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Title: Cauchy Sequences and Complete Metric Spaces
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Series: Functional Analysis
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YouTube-Title: Functional Analysis 5 | Cauchy Sequences and Complete Spaces
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Quiz: Test your knowledge
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Exercise Download PDF sheets
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Subtitle on GitHub: fa05_sub_eng.srt
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Timestamps
00:00 Introduction
02:30 Cauchy sequences
03:34 Complete metric spaces
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Subtitle in English
1 00:00:00,540 –> 00:00:02,400 Hello and welcome back to
2 00:00:02,410 –> 00:00:03,650 functional analysis.
3 00:00:03,769 –> 00:00:04,960 And as always, I want to
4 00:00:04,969 –> 00:00:06,400 thank all the nice people
5 00:00:06,409 –> 00:00:07,469 that support the channel
6 00:00:07,480 –> 00:00:08,789 on Steady or PayPal.
7 00:00:09,619 –> 00:00:11,159 Today’s part five is about
8 00:00:11,170 –> 00:00:13,020 Cauchy sequences and complete
9 00:00:13,029 –> 00:00:14,000 metric spaces.
10 00:00:14,979 –> 00:00:16,318 And as promised in the last
11 00:00:16,329 –> 00:00:18,209 video, we start here with
12 00:00:18,219 –> 00:00:19,159 an example.
13 00:00:19,719 –> 00:00:21,229 Our metric space has as a
14 00:00:21,239 –> 00:00:23,159 set the interval from 0
15 00:00:23,170 –> 00:00:24,780 to 3 with the normal
16 00:00:24,790 –> 00:00:26,319 metric for the real numbers.
17 00:00:26,969 –> 00:00:28,350 Hence this is something you
18 00:00:28,360 –> 00:00:29,190 already know.
19 00:00:30,180 –> 00:00:32,000 What you also know is that
20 00:00:32,009 –> 00:00:33,919 the subset 0 to 3, which
21 00:00:33,930 –> 00:00:35,360 is the whole space, is a
22 00:00:35,369 –> 00:00:37,240 closed set inside
23 00:00:37,250 –> 00:00:38,279 this metric space.
24 00:00:39,110 –> 00:00:40,630 For example, you could argue
25 00:00:40,639 –> 00:00:41,880 that the complement which
26 00:00:41,889 –> 00:00:43,369 is the empty set is clearly
27 00:00:43,380 –> 00:00:43,830 open.
28 00:00:44,630 –> 00:00:46,229 Or you can use the fact from
29 00:00:46,240 –> 00:00:47,630 the last video where you
30 00:00:47,639 –> 00:00:49,150 consider any convergence
31 00:00:49,159 –> 00:00:51,150 sequence with members from
32 00:00:51,159 –> 00:00:52,930 the subset. Being a
33 00:00:52,939 –> 00:00:54,430 convergent sequence means
34 00:00:54,439 –> 00:00:56,099 it has a limit which we call
35 00:00:56,110 –> 00:00:56,750 x tilde
36 00:00:56,759 –> 00:00:57,900 and of course, it has to
37 00:00:57,909 –> 00:00:58,650 lie in X.
38 00:00:59,430 –> 00:01:00,599 What we proved last time
39 00:01:00,610 –> 00:01:02,069 was now; the set is closed,
40 00:01:02,080 –> 00:01:03,849 if the limit is also in the
41 00:01:03,860 –> 00:01:05,800 subset. Of course here you
42 00:01:05,809 –> 00:01:06,970 see immediately that this
43 00:01:06,980 –> 00:01:08,720 is true, because the subset
44 00:01:08,730 –> 00:01:10,489 (0,3) is the whole space
45 00:01:10,519 –> 00:01:10,970 X.
46 00:01:11,809 –> 00:01:13,050 Now a question that often
47 00:01:13,059 –> 00:01:14,750 arises with such an example
48 00:01:14,760 –> 00:01:15,629 is the following.
49 00:01:16,550 –> 00:01:18,129 what is about the sequence
50 00:01:18,139 –> 00:01:19,319 1 over N?
51 00:01:19,919 –> 00:01:21,529 Clearly this is a sequence
52 00:01:21,540 –> 00:01:23,169 in X. First member is
53 00:01:23,180 –> 00:01:24,660 1, next one is 1/2
54 00:01:24,669 –> 00:01:25,919 and so on.
55 00:01:25,930 –> 00:01:27,199 You see all the members are
56 00:01:27,209 –> 00:01:29,190 positive, so they lie in
57 00:01:29,199 –> 00:01:31,160 this interval and the
58 00:01:31,169 –> 00:01:32,599 distance between two members
59 00:01:32,610 –> 00:01:33,519 of the sequence.
60 00:01:33,529 –> 00:01:34,779 So maybe let’s call them
61 00:01:34,790 –> 00:01:36,160 x_n, x_m.
62 00:01:36,480 –> 00:01:37,959 This number gets
63 00:01:37,970 –> 00:01:39,730 smaller and smaller when
64 00:01:39,739 –> 00:01:41,599 x and m get larger and
65 00:01:41,610 –> 00:01:42,120 larger.
66 00:01:42,889 –> 00:01:44,150 So we could write it in this
67 00:01:44,160 –> 00:01:44,550 way,
68 00:01:44,620 –> 00:01:45,650 but the important thing is
69 00:01:45,660 –> 00:01:47,029 that you see that you get
70 00:01:47,040 –> 00:01:48,959 closer and closer to something
71 00:01:48,970 –> 00:01:50,019 with the sequence.
72 00:01:50,589 –> 00:01:52,360 However, still, it does
73 00:01:52,370 –> 00:01:53,430 not converge.
74 00:01:54,160 –> 00:01:55,440 For our example, this is
75 00:01:55,449 –> 00:01:57,059 easy to see, because the
76 00:01:57,069 –> 00:01:58,760 only possible limit this
77 00:01:58,769 –> 00:01:59,699 sequence could have in the
78 00:01:59,709 –> 00:02:01,099 real numbers would be the
79 00:02:01,110 –> 00:02:01,959 number zero.
80 00:02:02,480 –> 00:02:04,260 However, zero is not in
81 00:02:04,269 –> 00:02:05,889 X, there is no number
82 00:02:05,900 –> 00:02:07,389 zero in our universe
83 00:02:07,400 –> 00:02:07,699 here.
84 00:02:08,419 –> 00:02:10,309 In summary, we found a metric
85 00:02:10,320 –> 00:02:11,770 space which owns a
86 00:02:11,779 –> 00:02:13,259 sequence, which should
87 00:02:13,270 –> 00:02:14,869 converge but there is no
88 00:02:14,880 –> 00:02:16,539 point in the space, where
89 00:02:16,550 –> 00:02:17,660 the sequence leads to.
90 00:02:18,500 –> 00:02:19,839 Therefore, one could say
91 00:02:19,850 –> 00:02:21,470 there is a hole in the space,
92 00:02:21,610 –> 00:02:23,080 but we will say the space
93 00:02:23,089 –> 00:02:24,149 is not complete.
94 00:02:25,050 –> 00:02:26,500 So now we have to give a
95 00:02:26,509 –> 00:02:28,050 formal definition for a
96 00:02:28,059 –> 00:02:29,550 complete metric space.
97 00:02:30,330 –> 00:02:32,220 First, for given metric space
98 00:02:32,229 –> 00:02:34,179 (X, d) sequences
99 00:02:34,190 –> 00:02:35,729 that fulfill this property
100 00:02:35,740 –> 00:02:37,389 here are called cauchy
101 00:02:37,399 –> 00:02:38,160 sequences.
102 00:02:39,100 –> 00:02:39,949 More concretely
103 00:02:39,960 –> 00:02:41,630 this means for all
104 00:02:41,639 –> 00:02:43,440 epsilon greater than zero,
105 00:02:44,270 –> 00:02:45,990 there exists an index capital
106 00:02:46,000 –> 00:02:47,979 N in N, such
107 00:02:47,990 –> 00:02:49,669 that for all indices
108 00:02:49,869 –> 00:02:51,839 n, m greater than
109 00:02:51,850 –> 00:02:53,410 this index N, we
110 00:02:53,419 –> 00:02:55,020 have that the
111 00:02:55,029 –> 00:02:56,289 distance between the members
112 00:02:56,300 –> 00:02:58,080 x_n, x_m is arbitrarily
113 00:02:58,089 –> 00:02:59,940 small, so less than epsilon.
114 00:03:00,729 –> 00:03:02,240 So this is the actual meaning
115 00:03:02,250 –> 00:03:03,479 of this limit before.
116 00:03:04,270 –> 00:03:05,210 What you should be able to
117 00:03:05,220 –> 00:03:06,869 show now is if you have a
118 00:03:06,880 –> 00:03:08,509 convergent sequence, then
119 00:03:08,520 –> 00:03:10,289 this thing is also fulfilled.
120 00:03:11,110 –> 00:03:12,729 The Cauchy sequence is therefore
121 00:03:12,740 –> 00:03:14,419 always a generalization of
122 00:03:14,429 –> 00:03:15,600 a convergence sequence.
123 00:03:16,330 –> 00:03:18,000 So you see there’s no limit,
124 00:03:18,009 –> 00:03:19,720 there’s no x tilde in the
125 00:03:19,729 –> 00:03:21,559 definition of a Cauchy sequence.
126 00:03:22,410 –> 00:03:23,899 Hence, it would be much nicer
127 00:03:23,910 –> 00:03:25,539 working with this definition
128 00:03:25,610 –> 00:03:26,910 than working with the one
129 00:03:26,919 –> 00:03:28,300 for the convergent sequences.
130 00:03:29,029 –> 00:03:30,289 However, we already know
131 00:03:30,300 –> 00:03:32,089 the problem. This does not
132 00:03:32,100 –> 00:03:33,089 work in general.
133 00:03:34,100 –> 00:03:35,589 Now the nice spaces where
134 00:03:35,600 –> 00:03:37,220 this actually works, we call
135 00:03:37,229 –> 00:03:37,820 complete.
136 00:03:38,679 –> 00:03:40,350 Therefore, we say here all
137 00:03:40,360 –> 00:03:42,100 Cauchy sequences converge.
138 00:03:43,029 –> 00:03:44,250 In other words, something
139 00:03:44,259 –> 00:03:46,110 like above can’t happen then
140 00:03:47,080 –> 00:03:48,220 and of course, you already
141 00:03:48,229 –> 00:03:49,910 know how to fix the example
142 00:03:49,919 –> 00:03:50,399 above.
143 00:03:51,029 –> 00:03:52,660 We could change the set X
144 00:03:52,729 –> 00:03:54,470 when we include zero and
145 00:03:54,479 –> 00:03:55,089 three.
146 00:03:55,110 –> 00:03:56,660 Then of course, now the
147 00:03:56,669 –> 00:03:57,899 space is complete.
148 00:03:58,869 –> 00:03:59,940 You might already see the
149 00:03:59,949 –> 00:04:01,740 general result here, because
150 00:04:01,750 –> 00:04:03,240 we consider metric spaces
151 00:04:03,250 –> 00:04:05,039 coming from the real numbers
152 00:04:05,050 –> 00:04:06,830 R with the same metric as
153 00:04:06,839 –> 00:04:07,259 R.
154 00:04:07,869 –> 00:04:09,639 Therefore, as long as the
155 00:04:09,649 –> 00:04:11,600 set X is closed in
156 00:04:11,610 –> 00:04:13,460 R, which means in a
157 00:04:13,470 –> 00:04:14,839 metric space (R, d) in
158 00:04:14,850 –> 00:04:16,839 this case, then what
159 00:04:16,850 –> 00:04:18,500 we get out is a complete
160 00:04:18,510 –> 00:04:19,358 metric space.
161 00:04:20,088 –> 00:04:21,678 However, that is just because
162 00:04:21,688 –> 00:04:23,049 we already knew that
163 00:04:23,058 –> 00:04:24,928 (R,d) is a complete metric
164 00:04:24,938 –> 00:04:25,389 space.
165 00:04:26,100 –> 00:04:27,570 You might already feel that
166 00:04:27,579 –> 00:04:29,079 this should work in general
167 00:04:29,119 –> 00:04:30,170 and you are correct.
168 00:04:30,570 –> 00:04:31,910 If you have a complete metric
169 00:04:31,920 –> 00:04:33,609 space, you can form new
170 00:04:33,619 –> 00:04:35,089 ones just by considering
171 00:04:35,100 –> 00:04:37,019 the closed subsets with the
172 00:04:37,029 –> 00:04:37,809 same metric.
173 00:04:38,609 –> 00:04:39,589 However, that’s something
174 00:04:39,600 –> 00:04:40,510 we can prove later.
175 00:04:40,519 –> 00:04:41,829 Let’s first look at another
176 00:04:41,839 –> 00:04:42,420 example.
177 00:04:43,079 –> 00:04:44,559 Now we don’t change the set.
178 00:04:44,570 –> 00:04:45,850 We still consider the interval
179 00:04:45,859 –> 00:04:47,410 0 to 3 where we exclude
180 00:04:47,420 –> 00:04:48,420 zero and three.
181 00:04:48,429 –> 00:04:50,049 But now we change the metric
182 00:04:50,059 –> 00:04:51,320 and I want to choose the
183 00:04:51,329 –> 00:04:52,399 discrete metric.
184 00:04:52,910 –> 00:04:54,260 So this is what we defined
185 00:04:54,269 –> 00:04:55,790 in part two as a discrete
186 00:04:55,799 –> 00:04:56,299 metric.
187 00:04:57,260 –> 00:04:59,000 Now my claim is this is
188 00:04:59,010 –> 00:05:00,619 also a complete metric
189 00:05:00,630 –> 00:05:01,149 space.
190 00:05:01,910 –> 00:05:03,059 Indeed, that’s not hard to
191 00:05:03,070 –> 00:05:04,809 show we just choose an arbitrary
192 00:05:04,920 –> 00:05:06,059 Cauchy sequence here.
193 00:05:06,799 –> 00:05:08,459 Then we know this whole thing
194 00:05:08,470 –> 00:05:09,739 here holds for all
195 00:05:09,750 –> 00:05:11,720 epsilon so that I can just
196 00:05:11,730 –> 00:05:13,679 take one and I want to
197 00:05:13,690 –> 00:05:15,649 have epsilon as one half.
198 00:05:16,510 –> 00:05:17,630 Then we know there is a
199 00:05:17,640 –> 00:05:19,250 capital N such that
200 00:05:19,260 –> 00:05:20,679 all indices that are
201 00:05:20,690 –> 00:05:22,549 created than N fulfill this
202 00:05:22,559 –> 00:05:22,890 one.
203 00:05:23,779 –> 00:05:25,079 This means that the distance
204 00:05:25,089 –> 00:05:26,959 between x_n and x_m is
205 00:05:26,970 –> 00:05:28,739 less than epsilon or in other
206 00:05:28,750 –> 00:05:30,510 words, less than one half.
207 00:05:31,309 –> 00:05:33,049 However, now we see in the
208 00:05:33,059 –> 00:05:34,529 definition of the metric
209 00:05:34,640 –> 00:05:36,290 that less than one half is
210 00:05:36,299 –> 00:05:37,739 only possible in the bottom
211 00:05:37,750 –> 00:05:38,170 case.
212 00:05:39,029 –> 00:05:40,160 Therefore, the distance is
213 00:05:40,170 –> 00:05:41,459 indeed zero,
214 00:05:42,320 –> 00:05:44,260 which again means that x_n
215 00:05:44,269 –> 00:05:45,899 is the same as x_m.
216 00:05:46,380 –> 00:05:47,859 So please note this works
217 00:05:47,869 –> 00:05:49,750 for almost all indices here.
218 00:05:49,779 –> 00:05:51,769 Only finitely many before
219 00:05:51,940 –> 00:05:53,019 could be different.
220 00:05:54,160 –> 00:05:55,500 Hence, it tells you that
221 00:05:55,510 –> 00:05:57,500 the sequence is indeed constant
222 00:05:57,510 –> 00:05:58,700 after the index N
223 00:06:00,010 –> 00:06:01,220 OK, maybe let’s look at the
224 00:06:01,230 –> 00:06:02,420 graph where we have here
225 00:06:02,429 –> 00:06:03,829 the indices and here the
226 00:06:03,839 –> 00:06:05,820 space X. At
227 00:06:05,829 –> 00:06:07,220 the beginning the sequence
228 00:06:07,230 –> 00:06:08,690 is allowed to do something.
229 00:06:09,880 –> 00:06:11,230 However, the value it reaches
230 00:06:11,239 –> 00:06:12,989 with the index N is then
231 00:06:13,000 –> 00:06:14,690 fixed for the entire time
232 00:06:14,700 –> 00:06:15,540 of the sequence.
233 00:06:16,649 –> 00:06:17,790 So you could say this is
234 00:06:17,799 –> 00:06:19,630 quite a boring sequence because
235 00:06:19,640 –> 00:06:20,869 it does not change anymore.
236 00:06:21,429 –> 00:06:23,040 However, for such a sequence,
237 00:06:23,049 –> 00:06:24,829 it’s easy to show that it
238 00:06:24,839 –> 00:06:26,549 is a convergence sequence
239 00:06:26,559 –> 00:06:27,950 with the limit x tilde
240 00:06:27,959 –> 00:06:29,929 as exactly this constant
241 00:06:29,940 –> 00:06:30,390 in the end.
242 00:06:31,269 –> 00:06:32,609 And now we have seen that
243 00:06:32,619 –> 00:06:34,209 these are indeed the only
244 00:06:34,220 –> 00:06:36,119 possible convergence sequences
245 00:06:36,130 –> 00:06:37,690 in a discrete metric space.
246 00:06:38,230 –> 00:06:39,279 And of course, we have also
247 00:06:39,290 –> 00:06:41,239 shown that this one is indeed
248 00:06:41,250 –> 00:06:42,690 a complete metric space.
249 00:06:43,589 –> 00:06:43,989 OK.
250 00:06:44,000 –> 00:06:44,859 Maybe that’s good enough
251 00:06:44,869 –> 00:06:45,609 for today.
252 00:06:45,640 –> 00:06:46,970 In the next video, we will
253 00:06:46,980 –> 00:06:48,750 look at some important complete
254 00:06:48,760 –> 00:06:49,679 metric spaces.
255 00:06:50,359 –> 00:06:51,540 So I hope I see you there
256 00:06:51,549 –> 00:06:52,540 and goodbye.
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Quiz Content
Q1: What does ‘completeness’ of a metric space mean?
A1: Every sequence is convergent.
A2: Every Cauchy sequence is convergent.
A3: Every bounded sequence is covergent.
A4: Every convergent sequence is a Cauchy sequence.
Q2: Let $(X,d)$ be a metric space given by $X = (0,4]$ and $d(x,y) = \frac{1}{2}|x-y|$. Is $(X,d)$ complete?
A1: Yes!
A2: No!
Q3: Let $(X,d)$ be a complete metric space and $A \subseteq X$ a closed set. Is the induced metric space $(A, d|_{ A \times A} )$ also complete?
A1: Yes!
A2: No!
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Last update: 2024-10
