general thing. Closed means you can not leave this set with sequence from inside
69 00:06:51,960 –> 00:06:58,796 and this is indeed equivalent. All the boundary points already lie inside the set itself.
70 00:06:58,996 –> 00:07:04,852 Ok, so this should not be a surprise and now we want to put this into the context of an operator.
71 00:07:05,052 –> 00:07:10,666 So first instead of z_n we can already write what we know about the graph.
72 00:07:10,866 –> 00:07:15,887 Namely that it always consists of a pair. (x_n, Tx_n).
73 00:07:16,243 –> 00:07:21,238 So a sequence inside the graph of T always has this form.
74 00:07:21,438 –> 00:07:26,483 Ok and now this sequence should converge to a point in our Cartesian product
75 00:07:27,257 –> 00:07:30,710 and this limit we can just call (x,y).
76 00:07:30,910 –> 00:07:38,591 Ok, but now this claim here tells us that this pair (x,y) also lies in the graph of T
77 00:07:38,791 –> 00:07:43,606 and by the definition of the graph, this exactly means 2 things.
78 00:07:43,806 –> 00:07:47,087 First our point x lies in the domain
79 00:07:47,287 –> 00:07:51,355 and second Tx is exactly y
80 00:07:52,000 –> 00:07:56,281 and there you should already see, this is exactly the conclusion in the claim above
81 00:07:56,481 –> 00:08:01,153 and now the only difference here is the assumption from before.
82 00:08:01,353 –> 00:08:06,070 However, now we can conclude by the definition of the norm in the Cartesian product
83 00:08:06,270 –> 00:08:11,324 that this convergence here is nothing else then this convergence.
84 00:08:11,524 –> 00:08:18,477 Indeed, you could say here, if you want to converge to the pair, you have to converge in each component
85 00:08:18,677 –> 00:08:23,818 and converging with the two components is exactly what we have written here
86 00:08:24,018 –> 00:08:28,584 and with that I would say the proof is good enough. We have this equivalence here
87 00:08:28,784 –> 00:08:36,281 and I would say this sequence definition here is what you should always have in mind when you talk about closed operators
88 00:08:36,481 –> 00:08:41,359 and then it’s also no problem at all for you to show the following.
89 00:08:41,559 –> 00:08:44,902 Namely let’s consider a bounded operator.
90 00:08:45,186 –> 00:08:49,926 In addition the domain of T should also be the whole space X.
91 00:08:50,126 –> 00:08:55,352 So this is an operator like we had it in our functional analysis course before
92 00:08:55,714 –> 00:09:02,138 and in fact the conclusion here is that such a standard bounded operator is a closed one.
93 00:09:02,338 –> 00:09:06,611 So you should see, this closed operator notion is so important,
94 00:09:06,811 –> 00:09:13,395 because it’s our generalization from the bounded operators to the unbounded operators, as we wanted.
95 00:09:13,595 –> 00:09:20,156 More precisely closed operators still have some nice properties the bounded operators had.
96 00:09:20,356 –> 00:09:24,811 However the details for that is something for the next videos.
97 00:09:25,011 –> 00:09:27,613 In particular we will look at examples,
98 00:09:27,813 –> 00:09:30,268 we will talk about closable operators
99 00:09:30,468 –> 00:09:34,413 and we will have a look at the so called closed graph theorem.
100 00:09:34,613 –> 00:09:39,271 So I really hope we meet again and have a nice day. Bye-bye!
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Last update: 2024-11
