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Title: Open Mapping Theorem
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Series: Functional Analysis
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YouTube-Title: Functional Analysis 26 | Open Mapping Theorem
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Bright video: https://youtu.be/o0vAbQFZnIo
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Subtitle on GitHub: fa26_sub_eng.srt
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Timestamps
00:00 Introduction
01:15 General example
02:47 Examples
03:47 Theorem
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Subtitle in English
1 00:00:00,379 –> 00:00:02,240 Hello and welcome back to
2 00:00:02,250 –> 00:00:03,619 functional analysis
3 00:00:03,779 –> 00:00:05,480 and as always, many thanks
4 00:00:05,489 –> 00:00:06,760 to all the nice people that
5 00:00:06,769 –> 00:00:08,279 support this channel on Steady
6 00:00:08,289 –> 00:00:09,119 or PayPal.
7 00:00:09,630 –> 00:00:11,199 This is part 26
8 00:00:11,319 –> 00:00:12,420 and I still want to show
9 00:00:12,430 –> 00:00:13,960 you more main results in
10 00:00:13,970 –> 00:00:15,680 the topic of functional analysis.
11 00:00:16,459 –> 00:00:17,790 Today, we will talk about
12 00:00:17,799 –> 00:00:19,450 the so called open mapping
13 00:00:19,459 –> 00:00:21,290 theorem, also known as the
14 00:00:21,299 –> 00:00:22,709 Banach-Schauder theorem.
15 00:00:23,469 –> 00:00:25,010 As the name suggests, the
16 00:00:25,020 –> 00:00:26,620 theorem is about open
17 00:00:26,629 –> 00:00:27,290 maps.
18 00:00:27,299 –> 00:00:28,530 Therefore, I first want to
19 00:00:28,540 –> 00:00:30,170 start explaining what an
20 00:00:30,180 –> 00:00:31,920 open map is. The
21 00:00:31,930 –> 00:00:33,549 notion openness we define
22 00:00:33,560 –> 00:00:34,860 for metric spaces.
23 00:00:34,869 –> 00:00:36,360 So therefore, let’s choose
24 00:00:36,369 –> 00:00:38,099 two metric spaces X and
25 00:00:38,110 –> 00:00:38,520 Y.
26 00:00:39,299 –> 00:00:40,549 Of course, now we want to
27 00:00:40,560 –> 00:00:42,430 consider a map from X
28 00:00:42,439 –> 00:00:43,020 to Y.
29 00:00:43,849 –> 00:00:45,630 Now such a map f gets the
30 00:00:45,639 –> 00:00:47,540 name open, if it sends
31 00:00:47,549 –> 00:00:49,290 open sets in X to
32 00:00:49,299 –> 00:00:50,509 open sets in Y.
33 00:00:51,520 –> 00:00:53,369 In other words, if we take
34 00:00:53,380 –> 00:00:55,290 any open set A in
35 00:00:55,299 –> 00:00:56,959 X, then the
36 00:00:56,970 –> 00:00:58,759 image f(A) should
37 00:00:58,770 –> 00:01:00,360 also be an open set.
38 00:01:00,919 –> 00:01:02,599 However, please don’t forget
39 00:01:02,610 –> 00:01:04,540 the notion openness is always
40 00:01:04,550 –> 00:01:05,919 given with respect to the
41 00:01:05,930 –> 00:01:07,480 corresponding metric space.
42 00:01:08,169 –> 00:01:08,589 OK.
43 00:01:08,599 –> 00:01:10,220 So this looks like a natural
44 00:01:10,230 –> 00:01:10,819 notion.
45 00:01:10,949 –> 00:01:12,290 Therefore, let’s immediately
46 00:01:12,300 –> 00:01:13,669 look at some examples.
47 00:01:14,720 –> 00:01:16,660 First, a very general example,
48 00:01:16,669 –> 00:01:18,120 because you might already
49 00:01:18,129 –> 00:01:19,699 recognize the connection
50 00:01:19,709 –> 00:01:21,400 to the definition of continuity
51 00:01:21,410 –> 00:01:21,720 here.
52 00:01:22,500 –> 00:01:23,860 So let’s fix a bijective
53 00:01:23,870 –> 00:01:25,370 map, which means we have
54 00:01:25,379 –> 00:01:27,120 an inverse map f to the
55 00:01:27,129 –> 00:01:28,410 power minus one.
56 00:01:28,940 –> 00:01:30,449 Of course, this one now goes
57 00:01:30,459 –> 00:01:32,220 from Y to X and we
58 00:01:32,230 –> 00:01:34,110 also choose it to be a continuous
59 00:01:34,120 –> 00:01:34,519 map.
60 00:01:35,300 –> 00:01:36,970 Having these two assumptions.
61 00:01:36,980 –> 00:01:38,830 The map f from X to Y
62 00:01:38,839 –> 00:01:40,610 is indeed an open map.
63 00:01:41,440 –> 00:01:43,029 This is easy to see when
64 00:01:43,040 –> 00:01:44,269 you look at the definition
65 00:01:44,279 –> 00:01:45,930 of the continuity of the
66 00:01:45,940 –> 00:01:46,709 inverse map.
67 00:01:47,720 –> 00:01:49,069 It simply means when you
68 00:01:49,080 –> 00:01:50,489 have an open set on the right-
69 00:01:50,500 –> 00:01:52,239 hand side, the preimage
70 00:01:52,250 –> 00:01:53,779 on the left-hand side is
71 00:01:53,790 –> 00:01:55,180 also an open set.
72 00:01:55,559 –> 00:01:56,860 Of course, everything looks
73 00:01:56,870 –> 00:01:58,300 very similar. Here
74 00:01:58,309 –> 00:01:59,589 you see we have the inverse
75 00:01:59,599 –> 00:02:01,519 map from Y to X.
76 00:02:02,019 –> 00:02:03,800 Then we take a subset A in
77 00:02:03,809 –> 00:02:05,419 the space X which is on the
78 00:02:05,430 –> 00:02:06,739 right-hand side of the inverse
79 00:02:06,750 –> 00:02:07,220 map
80 00:02:07,230 –> 00:02:08,339 but on the left-hand side
81 00:02:08,350 –> 00:02:09,820 in the picture. So don’t
82 00:02:09,830 –> 00:02:10,800 get confused here.
83 00:02:11,380 –> 00:02:13,270 Then we go to the preimage
84 00:02:13,279 –> 00:02:14,860 which is here again denoted
85 00:02:14,869 –> 00:02:16,419 by the power minus one.
86 00:02:17,190 –> 00:02:17,539 OK.
87 00:02:17,550 –> 00:02:18,520 You could think this is an
88 00:02:18,529 –> 00:02:20,220 overload for the notation, but
89 00:02:20,229 –> 00:02:21,880 indeed it fits together,
90 00:02:21,889 –> 00:02:23,660 because this is simply
91 00:02:23,669 –> 00:02:25,410 the image of A and
92 00:02:25,419 –> 00:02:25,839 f.
93 00:02:26,440 –> 00:02:27,759 This comes from the 1 to
94 00:02:27,770 –> 00:02:29,399 1 correspondence we have
95 00:02:29,410 –> 00:02:30,759 by our bijective map
96 00:02:30,770 –> 00:02:31,789 f.
97 00:02:31,860 –> 00:02:33,210 Now with this example, you
98 00:02:33,220 –> 00:02:34,919 should see that the definition
99 00:02:34,929 –> 00:02:36,440 of an open map looks
100 00:02:36,449 –> 00:02:37,770 similar to the one of a
101 00:02:37,779 –> 00:02:39,119 continuous map.
102 00:02:39,130 –> 00:02:40,380 However, it’s completely
103 00:02:40,389 –> 00:02:41,970 different, because it goes
104 00:02:41,979 –> 00:02:42,960 the other way around.
105 00:02:43,720 –> 00:02:44,220 OK.
106 00:02:44,229 –> 00:02:45,500 Then let’s write down some
107 00:02:45,509 –> 00:02:47,039 concrete examples of open
108 00:02:47,050 –> 00:02:47,419 maps.
109 00:02:48,199 –> 00:02:49,639 Maybe a simple example would
110 00:02:49,649 –> 00:02:51,600 be to look at a normal function
111 00:02:51,639 –> 00:02:53,110 between the real numbers.
112 00:02:53,669 –> 00:02:55,320 Of course R is just given
113 00:02:55,330 –> 00:02:56,559 with the standard metric,
114 00:02:56,570 –> 00:02:57,699 which means the absolute
115 00:02:57,710 –> 00:02:58,309 value
116 00:02:58,440 –> 00:03:00,279 and I want to send x to
117 00:03:00,289 –> 00:03:01,160 x cubed.
118 00:03:02,009 –> 00:03:03,639 This has to be an open map,
119 00:03:03,649 –> 00:03:05,300 because f is bijective
120 00:03:05,470 –> 00:03:06,669 and the inverse is
121 00:03:06,679 –> 00:03:07,520 continuous.
122 00:03:08,410 –> 00:03:10,000 In the same way, we can construct
123 00:03:10,009 –> 00:03:11,130 a counter example.
124 00:03:11,139 –> 00:03:12,570 Just look what happens when
125 00:03:12,580 –> 00:03:14,100 we send x to x
126 00:03:14,110 –> 00:03:14,720 squared.
127 00:03:15,179 –> 00:03:16,490 Of course, we can’t use the
128 00:03:16,500 –> 00:03:18,169 general example here, because
129 00:03:18,179 –> 00:03:19,690 it’s simply not bijective,
130 00:03:20,250 –> 00:03:21,529 but still it could be an
131 00:03:21,539 –> 00:03:22,289 open map.
132 00:03:22,809 –> 00:03:24,009 Of course, this is not the
133 00:03:24,020 –> 00:03:25,899 case. It’s not an open map.
134 00:03:25,910 –> 00:03:27,130 As you can see when
135 00:03:27,139 –> 00:03:28,850 considering the open
136 00:03:28,860 –> 00:03:30,679 set as the open interval
137 00:03:30,690 –> 00:03:32,169 minus 2 to 2.
138 00:03:32,839 –> 00:03:34,369 Then the image is simply
139 00:03:34,380 –> 00:03:36,100 the interval 0 to
140 00:03:36,110 –> 00:03:36,550 4
141 00:03:37,360 –> 00:03:38,779 and there you see this is
142 00:03:38,789 –> 00:03:40,710 not an open set in R.
143 00:03:41,369 –> 00:03:42,740 Now, with your new knowledge,
144 00:03:42,750 –> 00:03:44,460 what an open map is, we can
145 00:03:44,470 –> 00:03:46,440 finally state the open mapping
146 00:03:46,449 –> 00:03:47,039 theorem.
147 00:03:47,600 –> 00:03:48,899 The important ingredient
148 00:03:48,910 –> 00:03:50,039 you always should remember
149 00:03:50,050 –> 00:03:51,699 here is that we need Banach
150 00:03:51,710 –> 00:03:53,300 spaces on the left- and the
151 00:03:53,309 –> 00:03:54,179 right-hand side.
152 00:03:54,699 –> 00:03:55,949 So the completeness of the
153 00:03:55,960 –> 00:03:57,699 spaces is essential here.
154 00:03:58,500 –> 00:04:00,119 Now, for bounded linear
155 00:04:00,130 –> 00:04:02,100 operator T, where I use the
156 00:04:02,110 –> 00:04:03,380 notation with the curved
157 00:04:03,389 –> 00:04:05,070 B again, we
158 00:04:05,080 –> 00:04:06,289 have that it is
159 00:04:06,300 –> 00:04:07,270 surjective
160 00:04:08,039 –> 00:04:09,699 if and only if
161 00:04:09,710 –> 00:04:11,259 T is an open map
162 00:04:12,050 –> 00:04:13,889 and that is the famous
163 00:04:13,899 –> 00:04:15,350 open mapping theorem.
164 00:04:16,098 –> 00:04:17,759 It just tells you that
165 00:04:17,908 –> 00:04:19,778 surjectivity for a linear
166 00:04:19,829 –> 00:04:21,368 bounded operator between
167 00:04:21,378 –> 00:04:23,229 Banach spaces is a strong
168 00:04:23,239 –> 00:04:25,158 property, because
169 00:04:25,169 –> 00:04:26,928 it implies the openness.
170 00:04:27,670 –> 00:04:29,399 Indeed, the implication from
171 00:04:29,410 –> 00:04:31,040 right to left is not so
172 00:04:31,049 –> 00:04:32,600 surprising and easy to
173 00:04:32,609 –> 00:04:34,549 prove. The actual work
174 00:04:34,559 –> 00:04:36,109 one has to do from the left
175 00:04:36,119 –> 00:04:37,279 to the right-hand side.
176 00:04:37,910 –> 00:04:39,130 Therefore, the proof of the
177 00:04:39,140 –> 00:04:40,779 first implication is a good
178 00:04:40,790 –> 00:04:41,880 exercise for you,
179 00:04:42,000 –> 00:04:43,450 but for the other one, we
180 00:04:43,459 –> 00:04:44,790 need a whole other video.
181 00:04:45,410 –> 00:04:46,970 However, before we do that,
182 00:04:46,980 –> 00:04:48,670 I want to use the next video
183 00:04:48,679 –> 00:04:50,170 to show you an important
184 00:04:50,179 –> 00:04:51,910 consequence of the open mapping
185 00:04:51,920 –> 00:04:52,570 theorem.
186 00:04:53,040 –> 00:04:54,410 It’s called the bounded
187 00:04:54,420 –> 00:04:55,709 inverse theorem.
188 00:04:56,209 –> 00:04:57,730 Hence please watch the next
189 00:04:57,739 –> 00:04:59,609 video, because this consequence
190 00:04:59,619 –> 00:05:01,429 is very important in a lot
191 00:05:01,440 –> 00:05:02,359 of applications.
192 00:05:03,170 –> 00:05:04,510 So have a nice day and see
193 00:05:04,519 –> 00:05:04,989 you then.
194 00:05:05,000 –> 00:05:05,750 Bye.
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Last update: 2024-10
