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Title: Examples of Banach Spaces
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Series: Functional Analysis
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YouTube-Title: Functional Analysis 7 | Examples of Banach Spaces
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Bright video: https://youtu.be/GAkVaD1ihi4
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Dark video: https://youtu.be/qjrgiZOnqdE
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Ad-free video: Watch Vimeo video
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Quiz: Test your knowledge
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Exercise Download PDF sheets
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Subtitle on GitHub: fa07_sub_eng.srt
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Timestamps
00:00 Introduction
00:30 One-dimensional example
01:21 Zero-dimensional example
02:10 l^p-space
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Subtitle in English
1 00:00:00,449 –> 00:00:02,269 Hello and welcome back to
2 00:00:02,279 –> 00:00:03,490 functional analysis
3 00:00:03,559 –> 00:00:04,670 and as always, I want to
4 00:00:04,679 –> 00:00:06,099 thank all the nice people
5 00:00:06,110 –> 00:00:07,269 that support this channel
6 00:00:07,280 –> 00:00:08,779 on Steady or PayPal.
7 00:00:09,569 –> 00:00:11,140 Today we have part seven
8 00:00:11,149 –> 00:00:12,829 where we discuss examples
9 00:00:12,840 –> 00:00:14,060 of Banach spaces.
10 00:00:14,739 –> 00:00:16,040 Please recall the picture
11 00:00:16,049 –> 00:00:17,299 from the last time, where
12 00:00:17,309 –> 00:00:18,450 we learned that the Banach
13 00:00:18,489 –> 00:00:19,969 space is on the one hand,
14 00:00:19,979 –> 00:00:21,340 a real or complex vector
15 00:00:21,350 –> 00:00:21,920 space.
16 00:00:21,930 –> 00:00:23,170 And on the other hand, a
17 00:00:23,180 –> 00:00:24,920 complete metric space where
18 00:00:24,930 –> 00:00:26,659 the connection is given by
19 00:00:26,670 –> 00:00:27,280 a norm.
20 00:00:28,159 –> 00:00:29,899 Of course, now we can find
21 00:00:29,909 –> 00:00:31,420 a lot of examples.
22 00:00:32,209 –> 00:00:33,419 Let’s start with the simplest
23 00:00:33,430 –> 00:00:35,139 one, you can just consider
24 00:00:35,150 –> 00:00:36,380 the real number line.
25 00:00:37,380 –> 00:00:38,860 Therefore, we already know
26 00:00:38,889 –> 00:00:40,840 it’s a one-dimensional real
27 00:00:40,849 –> 00:00:41,720 vector space.
28 00:00:42,500 –> 00:00:43,919 And we also know that we
29 00:00:43,930 –> 00:00:45,459 can calculate lengths,
30 00:00:45,540 –> 00:00:47,180 if we consider the absolute
31 00:00:47,189 –> 00:00:48,799 value in the real numbers.
32 00:00:49,630 –> 00:00:50,819 Indeed with our
33 00:00:50,830 –> 00:00:52,400 definition, this is a
34 00:00:52,409 –> 00:00:54,400 norm and then
35 00:00:54,409 –> 00:00:56,180 you should know the associated
36 00:00:56,189 –> 00:00:57,959 metric is given by this
37 00:00:57,970 –> 00:00:58,599 formula.
38 00:00:59,509 –> 00:01:00,909 And that’s just the normal
39 00:01:00,919 –> 00:01:02,349 metric we have in the real
40 00:01:02,360 –> 00:01:02,930 numbers.
41 00:01:03,720 –> 00:01:05,319 And now from real analysis,
42 00:01:05,330 –> 00:01:06,809 you should know that all
43 00:01:06,819 –> 00:01:08,699 the Cauchy sequences indeed
44 00:01:08,709 –> 00:01:09,980 converge in R.
45 00:01:10,830 –> 00:01:12,760 In other words, R together
46 00:01:12,769 –> 00:01:14,180 with this norm is a Banach
47 00:01:14,370 –> 00:01:14,989 space.
48 00:01:15,980 –> 00:01:17,519 So this is our first
49 00:01:17,529 –> 00:01:19,129 and as you can see a very
50 00:01:19,139 –> 00:01:20,209 simple example.
51 00:01:20,839 –> 00:01:22,290 However, we can consider
52 00:01:22,300 –> 00:01:23,910 even a simpler example
53 00:01:24,760 –> 00:01:26,059 and you might already guess
54 00:01:26,069 –> 00:01:27,410 instead of a one-dimensional
55 00:01:27,419 –> 00:01:28,980 vector space, we choose a
56 00:01:28,989 –> 00:01:30,699 zero-dimensional vector space.
57 00:01:31,599 –> 00:01:32,900 This means that the zero
58 00:01:32,910 –> 00:01:34,769 vector is the only vector
59 00:01:34,779 –> 00:01:35,430 in the space.
60 00:01:36,279 –> 00:01:37,760 And of course, there’s only
61 00:01:37,769 –> 00:01:39,699 one norm we can define
62 00:01:39,709 –> 00:01:41,569 here, because the zero
63 00:01:41,580 –> 00:01:43,410 vector has to have length
64 00:01:43,419 –> 00:01:43,930 zero.
65 00:01:44,959 –> 00:01:46,300 Hence everything we need
66 00:01:46,309 –> 00:01:47,569 here is fulfilled.
67 00:01:47,739 –> 00:01:48,889 We have a Banach space.
68 00:01:50,069 –> 00:01:51,430 Of course, this is a very
69 00:01:51,440 –> 00:01:53,160 boring one and not very
70 00:01:53,169 –> 00:01:53,779 helpful.
71 00:01:54,589 –> 00:01:56,080 Indeed, often it’s a
72 00:01:56,089 –> 00:01:58,019 pathological one which means
73 00:01:58,029 –> 00:01:59,730 that some theorem we prove
74 00:01:59,739 –> 00:02:01,680 later for Banach spaces will
75 00:02:01,690 –> 00:02:03,260 not hold for this particular
76 00:02:03,269 –> 00:02:03,739 one here.
77 00:02:04,660 –> 00:02:06,050 Please keep that in mind
78 00:02:06,059 –> 00:02:07,489 if you want to use the zero
79 00:02:07,500 –> 00:02:09,449 Banach space as your example.
80 00:02:10,229 –> 00:02:11,820 However, now let’s continue
81 00:02:11,830 –> 00:02:13,300 with one of the most important
82 00:02:13,309 –> 00:02:14,059 examples.
83 00:02:14,919 –> 00:02:16,889 This is the L^p space written
84 00:02:16,899 –> 00:02:18,190 as L^p
85 00:02:18,380 –> 00:02:19,330 (N, F).
86 00:02:20,089 –> 00:02:21,729 N denotes the natural numbers
87 00:02:21,740 –> 00:02:23,130 and F is our field.
88 00:02:23,139 –> 00:02:25,070 So the real or complex numbers
89 00:02:25,759 –> 00:02:27,339 and p is a real number
90 00:02:27,350 –> 00:02:28,960 greater or equal than one,
91 00:02:29,039 –> 00:02:30,740 but it could also be infinity.
92 00:02:31,550 –> 00:02:33,339 However, infinity is a special
93 00:02:33,350 –> 00:02:34,699 case we consider later.
94 00:02:34,820 –> 00:02:36,600 So here I exclude it.
95 00:02:37,690 –> 00:02:39,020 Now, as I said, this
96 00:02:39,029 –> 00:02:40,949 L^p(N, F) is defined
97 00:02:40,960 –> 00:02:42,509 as all the sequences in
98 00:02:42,520 –> 00:02:44,470 F that fulfill one condition.
99 00:02:45,229 –> 00:02:46,690 So you immediately see the
100 00:02:46,699 –> 00:02:48,009 notation here makes sense.
101 00:02:48,020 –> 00:02:49,610 This is the domain and this
102 00:02:49,619 –> 00:02:51,009 is the codomain for the
103 00:02:51,020 –> 00:02:52,729 map given by the sequence.
104 00:02:53,500 –> 00:02:54,899 And Now this condition is
105 00:02:54,910 –> 00:02:56,559 given by the series where
106 00:02:56,570 –> 00:02:58,039 you add up all the sequence
107 00:02:58,050 –> 00:02:59,960 members and we do it in the
108 00:02:59,970 –> 00:03:01,720 absolute value and to the
109 00:03:01,729 –> 00:03:02,589 power p.
110 00:03:03,740 –> 00:03:05,080 And if this series now
111 00:03:05,089 –> 00:03:06,960 converges, which means it’s
112 00:03:06,970 –> 00:03:08,399 less than infinity, then
113 00:03:08,410 –> 00:03:09,990 the sequence is an L^p.
114 00:03:11,009 –> 00:03:12,449 Often from the context, it’s
115 00:03:12,460 –> 00:03:13,630 clear if we are dealing with
116 00:03:13,639 –> 00:03:15,229 real or complex numbers and
117 00:03:15,240 –> 00:03:16,789 what the index set of the
118 00:03:16,800 –> 00:03:17,820 sequences is.
119 00:03:17,830 –> 00:03:19,520 And therefore one just writes
120 00:03:19,529 –> 00:03:20,860 L^p in this case.
121 00:03:22,059 –> 00:03:23,029 Now, the first thing you
122 00:03:23,039 –> 00:03:24,440 should note is that if we
123 00:03:24,449 –> 00:03:25,869 ignore this condition and
124 00:03:25,880 –> 00:03:27,649 just consider the sequences,
125 00:03:27,759 –> 00:03:29,559 then they form a vector space,
126 00:03:30,240 –> 00:03:32,009 because adding and scaling
127 00:03:32,020 –> 00:03:33,729 is just given in a component
128 00:03:33,740 –> 00:03:34,529 wise sense.
129 00:03:35,449 –> 00:03:36,750 Therefore, here we just have
130 00:03:36,759 –> 00:03:38,410 to show that L^p is a
131 00:03:38,419 –> 00:03:40,279 subspace of this bigger vector
132 00:03:40,289 –> 00:03:40,779 space.
133 00:03:41,990 –> 00:03:43,729 However, what we also want
134 00:03:43,740 –> 00:03:45,229 is a norm for this vector
135 00:03:45,240 –> 00:03:47,100 space and
136 00:03:47,110 –> 00:03:48,710 usually one just uses a
137 00:03:48,720 –> 00:03:50,369 p as an index.
138 00:03:51,250 –> 00:03:52,550 Now if we give the whole
139 00:03:52,559 –> 00:03:54,199 sequence, the name x,
140 00:03:54,500 –> 00:03:56,350 then we can calculate
141 00:03:56,360 –> 00:03:58,190 the p-norm of x
142 00:03:58,199 –> 00:04:00,130 by using this number because
143 00:04:00,139 –> 00:04:01,419 we know it’s finite.
144 00:04:01,929 –> 00:04:03,130 However, you might already
145 00:04:03,139 –> 00:04:04,460 see that the second property
146 00:04:04,470 –> 00:04:05,729 of a norm is not fulfilled.
147 00:04:05,740 –> 00:04:07,559 In this case, we also need
148 00:04:07,570 –> 00:04:09,110 the p-th root outside.
149 00:04:10,080 –> 00:04:11,910 If you set p equals to two,
150 00:04:12,020 –> 00:04:13,380 you will recognize something
151 00:04:13,389 –> 00:04:15,039 similar to the euclidean
152 00:04:15,050 –> 00:04:16,160 norm in (R, d).
153 00:04:17,048 –> 00:04:18,630 Indeed, with similar arguments,
154 00:04:18,640 –> 00:04:20,260 we can show that this now
155 00:04:20,269 –> 00:04:21,459 gives us a norm.
156 00:04:22,350 –> 00:04:23,739 I emphasize that this is
157 00:04:23,750 –> 00:04:25,200 not trivial to show that.
158 00:04:25,260 –> 00:04:26,850 And indeed, I skip that here,
159 00:04:26,859 –> 00:04:28,059 but we will do it later.
160 00:04:29,200 –> 00:04:29,470 Here,
161 00:04:29,480 –> 00:04:30,579 in this video, I want to
162 00:04:30,589 –> 00:04:32,549 focus how we get the completeness
163 00:04:32,559 –> 00:04:33,339 of the space.
164 00:04:34,170 –> 00:04:35,589 Hence, our claim is
165 00:04:35,600 –> 00:04:37,109 L^p together with this
166 00:04:37,119 –> 00:04:38,899 norm is a Banach space.
167 00:04:39,700 –> 00:04:40,980 The first part of the proof
168 00:04:40,989 –> 00:04:42,910 has to be that L^p is an
169 00:04:42,920 –> 00:04:44,700 F-vector space and that this
170 00:04:44,709 –> 00:04:46,589 p-norm is indeed a norm
171 00:04:46,600 –> 00:04:47,079 on it.
172 00:04:47,559 –> 00:04:49,239 I already told you we don’t
173 00:04:49,250 –> 00:04:49,839 do it here.
174 00:04:49,850 –> 00:04:51,549 We assume it, we will prove
175 00:04:51,559 –> 00:04:52,089 it later.
176 00:04:52,619 –> 00:04:53,769 That’s just because we need
177 00:04:53,779 –> 00:04:55,239 a lot of technical details
178 00:04:55,250 –> 00:04:55,959 for this
179 00:04:55,989 –> 00:04:57,779 and I think it will distract
180 00:04:57,790 –> 00:04:59,429 us from the important completeness
181 00:04:59,440 –> 00:05:00,679 part of the proof here.
182 00:05:01,339 –> 00:05:02,559 Now, in order to show the
183 00:05:02,570 –> 00:05:04,070 completeness, we have to
184 00:05:04,079 –> 00:05:05,720 choose an arbitrary
185 00:05:05,950 –> 00:05:06,839 Cauchy sequence.
186 00:05:07,660 –> 00:05:09,079 So this might be hard to
187 00:05:09,089 –> 00:05:09,600 understand.
188 00:05:09,609 –> 00:05:11,600 Now the green x (red x [dark version]) is
189 00:05:11,609 –> 00:05:13,410 a sequence in F and
190 00:05:13,420 –> 00:05:14,920 now we have to consider
191 00:05:14,929 –> 00:05:16,380 sequences of
192 00:05:16,390 –> 00:05:17,200 sequences.
193 00:05:18,059 –> 00:05:19,230 Therefore, I choose here
194 00:05:19,239 –> 00:05:21,149 an upper index k to
195 00:05:21,160 –> 00:05:23,149 denote the different sequences
196 00:05:23,160 –> 00:05:25,029 we have in our Cauchy sequence.
197 00:05:25,750 –> 00:05:27,579 And now we have to show that
198 00:05:27,589 –> 00:05:29,160 all the sequences here in
199 00:05:29,170 –> 00:05:31,149 a row converge to another
200 00:05:31,160 –> 00:05:32,410 sequence, a limit.
201 00:05:33,529 –> 00:05:33,989 OK,
202 00:05:34,000 –> 00:05:35,519 to get the right idea, let’s
203 00:05:35,529 –> 00:05:37,470 write down some of the sequences.
204 00:05:38,410 –> 00:05:39,970 So here you see our Cauchy
205 00:05:39,980 –> 00:05:41,470 sequence which goes from
206 00:05:41,480 –> 00:05:42,320 top to bottom.
207 00:05:43,230 –> 00:05:44,859 Now, for example, our x(1)
208 00:05:44,869 –> 00:05:46,290 is a sequence in L^p,
209 00:05:46,420 –> 00:05:48,079 so it consists of real or
210 00:05:48,089 –> 00:05:49,630 complex numbers in a row.
211 00:05:50,290 –> 00:05:51,350 And the same we can write
212 00:05:51,359 –> 00:05:53,209 down for x(2), x(3) and
213 00:05:53,220 –> 00:05:53,709 so on.
214 00:05:54,410 –> 00:05:55,500 Now, if you look at this
215 00:05:55,510 –> 00:05:57,230 picture, you see a lot of
216 00:05:57,239 –> 00:05:59,109 numbers in F in two different
217 00:05:59,119 –> 00:06:00,799 directions from left to
218 00:06:00,809 –> 00:06:02,500 right and from top to bottom.
219 00:06:03,269 –> 00:06:04,970 And please keep in mind in
220 00:06:04,980 –> 00:06:06,929 the end, we want the limit
221 00:06:06,940 –> 00:06:08,010 here on the left.
222 00:06:09,000 –> 00:06:10,459 In order to find this, we
223 00:06:10,470 –> 00:06:11,799 can just look at the different
224 00:06:11,809 –> 00:06:13,640 sequences we have here from
225 00:06:13,649 –> 00:06:14,549 top to bottom.
226 00:06:15,670 –> 00:06:16,989 So maybe you just take the
227 00:06:17,000 –> 00:06:18,529 fourth column here and then
228 00:06:18,540 –> 00:06:19,890 we look at the difference
229 00:06:19,899 –> 00:06:21,369 between two members of the
230 00:06:21,380 –> 00:06:22,010 sequence.
231 00:06:22,890 –> 00:06:24,470 So maybe we use k
232 00:06:24,600 –> 00:06:26,559 and l for the corresponding
233 00:06:26,570 –> 00:06:27,170 index.
234 00:06:28,040 –> 00:06:29,489 Now we want to put in the
235 00:06:29,500 –> 00:06:31,230 information we already have,
236 00:06:31,540 –> 00:06:32,910 which means we consider here
237 00:06:32,920 –> 00:06:34,079 the absolute value to the
238 00:06:34,089 –> 00:06:35,010 power of p
239 00:06:35,700 –> 00:06:37,420 and then we know this is
240 00:06:37,429 –> 00:06:39,220 less or equal than
241 00:06:39,230 –> 00:06:40,570 summing up all other
242 00:06:40,579 –> 00:06:41,480 possibilities.
243 00:06:42,450 –> 00:06:43,809 That just means that we have
244 00:06:43,820 –> 00:06:45,529 here all the other
245 00:06:45,540 –> 00:06:46,549 columns as well.
246 00:06:47,690 –> 00:06:49,519 Of course, we do that because
247 00:06:49,529 –> 00:06:51,269 here we have our
248 00:06:51,279 –> 00:06:52,579 p-norm to the power p
249 00:06:53,420 –> 00:06:55,119 and there we find our original
250 00:06:55,130 –> 00:06:56,600 sequences again where we
251 00:06:56,609 –> 00:06:58,320 know that they form a Cauchy
252 00:06:58,329 –> 00:06:58,920 sequence.
253 00:06:59,779 –> 00:07:01,200 Don’t worry, I’ll remind
254 00:07:01,209 –> 00:07:02,690 you what it means to be a
255 00:07:02,820 –> 00:07:03,750 Cauchy sequence here.
256 00:07:04,420 –> 00:07:06,209 For all epsilon greater zero,
257 00:07:06,220 –> 00:07:07,950 we find an index and
258 00:07:07,959 –> 00:07:09,209 maybe we call it capital
259 00:07:09,220 –> 00:07:11,070 K such that
260 00:07:11,079 –> 00:07:12,149 for all indices
261 00:07:12,160 –> 00:07:13,750 afterwards k and
262 00:07:13,760 –> 00:07:15,679 l we have that
263 00:07:15,690 –> 00:07:16,799 the distance between two
264 00:07:16,809 –> 00:07:18,679 members here is less than
265 00:07:18,690 –> 00:07:19,350 epsilon.
266 00:07:20,700 –> 00:07:22,390 Now, if you look at the inequality
267 00:07:22,399 –> 00:07:24,220 above, you see that this
268 00:07:24,230 –> 00:07:26,200 thing here is always greater
269 00:07:26,209 –> 00:07:27,279 than just looking at the
270 00:07:27,290 –> 00:07:29,119 absolute value in one column.
271 00:07:30,000 –> 00:07:31,279 Let’s put that in here.
272 00:07:31,290 –> 00:07:32,959 And then you can see this
273 00:07:32,970 –> 00:07:34,799 sequence here of normal
274 00:07:34,809 –> 00:07:36,070 numbers in F.
275 00:07:36,079 –> 00:07:37,670 So this column here is
276 00:07:37,679 –> 00:07:39,500 also a Cauchy sequence in
277 00:07:39,510 –> 00:07:39,940 F.
278 00:07:40,859 –> 00:07:42,390 The absolute value is just the normal
279 00:07:42,399 –> 00:07:43,540 norm in F.
280 00:07:43,549 –> 00:07:44,559 So the real numbers or the
281 00:07:44,570 –> 00:07:45,399 complex numbers.
282 00:07:46,140 –> 00:07:47,390 And we already discussed
283 00:07:47,399 –> 00:07:48,880 this in the first example,
284 00:07:48,929 –> 00:07:50,399 this is a Banach space.
285 00:07:51,470 –> 00:07:53,119 In other words, this
286 00:07:53,130 –> 00:07:54,540 sequence has a limit
287 00:07:55,429 –> 00:07:56,609 and a suitable name should
288 00:07:56,619 –> 00:07:58,589 be x tilde with an index
289 00:07:58,600 –> 00:07:59,010 four.
290 00:08:00,010 –> 00:08:01,910 And this limit in F is what
291 00:08:01,920 –> 00:08:03,829 we find if we go from top
292 00:08:03,839 –> 00:08:05,410 to bottom in the fourth
293 00:08:05,420 –> 00:08:07,109 column, of
294 00:08:07,119 –> 00:08:08,450 course, the four was just
295 00:08:08,459 –> 00:08:09,250 an example.
296 00:08:09,260 –> 00:08:11,029 You can do all the things
297 00:08:11,040 –> 00:08:12,350 here with another column
298 00:08:12,359 –> 00:08:12,940 as well.
299 00:08:12,950 –> 00:08:14,510 So we can introduce an
300 00:08:14,519 –> 00:08:16,100 index m for example,
301 00:08:16,820 –> 00:08:18,079 and then I can change it
302 00:08:18,089 –> 00:08:19,500 here everywhere.
303 00:08:20,209 –> 00:08:21,899 Now, this means for all the
304 00:08:21,910 –> 00:08:23,320 columns here, we find the
305 00:08:23,329 –> 00:08:25,400 corresponding limit x tilde_m
306 00:08:25,409 –> 00:08:26,739 we have here at the
307 00:08:26,750 –> 00:08:27,200 bottom.
308 00:08:28,100 –> 00:08:29,850 So indeed, what we get here
309 00:08:29,859 –> 00:08:31,540 on the right is a new
310 00:08:31,549 –> 00:08:33,159 sequence with numbers in
311 00:08:33,169 –> 00:08:33,549 F.
312 00:08:34,780 –> 00:08:36,140 Of course, we can and we
313 00:08:36,150 –> 00:08:37,429 should give it the name x
314 00:08:37,469 –> 00:08:38,969 tilde, but we don’t know
315 00:08:38,979 –> 00:08:40,530 yet if it’s the limit here
316 00:08:40,539 –> 00:08:41,359 on the left.
317 00:08:42,299 –> 00:08:43,669 This is what we still have
318 00:08:43,679 –> 00:08:44,408 to show now.
319 00:08:45,369 –> 00:08:45,760 OK.
320 00:08:45,770 –> 00:08:47,659 So the plan is to show that
321 00:08:47,669 –> 00:08:49,659 x^(k) minus x tilde
322 00:08:49,669 –> 00:08:51,619 in the p-norm goes to zero.
323 00:08:52,260 –> 00:08:53,599 Or in other words, for an
324 00:08:53,609 –> 00:08:55,489 arbitrary epsilon, this should
325 00:08:55,500 –> 00:08:56,929 be smaller than epsilon.
326 00:08:57,770 –> 00:08:58,119 OK.
327 00:08:58,130 –> 00:08:59,299 So then let’s write down
328 00:08:59,309 –> 00:09:00,229 the p-norm again.
329 00:09:01,099 –> 00:09:02,729 So we have this series and
330 00:09:02,739 –> 00:09:04,309 we need the p-th root again.
331 00:09:04,450 –> 00:09:05,590 But it’s easier to write
332 00:09:05,599 –> 00:09:06,479 the power of p
333 00:09:06,489 –> 00:09:07,390 on the other side.
334 00:09:08,250 –> 00:09:09,559 This just means that we used
335 00:09:09,570 –> 00:09:11,469 to p-th root just in the end.
336 00:09:12,250 –> 00:09:13,400 The first idea now would
337 00:09:13,409 –> 00:09:15,090 be to push the limit process
338 00:09:15,099 –> 00:09:15,940 here outside.
339 00:09:15,950 –> 00:09:17,909 So we write the limit of
340 00:09:17,919 –> 00:09:19,219 N to infinity.
341 00:09:20,140 –> 00:09:21,530 So you see it’s completely
342 00:09:21,539 –> 00:09:22,159 the same.
343 00:09:22,169 –> 00:09:23,710 But now we can discuss this
344 00:09:23,719 –> 00:09:25,359 finite sum here on the right.
345 00:09:26,130 –> 00:09:27,580 However, we know that this
346 00:09:27,590 –> 00:09:29,039 x tilde_n is
347 00:09:29,049 –> 00:09:30,239 also a limit.
348 00:09:30,710 –> 00:09:32,280 Hence, we can push that limit
349 00:09:32,289 –> 00:09:34,260 out as well because the absolute
350 00:09:34,270 –> 00:09:35,989 value is a continuous function
351 00:09:36,000 –> 00:09:37,119 and we just have a finite
352 00:09:37,130 –> 00:09:37,679 sum here.
353 00:09:38,510 –> 00:09:39,830 Now, if you look back, you
354 00:09:39,840 –> 00:09:41,460 will see that this thing
355 00:09:41,469 –> 00:09:42,969 here is what we can keep
356 00:09:42,979 –> 00:09:44,729 as small as we want.
357 00:09:45,599 –> 00:09:46,900 We use again that we have
358 00:09:46,909 –> 00:09:48,320 here the property of the
359 00:09:48,530 –> 00:09:49,349 Cauchy sequence.
360 00:09:50,400 –> 00:09:51,659 Therefore, for our given
361 00:09:51,669 –> 00:09:53,400 epsilon, we choose our capital
362 00:09:53,409 –> 00:09:55,140 K such that we have this
363 00:09:56,039 –> 00:09:56,799 at this point.
364 00:09:56,809 –> 00:09:58,080 In the proof, we can’t know
365 00:09:58,090 –> 00:09:59,640 how small this really should
366 00:09:59,650 –> 00:10:00,099 be.
367 00:10:00,109 –> 00:10:01,559 Therefore, to be careful,
368 00:10:01,570 –> 00:10:03,229 maybe we introduce an epsilon
369 00:10:03,239 –> 00:10:05,080 prime and define it later
370 00:10:05,090 –> 00:10:05,469 on.
371 00:10:06,140 –> 00:10:07,599 Now, what we know from above
372 00:10:07,609 –> 00:10:09,599 is that this whole thing
373 00:10:09,840 –> 00:10:11,609 is less than epsilon prime
374 00:10:11,619 –> 00:10:12,559 to the power p
375 00:10:13,520 –> 00:10:15,330 as long as k and l
376 00:10:15,340 –> 00:10:16,760 are bigger than the capital
377 00:10:16,770 –> 00:10:17,219 K.
378 00:10:17,229 –> 00:10:18,539 It’s the same reasoning as
379 00:10:18,549 –> 00:10:18,900 here.
380 00:10:20,309 –> 00:10:20,679 OK.
381 00:10:20,690 –> 00:10:21,979 So this is good information
382 00:10:21,989 –> 00:10:23,799 for us, because it tells you
383 00:10:23,809 –> 00:10:24,789 we have here the non-negative
384 00:10:25,270 –> 00:10:26,880 numbers that are bounded
385 00:10:26,890 –> 00:10:28,679 from above by this constant.
386 00:10:29,210 –> 00:10:30,640 Of course, then comes some
387 00:10:30,650 –> 00:10:31,710 limit process,
388 00:10:31,750 –> 00:10:33,309 but we know in the worst
389 00:10:33,320 –> 00:10:34,869 case, this limit is
390 00:10:34,880 –> 00:10:36,599 exactly this upper bound.
391 00:10:37,359 –> 00:10:39,320 So in summary, we know for
392 00:10:39,330 –> 00:10:40,859 all k greater or equal capital
393 00:10:40,869 –> 00:10:42,739 K, this whole thing
394 00:10:42,750 –> 00:10:44,609 is less or equal
395 00:10:44,619 –> 00:10:45,780 than epsilon prime to the
396 00:10:45,789 –> 00:10:46,580 power p.
397 00:10:47,440 –> 00:10:48,869 And now is the point to take
398 00:10:48,880 –> 00:10:50,380 the pth root because we get
399 00:10:50,390 –> 00:10:52,119 rid of these two p’s.
400 00:10:53,169 –> 00:10:54,710 And now to get to the correct
401 00:10:54,719 –> 00:10:56,349 result, we need epsilon prime
402 00:10:56,359 –> 00:10:58,229 to be less than epsilon itself,
403 00:10:58,559 –> 00:11:00,109 which means we can define
404 00:11:00,119 –> 00:11:01,909 epsilon prime, for example,
405 00:11:02,030 –> 00:11:03,390 as epsilon half.
406 00:11:04,440 –> 00:11:05,599 And now in the end, you can
407 00:11:05,609 –> 00:11:07,150 put everything together for
408 00:11:07,159 –> 00:11:08,840 an arbitrarily chosen epsilon
409 00:11:08,849 –> 00:11:10,169 greater zero, you find a
410 00:11:10,179 –> 00:11:12,049 capital K such that for
411 00:11:12,059 –> 00:11:13,950 all indices after the capital
412 00:11:13,960 –> 00:11:15,380 K, you find the
413 00:11:15,390 –> 00:11:16,940 inequality here, the
414 00:11:16,950 –> 00:11:18,489 distance between x^(k)
415 00:11:18,500 –> 00:11:20,359 minus x tilde is less
416 00:11:20,369 –> 00:11:22,349 than epsilon and there we
417 00:11:22,359 –> 00:11:23,830 have it x tilde is
418 00:11:23,840 –> 00:11:25,469 indeed the limit of
419 00:11:25,479 –> 00:11:27,190 x^(k) with respect to the
420 00:11:27,200 –> 00:11:27,619 p-norm.
421 00:11:28,909 –> 00:11:30,299 The only thing missing is
422 00:11:30,309 –> 00:11:32,090 the explanation why x tilde
423 00:11:32,099 –> 00:11:33,539 is also an element in
424 00:11:33,549 –> 00:11:34,130 L^p.
425 00:11:34,809 –> 00:11:36,289 However, this now follows
426 00:11:36,299 –> 00:11:38,090 from all we know. We know
427 00:11:38,099 –> 00:11:39,650 here that the difference
428 00:11:39,659 –> 00:11:40,729 is an L^p
429 00:11:41,000 –> 00:11:42,669 and we also know that x^(k)
430 00:11:42,679 –> 00:11:43,570 is an L^p.
431 00:11:44,400 –> 00:11:45,669 Therefore, if you write x
432 00:11:45,679 –> 00:11:47,049 tilde in this way, which
433 00:11:47,059 –> 00:11:48,969 means you subtract x^(k) and
434 00:11:48,979 –> 00:11:49,760 add it again.
435 00:11:49,859 –> 00:11:51,729 Then you see the linear combination,
436 00:11:52,710 –> 00:11:54,609 which means here’s one part
437 00:11:54,619 –> 00:11:56,539 in L^p and here’s one
438 00:11:56,549 –> 00:11:57,530 part in L^p.
439 00:11:58,690 –> 00:12:00,159 Hence also the sum is an
440 00:12:00,169 –> 00:12:02,070 L^p because we are dealing
441 00:12:02,080 –> 00:12:03,179 with a vector space.
442 00:12:04,109 –> 00:12:05,239 That’s the part of the proof
443 00:12:05,250 –> 00:12:06,409 at the beginning where we
444 00:12:06,419 –> 00:12:07,539 skipped the details,
445 00:12:07,549 –> 00:12:08,780 but here we can use it.
446 00:12:09,570 –> 00:12:11,049 Of course, we will discuss
447 00:12:11,059 –> 00:12:12,690 this even further in upcoming
448 00:12:12,700 –> 00:12:13,489 videos,
449 00:12:13,500 –> 00:12:14,979 but at this point, I really
450 00:12:14,989 –> 00:12:16,830 wanted to show you an important
451 00:12:16,849 –> 00:12:18,549 and non-trivial example of
452 00:12:18,559 –> 00:12:19,349 a Banach space.
453 00:12:20,200 –> 00:12:20,599 OK,
454 00:12:20,609 –> 00:12:21,580 I hope I see you in the next
455 00:12:21,590 –> 00:12:23,369 videos where I first explain
456 00:12:23,380 –> 00:12:24,869 more concepts in functional
457 00:12:24,880 –> 00:12:25,580 analysis
458 00:12:25,590 –> 00:12:27,099 and afterwards we go into
459 00:12:27,109 –> 00:12:28,940 more details for examples.
460 00:12:29,919 –> 00:12:31,440 So thanks for listening and
461 00:12:31,450 –> 00:12:32,210 see you later.
462 00:12:32,219 –> 00:12:32,859 Bye.
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Quiz Content
Q1: Is $\mathbb{R}$ given with $| x | = 2 |x|$ a Banach space?
A1: Yes!
A2: No!
A3: One needs more information.
Q2: Is $\mathbb{R}^2$ given with $\left| \binom{x_1}{x_2} \right| = \max(x_1, x_2)$ a Banach space?
A1: No!
A2: Yes!
A3: One needs more information.
Q3: Is $\mathbb{R}^2$ given with $\left| \binom{x_1}{x_2} \right| = |x_1| + |x_2|$ a Banach space?
A1: Yes!
A2: No!
A3: One needs more information.
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Last update: 2024-10
