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Title: Example
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Series: Unbounded Operators
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YouTube-Title: Unbounded Operators 5 | Example
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Bright video: https://youtu.be/9BG1TjzyL-E
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Subtitle on GitHub: uo05_sub_eng.srt
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Timestamps (n/a)
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Subtitle in English
1 00:00:00,449 –> 00:00:02,109 Hello and welcome back
2 00:00:02,119 –> 00:00:04,000 to unbounded operators,
3 00:00:04,019 –> 00:00:05,260 the video series where we
4 00:00:05,269 –> 00:00:06,880 talk a lot about some nice
5 00:00:06,889 –> 00:00:09,179 topics of functional analysis.
6 00:00:09,510 –> 00:00:12,630 And indeed, in today’s part 5 we will continue with
7 00:00:12,640 –> 00:00:14,539 the closable operators by
8 00:00:14,550 –> 00:00:16,068 looking at an example.
9 00:00:16,629 –> 00:00:17,879 But you might already know
10 00:00:17,889 –> 00:00:19,190 before we can do that,
11 00:00:19,200 –> 00:00:20,639 I first want to thank all
12 00:00:20,649 –> 00:00:21,889 the nice people who support
13 00:00:21,899 –> 00:00:23,059 the channel on Steady here
14 00:00:23,069 –> 00:00:24,870 on YouTube or on Patreon.
15 00:00:25,610 –> 00:00:26,950 And please don’t forget on
16 00:00:26,959 –> 00:00:27,950 these platforms
17 00:00:27,959 –> 00:00:29,309 you find PDF versions,
18 00:00:29,319 –> 00:00:32,939 quizzes and early access to all new videos.
19 00:00:33,580 –> 00:00:34,000 OK.
20 00:00:34,009 –> 00:00:35,310 Then let’s start this video
21 00:00:35,319 –> 00:00:36,970 here by considering a
22 00:00:36,979 –> 00:00:38,950 closable operator T
23 00:00:39,560 –> 00:00:40,779 and there you already know
24 00:00:40,790 –> 00:00:42,290 for the definition, we need
25 00:00:42,299 –> 00:00:44,790 two normed spaces X and Y.
26 00:00:45,169 –> 00:00:46,220 And at this point, I can
27 00:00:46,229 –> 00:00:47,849 already tell you in order
28 00:00:47,860 –> 00:00:49,580 to get nice theorems
29 00:00:49,590 –> 00:00:51,380 out, we have to assume that
30 00:00:51,389 –> 00:00:52,729 X and Y are Banach
31 00:00:52,750 –> 00:00:53,549 spaces.
32 00:00:53,659 –> 00:00:54,909 So complete normed
33 00:00:54,919 –> 00:00:55,740 spaces.
34 00:00:56,500 –> 00:00:58,439 In fact, also in this example
35 00:00:58,450 –> 00:01:00,209 today, we will take Banach
36 00:01:00,240 –> 00:01:00,959 spaces.
37 00:01:01,770 –> 00:01:03,180 But let’s first write down
38 00:01:03,189 –> 00:01:04,459 the general sequence
39 00:01:04,470 –> 00:01:06,080 characterization for a
40 00:01:06,089 –> 00:01:07,419 closable operator.
41 00:01:08,110 –> 00:01:09,680 There, we just have to look
42 00:01:09,690 –> 00:01:11,129 at sequences that
43 00:01:11,139 –> 00:01:12,769 converge to the origin
44 00:01:13,400 –> 00:01:14,819 and they have to have the
45 00:01:14,830 –> 00:01:16,139 property that the
46 00:01:16,150 –> 00:01:17,449 images also
47 00:01:17,459 –> 00:01:19,180 converge and
48 00:01:19,190 –> 00:01:21,059 then the conclusion is that
49 00:01:21,069 –> 00:01:22,860 only the zero vector in
50 00:01:22,870 –> 00:01:24,739 Y is possible for this
51 00:01:24,750 –> 00:01:25,260 limit.
52 00:01:26,000 –> 00:01:27,279 Therefore, please remember
53 00:01:27,290 –> 00:01:28,699 here, if we have this
54 00:01:28,709 –> 00:01:30,580 implication, we know that
55 00:01:30,589 –> 00:01:32,430 the operator T is closable.
56 00:01:33,410 –> 00:01:34,540 And this is exactly what
57 00:01:34,550 –> 00:01:36,099 we now apply to the
58 00:01:36,110 –> 00:01:37,650 following example.
59 00:01:38,139 –> 00:01:39,319 And in order to make it very
60 00:01:39,330 –> 00:01:41,089 concrete, let’s choose a
61 00:01:41,099 –> 00:01:42,919 very nice Banach space for
62 00:01:42,930 –> 00:01:44,900 X. Namely
63 00:01:44,910 –> 00:01:46,739 X should be given as lower
64 00:01:46,750 –> 00:01:47,940 case l^2.
65 00:01:48,680 –> 00:01:50,099 So you know this is a space
66 00:01:50,110 –> 00:01:51,629 of sequences with
67 00:01:51,639 –> 00:01:53,400 index set N and
68 00:01:53,410 –> 00:01:54,809 entries in C.
69 00:01:55,650 –> 00:01:56,989 And please recall from the
70 00:01:57,000 –> 00:01:58,699 functional analysis series
71 00:01:58,760 –> 00:02:00,230 that this is a well defined
72 00:02:00,239 –> 00:02:01,150 Banach space.
73 00:02:01,830 –> 00:02:03,569 Moreover, it’s even a Hilbert
74 00:02:03,580 –> 00:02:04,940 space, but this is not so
75 00:02:04,949 –> 00:02:06,129 important right now.
76 00:02:06,849 –> 00:02:07,349 OK.
77 00:02:07,370 –> 00:02:09,089 In addition, let’s also fix
78 00:02:09,100 –> 00:02:10,660 the canonical unit vectors
79 00:02:10,669 –> 00:02:13,559 denoted by e_1, e_2, e_3 and so on.
80 00:02:14,470 –> 00:02:15,889 Now, if you don’t know them,
81 00:02:15,899 –> 00:02:17,369 then maybe very quickly,
82 00:02:17,380 –> 00:02:18,550 e_2 is given as the
83 00:02:18,559 –> 00:02:19,479 sequence
84 00:02:19,850 –> 00:02:22,660 0100
85 00:02:22,669 –> 00:02:23,410 and so on.
86 00:02:23,850 –> 00:02:25,309 In other words, everything
87 00:02:25,320 –> 00:02:26,929 is 0 in the sequence
88 00:02:26,940 –> 00:02:28,649 except the second position,
89 00:02:28,660 –> 00:02:29,610 which is a 1.
90 00:02:30,389 –> 00:02:31,770 So you see this is exactly
91 00:02:31,779 –> 00:02:33,009 the same as you would have
92 00:02:33,020 –> 00:02:34,110 it in a finite dimensional
93 00:02:34,119 –> 00:02:35,860 vector space, but now in
94 00:02:35,869 –> 00:02:37,279 infinite dimensions.
95 00:02:37,990 –> 00:02:38,429 OK.
96 00:02:38,440 –> 00:02:39,699 And now I want the operator
97 00:02:39,710 –> 00:02:41,490 T going from X to
98 00:02:41,500 –> 00:02:42,940 C and
99 00:02:42,949 –> 00:02:44,919 obviously C is also a normed
100 00:02:44,929 –> 00:02:46,339 space, even a Banach
101 00:02:46,520 –> 00:02:47,039 space.
102 00:02:47,839 –> 00:02:49,259 Then in the next step, I
103 00:02:49,270 –> 00:02:50,589 want to define the domain
104 00:02:50,600 –> 00:02:52,410 of T and this
105 00:02:52,419 –> 00:02:54,179 should be given as the span
106 00:02:54,190 –> 00:02:55,520 of the canonical unit
107 00:02:55,529 –> 00:02:56,220 vectors.
108 00:02:57,139 –> 00:02:58,630 This means all the linear
109 00:02:58,639 –> 00:03:00,220 combinations of the canonical
110 00:03:00,229 –> 00:03:01,919 unit vectors lie in the
111 00:03:01,929 –> 00:03:02,960 domain of T.
112 00:03:03,639 –> 00:03:04,979 And here please note this
113 00:03:04,990 –> 00:03:06,660 is not equal to the whole
114 00:03:06,669 –> 00:03:08,300 space X, but it’s a
115 00:03:08,309 –> 00:03:09,720 dense subset in it.
116 00:03:10,520 –> 00:03:11,000 OK.
117 00:03:11,009 –> 00:03:12,500 And now this domain makes
118 00:03:12,509 –> 00:03:14,139 it very easy to write
119 00:03:14,149 –> 00:03:15,949 down a nice definition for
120 00:03:15,960 –> 00:03:16,960 an operator T.
121 00:03:17,570 –> 00:03:19,550 Namely we just have to say
122 00:03:19,559 –> 00:03:21,539 what D does to a canonical
123 00:03:21,550 –> 00:03:22,320 unit vector.
124 00:03:23,089 –> 00:03:24,350 And for example, we could
125 00:03:24,360 –> 00:03:25,639 just say take the
126 00:03:25,649 –> 00:03:27,509 corresponding index of the
127 00:03:27,520 –> 00:03:28,320 unit vector.
128 00:03:29,119 –> 00:03:30,759 Therefore T would
129 00:03:30,770 –> 00:03:33,300 send e_2 to the number 2
130 00:03:33,830 –> 00:03:34,940 and now, since we want a
131 00:03:34,949 –> 00:03:36,729 linear map, the extension
132 00:03:36,740 –> 00:03:38,440 to the whole span here is
133 00:03:38,449 –> 00:03:39,570 uniquely given.
134 00:03:40,020 –> 00:03:41,649 But for the sake of completeness,
135 00:03:41,660 –> 00:03:43,000 let’s write it down.
136 00:03:43,729 –> 00:03:45,589 So let’s say we have a finite
137 00:03:45,600 –> 00:03:48,719 linear combination lambda_j times e_j.
138 00:03:49,190 –> 00:03:50,490 So please note this here
139 00:03:50,500 –> 00:03:52,070 is always a finite sum,
140 00:03:52,080 –> 00:03:53,110 because this is what the
141 00:03:53,119 –> 00:03:54,089 span means.
142 00:03:54,100 –> 00:03:56,080 It means the set of the finite
143 00:03:56,089 –> 00:03:57,990 linear combinations of these
144 00:03:58,000 –> 00:03:58,710 vectors.
145 00:03:59,270 –> 00:04:00,279 And therefore the mapping
146 00:04:00,289 –> 00:04:01,559 here is totally clear, we
147 00:04:01,570 –> 00:04:04,500 just add up lambda_j times j
148 00:04:05,289 –> 00:04:06,880 This gives us a complex number
149 00:04:06,889 –> 00:04:08,979 which is the output of T.
150 00:04:09,619 –> 00:04:10,029 OK.
151 00:04:10,039 –> 00:04:11,160 And the first question from
152 00:04:11,169 –> 00:04:12,960 my side would be is
153 00:04:12,970 –> 00:04:14,529 this an unbounded or a
154 00:04:14,539 –> 00:04:15,660 bounded operator?
155 00:04:16,250 –> 00:04:17,750 To answer that we can just
156 00:04:17,760 –> 00:04:19,559 calculate the operator norm
157 00:04:19,570 –> 00:04:20,343 of T.
158 00:04:20,759 –> 00:04:22,609 By definition, this is the
159 00:04:22,619 –> 00:04:24,410 supremum over all
160 00:04:24,420 –> 00:04:26,760 inputs X with norm 1.
162 00:04:27,459 –> 00:04:29,220 And then we look at the norm
163 00:04:29,230 –> 00:04:30,480 of the images.
164 00:04:31,000 –> 00:04:32,779 Which in this case is the
165 00:04:32,790 –> 00:04:33,890 norm of C.
166 00:04:34,700 –> 00:04:36,040 But of course, the norm in
167 00:04:36,049 –> 00:04:37,820 C is just the absolute
168 00:04:37,829 –> 00:04:39,709 value of complex numbers.
169 00:04:40,230 –> 00:04:41,649 And now let’s say instead
170 00:04:41,660 –> 00:04:43,459 of every vector x, we just
171 00:04:43,470 –> 00:04:45,089 put in the canonical unit
172 00:04:45,100 –> 00:04:46,980 vectors. Then the
173 00:04:46,989 –> 00:04:48,880 whole supremum cannot get
174 00:04:48,890 –> 00:04:49,440 bigger.
175 00:04:49,760 –> 00:04:51,450 Then everything is much simpler.
176 00:04:51,459 –> 00:04:52,850 We just have to calculate
177 00:04:52,859 –> 00:04:54,279 the absolute value of
178 00:04:54,290 –> 00:04:55,200 Te_j.
179 00:04:55,700 –> 00:04:56,869 But by the definition of
180 00:04:56,880 –> 00:04:58,600 the operator T, we already
181 00:04:58,609 –> 00:05:00,770 know this is just the number j.
182 00:05:01,450 –> 00:05:02,859 And at this point, we already
183 00:05:02,869 –> 00:05:04,470 see the supremum over
184 00:05:04,480 –> 00:05:06,269 j is not finite.
185 00:05:06,940 –> 00:05:08,480 Therefore, the operator norm
186 00:05:08,489 –> 00:05:10,279 of T can also not be
187 00:05:10,290 –> 00:05:10,959 finite.
188 00:05:11,660 –> 00:05:12,869 So the first conclusion here
189 00:05:12,880 –> 00:05:14,660 is we have an unbounded
190 00:05:14,670 –> 00:05:15,359 operator.
191 00:05:16,089 –> 00:05:17,579 Therefore my next question
192 00:05:17,589 –> 00:05:19,149 would be, is it at least a
193 00:05:19,160 –> 00:05:20,660 closable operator?
194 00:05:21,109 –> 00:05:22,500 And in order to answer this
195 00:05:22,510 –> 00:05:24,380 question, we will use our
196 00:05:24,390 –> 00:05:26,160 sequence property for closable
197 00:05:26,170 –> 00:05:27,820 operators from before.
198 00:05:28,369 –> 00:05:29,859 And we can combine that with
199 00:05:29,869 –> 00:05:31,179 the fact that we have an
200 00:05:31,190 –> 00:05:32,660 operator which is not
201 00:05:32,670 –> 00:05:33,980 continuous at the
202 00:05:33,989 –> 00:05:34,690 origin.
203 00:05:35,089 –> 00:05:36,640 So please never forget for
204 00:05:36,649 –> 00:05:38,450 linear maps, boundedness
205 00:05:38,459 –> 00:05:39,959 and continuity are
206 00:05:39,970 –> 00:05:40,720 connected.
207 00:05:41,470 –> 00:05:41,920 OK.
208 00:05:41,929 –> 00:05:43,250 Then let’s use this fact
209 00:05:43,260 –> 00:05:44,799 here for a sequence that
210 00:05:44,809 –> 00:05:46,079 converges to zero.
211 00:05:46,739 –> 00:05:48,309 So what we can do is to choose
212 00:05:48,320 –> 00:05:50,040 a sequence inside the
213 00:05:50,049 –> 00:05:52,019 domain of T that converges
214 00:05:52,029 –> 00:05:53,019 to the origin.
215 00:05:53,309 –> 00:05:54,989 And now we know that we have
216 00:05:55,000 –> 00:05:56,679 at least one sequence with
217 00:05:56,690 –> 00:05:58,040 the property, that the
218 00:05:58,049 –> 00:05:59,839 images don’t converge to
219 00:05:59,850 –> 00:06:00,359 zero.
220 00:06:01,049 –> 00:06:02,980 The operator T is not continuous
221 00:06:02,989 –> 00:06:03,690 at zero.
222 00:06:03,769 –> 00:06:05,070 So this is possible
223 00:06:05,760 –> 00:06:07,350 however, still it could
224 00:06:07,359 –> 00:06:09,170 happen that the images here
225 00:06:09,179 –> 00:06:11,070 accumulate at zero.
226 00:06:11,510 –> 00:06:13,190 But we also know that for
227 00:06:13,200 –> 00:06:14,790 infinitely many sequence
228 00:06:14,799 –> 00:06:16,500 members that does not happen.
229 00:06:17,000 –> 00:06:18,459 So it’s possible to choose
230 00:06:18,470 –> 00:06:20,369 a safety distant epsilon
231 00:06:20,380 –> 00:06:21,269 from zero
232 00:06:22,160 –> 00:06:23,739 and a subsequence
233 00:06:23,750 –> 00:06:25,100 x_nk
234 00:06:25,480 –> 00:06:27,239 such that the images
235 00:06:27,250 –> 00:06:29,019 stay away from zero
236 00:06:29,029 –> 00:06:30,769 with distance epsilon.
237 00:06:31,290 –> 00:06:32,709 So the absolute value is
238 00:06:32,720 –> 00:06:34,660 greater or equal than our
239 00:06:34,670 –> 00:06:35,260 epsilon.
240 00:06:35,899 –> 00:06:37,220 Therefore, now we can
241 00:06:37,230 –> 00:06:39,029 conclude it’s allowed to
242 00:06:39,040 –> 00:06:40,720 divide by this number.
243 00:06:41,209 –> 00:06:42,829 Hence, we can define a new
244 00:06:42,839 –> 00:06:44,679 sequence and let’s call it
245 00:06:44,690 –> 00:06:45,640 z_k.
246 00:06:46,079 –> 00:06:48,000 So we take x_nk and then
247 00:06:48,010 –> 00:06:49,950 we divide this by the number
248 00:06:49,959 –> 00:06:50,869 Tx_nk.
249 00:06:51,540 –> 00:06:53,000 So this one is definitely
250 00:06:53,010 –> 00:06:54,850 a well-defined element in
251 00:06:54,859 –> 00:06:55,959 the domain of T.
252 00:06:56,609 –> 00:06:58,549 And moreover, if we increase
253 00:06:58,559 –> 00:07:00,269 k, we also see that
254 00:07:00,279 –> 00:07:02,250 this sequence tends to zero.
255 00:07:02,839 –> 00:07:04,450 So in this regard, it works
256 00:07:04,459 –> 00:07:06,070 the same as our original
257 00:07:06,079 –> 00:07:07,309 sequence x_n.
258 00:07:08,140 –> 00:07:10,049 However, for the images,
259 00:07:10,059 –> 00:07:11,850 something different happens,
260 00:07:12,510 –> 00:07:13,970 namely, we immediately
261 00:07:13,980 –> 00:07:16,269 see that we divide 1 by 1.
262 00:07:16,279 –> 00:07:17,769 So the outcome is always
263 00:07:17,779 –> 00:07:19,750
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264 00:07:19,760 –> 00:07:21,549 k from N we choose.
265 00:07:22,220 –> 00:07:23,730 And with that, we now have
266 00:07:23,739 –> 00:07:25,579 our counter example for the
267 00:07:25,589 –> 00:07:27,570 characterization for a closable
268 00:07:27,709 –> 00:07:28,940 operator with
269 00:07:28,950 –> 00:07:29,720 sequences.
270 00:07:30,500 –> 00:07:31,649 In fact, we can immediately
271 00:07:31,660 –> 00:07:32,619 compare it here.
272 00:07:32,670 –> 00:07:34,519 Now we have a sequence in
273 00:07:34,529 –> 00:07:36,130 the domain of T which
274 00:07:36,140 –> 00:07:37,579 converges to zero
275 00:07:37,839 –> 00:07:39,470 and the images also
276 00:07:39,480 –> 00:07:40,859 converge to a y,
277 00:07:41,429 –> 00:07:43,220 in particular, the y in
278 00:07:43,230 –> 00:07:45,620 our case is given by 1
279 00:07:46,019 –> 00:07:47,660 and that’s already the contradiction
280 00:07:47,670 –> 00:07:49,630 because y has to be equal
281 00:07:49,640 –> 00:07:51,510 to zero if the operator
282 00:07:51,519 –> 00:07:52,399 T is closable.
283 00:07:53,209 –> 00:07:54,700 And with that, we can make
284 00:07:54,709 –> 00:07:56,700 our conclusion T is not
285 00:07:56,709 –> 00:07:58,040 a closable operator.
286 00:07:58,730 –> 00:08:00,649 And this also includes that
287 00:08:00,660 –> 00:08:02,269 T is not a closed operator
288 00:08:02,279 –> 00:08:02,910 as well.
289 00:08:03,459 –> 00:08:04,989 Therefore, this is important
290 00:08:05,000 –> 00:08:06,510 to remember in infinite
291 00:08:06,519 –> 00:08:08,290 dimensions, we can easily
292 00:08:08,299 –> 00:08:09,579 define unbounded
293 00:08:09,589 –> 00:08:11,290 operators that are not even
294 00:08:11,299 –> 00:08:11,690 closable.
295 00:08:12,589 –> 00:08:13,910 But I can already tell you
296 00:08:13,920 –> 00:08:15,410 in this course, we will also
297 00:08:15,420 –> 00:08:16,989 see a lot of closed
298 00:08:17,000 –> 00:08:17,709 operators.
299 00:08:17,720 –> 00:08:19,549 So a lot of positive examples.
300 00:08:20,369 –> 00:08:20,790 OK.
301 00:08:20,799 –> 00:08:21,790 Then I really hope we meet
302 00:08:21,799 –> 00:08:23,190 again and have a nice day.
303 00:08:23,200 –> 00:08:23,910 Bye bye.
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Last update: 2024-11
