
Title: CauchySchwarz Inequality

Series: Functional Analysis

YouTubeTitle: Functional Analysis 10  CauchySchwarz Inequality

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Timestamps
00:00 Introduction 00:25 CauchySchwarz inequality 02:00 Proof 08:15 Triangle inequality for the norm 
Subtitle in English
1 00:00:00,479 –> 00:00:02,369 Hello and welcome back to
2 00:00:02,380 –> 00:00:03,509 functional analysis.
3 00:00:03,599 –> 00:00:05,280 And first many thanks to
4 00:00:05,289 –> 00:00:06,380 all the nice people that
5 00:00:06,389 –> 00:00:07,769 support me on study or
6 00:00:07,780 –> 00:00:08,489 paypal.
7 00:00:09,180 –> 00:00:10,960 And I can also tell you that
8 00:00:10,970 –> 00:00:12,470 now you find the PDF
9 00:00:12,479 –> 00:00:14,350 versions of all new videos
10 00:00:14,359 –> 00:00:15,600 in the description below.
11 00:00:16,430 –> 00:00:18,180 So this is part 10 today
12 00:00:18,190 –> 00:00:19,739 and we will still talk about
13 00:00:19,750 –> 00:00:20,780 inner products.
14 00:00:21,719 –> 00:00:23,010 In particular, I want to
15 00:00:23,020 –> 00:00:24,590 prove the Cauchy Schwarz
16 00:00:24,610 –> 00:00:25,430 inequality
17 00:00:26,379 –> 00:00:27,319 depending where you come
18 00:00:27,329 –> 00:00:27,809 from.
19 00:00:27,829 –> 00:00:29,190 The equality is known by
20 00:00:29,200 –> 00:00:30,280 different names.
21 00:00:30,290 –> 00:00:31,700 But I stay here with the
22 00:00:31,709 –> 00:00:32,598 most common one.
23 00:00:33,270 –> 00:00:34,330 And here it is named after the
24 00:00:34,340 –> 00:00:35,770 French mathematician Cauchy
25 00:00:36,119 –> 00:00:37,659 and a German mathematician
26 00:00:37,669 –> 00:00:38,119 Schwarz.
27 00:00:39,060 –> 00:00:40,569 Indeed, this inequality is
28 00:00:40,580 –> 00:00:42,270 very important because it
29 00:00:42,279 –> 00:00:44,049 holds for all inner products.
30 00:00:44,560 –> 00:00:45,770 Therefore, let’s choose an
31 00:00:45,779 –> 00:00:47,729 F vector space X and an
32 00:00:47,740 –> 00:00:48,529 inner product.
33 00:00:49,369 –> 00:00:50,790 And we also consider the
34 00:00:50,799 –> 00:00:52,369 corresponding norm which
35 00:00:52,380 –> 00:00:53,459 is the square root of the
36 00:00:53,470 –> 00:00:54,229 inner product.
37 00:00:54,880 –> 00:00:56,650 And then with these notations,
38 00:00:56,659 –> 00:00:58,119 the following inequality
39 00:00:58,130 –> 00:01:00,020 holds, we
40 00:01:00,029 –> 00:01:01,389 look at the inner product
41 00:01:01,400 –> 00:01:02,630 X with Y.
42 00:01:03,400 –> 00:01:04,489 And because it could be in
43 00:01:04,500 –> 00:01:05,980 general a complex number,
44 00:01:05,989 –> 00:01:07,120 we need the absolute value
45 00:01:07,129 –> 00:01:07,489 here.
46 00:01:08,239 –> 00:01:09,849 And then we can say this
47 00:01:09,860 –> 00:01:11,819 is less or equal than the
48 00:01:11,830 –> 00:01:13,419 two norms multiplied.
49 00:01:14,339 –> 00:01:15,580 And that’s the whole Cauchy
50 00:01:15,610 –> 00:01:16,919 Schwarz inequality.
51 00:01:17,930 –> 00:01:19,510 Now we can visualize it with
52 00:01:19,519 –> 00:01:21,139 the same picture we had when
53 00:01:21,150 –> 00:01:22,339 we started with the inner
54 00:01:22,349 –> 00:01:24,059 products in this
55 00:01:24,069 –> 00:01:25,510 multiplication X with
56 00:01:25,519 –> 00:01:27,199 Y only the parallel
57 00:01:27,209 –> 00:01:28,900 part of Y, the yellow part
58 00:01:28,910 –> 00:01:30,050 here should matter.
59 00:01:30,779 –> 00:01:32,160 Then this picture tells you,
60 00:01:32,169 –> 00:01:33,319 yeah, of course, the length
61 00:01:33,330 –> 00:01:34,870 of the yellow part is less
62 00:01:34,879 –> 00:01:36,199 or equal than the length
63 00:01:36,209 –> 00:01:37,400 of the wet arrow here.
64 00:01:38,209 –> 00:01:39,669 With this in mind, we also
65 00:01:39,680 –> 00:01:41,589 get a result in which cases
66 00:01:41,599 –> 00:01:43,120 the equality here holds
67 00:01:43,900 –> 00:01:45,290 indeed, this should only
68 00:01:45,300 –> 00:01:46,730 be the case when the arrows
69 00:01:46,739 –> 00:01:48,169 go in the same direction.
70 00:01:48,989 –> 00:01:50,760 In other words, X and Y
71 00:01:50,769 –> 00:01:52,419 are linearly dependent
72 00:01:52,430 –> 00:01:53,099 vectors.
73 00:01:53,739 –> 00:01:54,199 OK.
74 00:01:54,209 –> 00:01:55,419 Now the picture gets us in
75 00:01:55,430 –> 00:01:56,919 the correct direction, but
76 00:01:56,930 –> 00:01:58,230 we don’t have any choice.
77 00:01:58,239 –> 00:01:59,910 We need to prove the inequality.
78 00:01:59,919 –> 00:02:00,319 Now.
79 00:02:01,160 –> 00:02:01,529 OK.
80 00:02:01,540 –> 00:02:02,709 So let’s start with an easy
81 00:02:02,720 –> 00:02:04,400 case, let’s call it the first
82 00:02:04,410 –> 00:02:06,389 case where X is the zero
83 00:02:06,400 –> 00:02:06,860 vector.
84 00:02:07,809 –> 00:02:09,270 Of course, there we know
85 00:02:09,288 –> 00:02:10,839 the left hand side has to
86 00:02:10,850 –> 00:02:11,589 be zero
87 00:02:12,830 –> 00:02:14,410 simply by the linearity,
88 00:02:14,419 –> 00:02:15,940 we can just pull out the
89 00:02:15,949 –> 00:02:16,990 factor zero.
90 00:02:17,660 –> 00:02:18,679 And of course, the right
91 00:02:18,690 –> 00:02:20,309 hand side is also zero
92 00:02:20,380 –> 00:02:22,190 because the norm of X has
93 00:02:22,199 –> 00:02:23,089 to be zero here.
94 00:02:23,929 –> 00:02:25,440 In particular, the general
95 00:02:25,449 –> 00:02:26,970 inequality is
96 00:02:26,979 –> 00:02:28,649 obviously fulfilled for this
97 00:02:28,660 –> 00:02:29,369 simple case.
98 00:02:29,990 –> 00:02:31,570 Now you might already guess
99 00:02:31,580 –> 00:02:33,220 the actual interesting case
100 00:02:33,229 –> 00:02:34,559 would be the second one
101 00:02:35,320 –> 00:02:36,869 here X is not a zero
102 00:02:36,880 –> 00:02:38,419 vector, which means we can
103 00:02:38,429 –> 00:02:40,270 divide by the norm of X.
104 00:02:41,029 –> 00:02:42,789 And that’s what we do immediately,
105 00:02:42,800 –> 00:02:44,660 I want to define X hat
106 00:02:44,710 –> 00:02:46,160 as the normalized vector
107 00:02:46,169 –> 00:02:48,039 X, this means
108 00:02:48,050 –> 00:02:50,039 that X hat has length one.
109 00:02:50,050 –> 00:02:51,539 So it just gives the direction
110 00:02:51,550 –> 00:02:53,300 of the vector X and
111 00:02:53,309 –> 00:02:54,500 then we calculate the inner
112 00:02:54,509 –> 00:02:56,070 product with X hat and
113 00:02:56,080 –> 00:02:57,940 Y and go into the
114 00:02:57,949 –> 00:02:59,500 direction of X hat.
115 00:03:00,850 –> 00:03:02,070 Now, you might recognize
116 00:03:02,080 –> 00:03:03,660 that because if we have
117 00:03:03,669 –> 00:03:05,389 considered the normal Euclidean
118 00:03:05,399 –> 00:03:07,070 geometry in the plane, this
119 00:03:07,080 –> 00:03:08,240 picture would be completely
120 00:03:08,250 –> 00:03:09,009 correct.
121 00:03:09,199 –> 00:03:10,860 And this expression is
122 00:03:10,869 –> 00:03:12,830 exactly this orange line
123 00:03:13,559 –> 00:03:14,699 by our abstraction.
124 00:03:14,710 –> 00:03:16,179 This should be the same here.
125 00:03:16,190 –> 00:03:17,520 So let’s call the orange arrow
126 00:03:17,759 –> 00:03:19,009 just y
127 00:03:19,020 –> 00:03:20,899 parallel in the
128 00:03:20,910 –> 00:03:22,039 Euclidean geometry.
129 00:03:22,050 –> 00:03:23,940 This is known as the orthogonal
130 00:03:23,949 –> 00:03:25,800 projection of Y on to
131 00:03:25,809 –> 00:03:26,470 X.
132 00:03:27,309 –> 00:03:28,550 So it makes sense to do the
133 00:03:28,559 –> 00:03:30,020 same for a general in our
134 00:03:30,029 –> 00:03:30,619 product.
135 00:03:31,460 –> 00:03:33,119 Now the gray line here is
136 00:03:33,130 –> 00:03:34,639 the orthogonal part which
137 00:03:34,649 –> 00:03:35,850 we also can calculate.
138 00:03:35,860 –> 00:03:37,630 Now this is
139 00:03:37,639 –> 00:03:39,050 simply given by Y
140 00:03:39,059 –> 00:03:40,729 minus the parallel part.
141 00:03:41,520 –> 00:03:41,880 OK.
142 00:03:41,889 –> 00:03:43,220 So we defined some vectors
143 00:03:43,229 –> 00:03:44,300 we want to deal with.
144 00:03:44,449 –> 00:03:46,380 But now we need an idea how
145 00:03:46,389 –> 00:03:47,850 to get to the inequality.
146 00:03:48,580 –> 00:03:50,100 Indeed, the whole idea is
147 00:03:50,110 –> 00:03:51,160 that we can calculate the
148 00:03:51,169 –> 00:03:52,940 length or the norm of
149 00:03:52,949 –> 00:03:54,240 this or formal part
150 00:03:55,179 –> 00:03:56,910 simply because we know the
151 00:03:56,919 –> 00:03:58,830 norm can’t be negative.
152 00:03:59,830 –> 00:04:01,399 Now, on the line we can put
153 00:04:01,410 –> 00:04:03,389 in the definition and then
154 00:04:03,399 –> 00:04:05,149 of course, also the definition
155 00:04:05,160 –> 00:04:06,619 of Y parallel.
156 00:04:08,070 –> 00:04:09,169 Now if you see something
157 00:04:09,179 –> 00:04:11,089 like this calculating norms,
158 00:04:11,100 –> 00:04:12,570 but you have an inner product,
159 00:04:12,600 –> 00:04:14,490 it’s better to use squares.
160 00:04:15,149 –> 00:04:16,769 So square everything and
161 00:04:16,779 –> 00:04:18,309 the square roots will vanish.
162 00:04:19,248 –> 00:04:20,878 And now you can see on the
163 00:04:20,889 –> 00:04:22,169 right hand side, we have
164 00:04:22,178 –> 00:04:23,868 this long vector in the middle,
165 00:04:24,058 –> 00:04:25,658 in both components of the
166 00:04:25,669 –> 00:04:26,378 inner product.
167 00:04:27,299 –> 00:04:28,720 So it looks like this.
168 00:04:29,989 –> 00:04:31,170 And in order to simplify
169 00:04:31,179 –> 00:04:32,500 this, we will use the
170 00:04:32,510 –> 00:04:33,410 linearity.
171 00:04:34,209 –> 00:04:35,390 And this linearity in the
172 00:04:35,399 –> 00:04:36,950 second component means we
173 00:04:36,959 –> 00:04:38,649 can pull out this minus
174 00:04:38,660 –> 00:04:39,350 sign here.
175 00:04:39,929 –> 00:04:41,149 So we have here a product
176 00:04:41,160 –> 00:04:43,000 with Y minus the other
177 00:04:43,010 –> 00:04:43,380 part.
178 00:04:44,119 –> 00:04:45,390 In the next step, we want
179 00:04:45,399 –> 00:04:46,730 to pull out the minus sign
180 00:04:46,739 –> 00:04:48,570 here And here, and we
181 00:04:48,579 –> 00:04:50,510 know we can do that because
182 00:04:50,519 –> 00:04:51,679 the inner product is
183 00:04:51,690 –> 00:04:53,250 conjugate linear in the
184 00:04:53,260 –> 00:04:54,369 first argument.
185 00:04:54,989 –> 00:04:56,160 So minus signs are not a
186 00:04:56,170 –> 00:04:56,769 problem.
187 00:04:56,779 –> 00:04:58,230 And scalars get a complex
188 00:04:58,239 –> 00:04:58,989 conjugation.
189 00:04:59,570 –> 00:05:00,839 So that’s the first part
190 00:05:00,850 –> 00:05:01,750 and the second part would
191 00:05:01,760 –> 00:05:03,480 be this vector with y?
192 00:05:04,239 –> 00:05:05,380 And of course, now we do
193 00:05:05,390 –> 00:05:06,470 the same with the second
194 00:05:06,480 –> 00:05:07,480 inner product here.
195 00:05:08,829 –> 00:05:09,179 OK.
196 00:05:09,190 –> 00:05:11,049 Here we have Y with the vector
197 00:05:11,059 –> 00:05:12,100 on the right hand side.
198 00:05:12,140 –> 00:05:13,730 And now we have minus minus.
199 00:05:13,739 –> 00:05:15,450 So plus the rest.
200 00:05:16,399 –> 00:05:17,600 In fact, here we have the
201 00:05:17,609 –> 00:05:19,019 same vector left and right
202 00:05:19,029 –> 00:05:19,959 in the inner product.
203 00:05:20,619 –> 00:05:21,679 Therefore, now comes the
204 00:05:21,690 –> 00:05:23,160 part where we can rewrite
205 00:05:23,170 –> 00:05:23,630 everything.
206 00:05:23,640 –> 00:05:24,109 Again.
207 00:05:24,640 –> 00:05:26,070 The first thing is the norm
208 00:05:26,079 –> 00:05:27,190 of Y squared.
209 00:05:27,970 –> 00:05:28,880 For the two parts in the
210 00:05:28,890 –> 00:05:30,549 middle, we see they are almost
211 00:05:30,559 –> 00:05:31,899 the same, the one is the
212 00:05:31,910 –> 00:05:33,260 complex conjugate of the
213 00:05:33,269 –> 00:05:33,660 other one.
214 00:05:34,390 –> 00:05:35,440 So we can write it in this
215 00:05:35,450 –> 00:05:36,839 way with parentheses, we
216 00:05:36,850 –> 00:05:38,209 put a bar over the second
217 00:05:38,220 –> 00:05:38,640 part.
218 00:05:39,459 –> 00:05:41,079 And finally, the last part
219 00:05:41,089 –> 00:05:42,760 is the norm of this vector
220 00:05:42,769 –> 00:05:43,440 squared.
221 00:05:44,380 –> 00:05:45,760 In the next equality here,
222 00:05:45,769 –> 00:05:47,079 I want to simplify that even
223 00:05:47,089 –> 00:05:48,709 more because you see
224 00:05:48,720 –> 00:05:50,450 here’s a complex number plus
225 00:05:50,459 –> 00:05:51,799 the complex conjugate of
226 00:05:51,809 –> 00:05:53,350 this number, which means
227 00:05:53,359 –> 00:05:54,910 this is two times the real
228 00:05:54,920 –> 00:05:56,559 part of this complex
229 00:05:56,570 –> 00:05:58,359 number, which means that
230 00:05:58,369 –> 00:05:59,869 we can write it like this.
231 00:06:00,790 –> 00:06:01,970 And now in the third part,
232 00:06:01,980 –> 00:06:03,500 we can pull out the scalar
233 00:06:03,940 –> 00:06:05,149 which means it comes out
234 00:06:05,160 –> 00:06:06,850 with the absolute value squared,
235 00:06:07,649 –> 00:06:09,299 you see it remains the norm
236 00:06:09,309 –> 00:06:10,799 of x hat where we already
237 00:06:10,809 –> 00:06:12,279 know this is one.
238 00:06:12,529 –> 00:06:14,339 Now, finally, I want to simplify
239 00:06:14,350 –> 00:06:15,769 the middle part here, which
240 00:06:15,779 –> 00:06:16,839 to be honest, we could have
241 00:06:16,850 –> 00:06:17,750 done before.
242 00:06:17,760 –> 00:06:18,970 But then we would have done
243 00:06:18,980 –> 00:06:19,880 it two times.
244 00:06:20,649 –> 00:06:22,190 I want to show you visually
245 00:06:22,200 –> 00:06:23,350 what we want to do.
246 00:06:23,540 –> 00:06:25,230 We pull this scalar out
247 00:06:25,239 –> 00:06:26,799 from the first argument of
248 00:06:26,809 –> 00:06:28,510 the inner product, but
249 00:06:28,519 –> 00:06:29,980 then it gets a complex
250 00:06:29,989 –> 00:06:30,679 conjugation.
251 00:06:31,549 –> 00:06:32,799 So let’s fill in the gaps
252 00:06:32,809 –> 00:06:33,220 again.
253 00:06:33,230 –> 00:06:34,529 And then you see we have
254 00:06:34,540 –> 00:06:36,100 the same complex number left
255 00:06:36,109 –> 00:06:36,739 and right.
256 00:06:36,750 –> 00:06:38,130 But the one is the complex
257 00:06:38,140 –> 00:06:39,480 conjugate of the other one.
258 00:06:40,290 –> 00:06:41,510 In other words, it’s the
259 00:06:41,519 –> 00:06:42,970 absolute value of the complex
260 00:06:42,980 –> 00:06:44,149 number squared.
261 00:06:44,880 –> 00:06:46,359 And since that is clearly
262 00:06:46,369 –> 00:06:47,940 a real number, we can omit
263 00:06:47,950 –> 00:06:48,959 the real part here.
264 00:06:49,769 –> 00:06:50,989 So in summary, we have the
265 00:06:51,000 –> 00:06:52,899 norm of Y squared minus two
266 00:06:52,910 –> 00:06:54,309 times this number,
267 00:06:54,910 –> 00:06:56,500 but also plus one
268 00:06:56,510 –> 00:06:57,869 times the same number.
269 00:06:58,869 –> 00:07:00,089 Therefore, we can put both
270 00:07:00,100 –> 00:07:01,420 things together and have
271 00:07:01,429 –> 00:07:02,600 a simple result here,
272 00:07:03,410 –> 00:07:05,369 the norm of Y squared minus
273 00:07:05,380 –> 00:07:06,720 the inner product in absolute
274 00:07:06,730 –> 00:07:07,540 value squared.
275 00:07:08,359 –> 00:07:09,720 So this is our right hand
276 00:07:09,730 –> 00:07:11,529 side and on the left, we
277 00:07:11,540 –> 00:07:12,339 have the zero.
278 00:07:13,600 –> 00:07:14,739 Hence, in the next step,
279 00:07:14,750 –> 00:07:15,660 I want to bring the inner
280 00:07:15,670 –> 00:07:17,339 product to the other side.
281 00:07:18,380 –> 00:07:20,079 Now, you can see we are almost
282 00:07:20,089 –> 00:07:21,670 there with our inequality.
283 00:07:21,679 –> 00:07:22,859 The only thing missing is
284 00:07:22,869 –> 00:07:23,929 that we have to translate
285 00:07:23,940 –> 00:07:24,959 back X hat
286 00:07:25,820 –> 00:07:27,119 which is not so hard to do
287 00:07:27,130 –> 00:07:28,920 because we define it as X
288 00:07:28,929 –> 00:07:30,670 divided by the norm of X.
289 00:07:31,359 –> 00:07:32,399 Of course, we do the same
290 00:07:32,410 –> 00:07:34,190 as always, we pull out one
291 00:07:34,200 –> 00:07:35,799 divided by the norm of X.
292 00:07:36,359 –> 00:07:37,959 Hence, that’s what we get.
293 00:07:37,970 –> 00:07:39,239 And you see, we just have
294 00:07:39,250 –> 00:07:40,839 to multiply with the norm
295 00:07:40,850 –> 00:07:42,600 of X squared on both sides.
296 00:07:43,679 –> 00:07:45,010 And then taking the square
297 00:07:45,019 –> 00:07:46,579 root on both sides, we get
298 00:07:46,589 –> 00:07:48,070 out our Cauchy Schwarz
299 00:07:48,079 –> 00:07:48,980 inequality.
300 00:07:49,549 –> 00:07:51,130 Well, that’s the whole proof
301 00:07:51,140 –> 00:07:52,279 of the inequality.
302 00:07:52,510 –> 00:07:54,170 I will skip the part about
303 00:07:54,179 –> 00:07:55,970 the equality now because
304 00:07:55,980 –> 00:07:57,790 I want to use time to show
305 00:07:57,799 –> 00:07:58,950 you that the symbol we use
306 00:07:58,959 –> 00:08:00,600 here is indeed the norm.
307 00:08:00,609 –> 00:08:02,059 So it fulfills the triangle
308 00:08:02,070 –> 00:08:03,029 inequality.
309 00:08:03,190 –> 00:08:04,700 And we will do that by
310 00:08:04,709 –> 00:08:06,220 using the Cauchy Schwarz
311 00:08:06,230 –> 00:08:07,059 inequality.
312 00:08:07,920 –> 00:08:09,880 In fact, the triangle inequality
313 00:08:09,890 –> 00:08:11,200 is most of the time, the
314 00:08:11,209 –> 00:08:12,820 hardest part of a proof that
315 00:08:12,829 –> 00:08:13,989 something is a norm.
316 00:08:15,179 –> 00:08:16,779 So let’s do that very quickly.
317 00:08:16,790 –> 00:08:18,059 Of course, we square the
318 00:08:18,070 –> 00:08:18,839 norm again
319 00:08:19,820 –> 00:08:21,119 because then we don’t need
320 00:08:21,130 –> 00:08:22,369 the square root, we can just
321 00:08:22,380 –> 00:08:24,350 write X plus YX
322 00:08:24,359 –> 00:08:26,000 plus Y in the inner product.
323 00:08:26,420 –> 00:08:28,019 The linearity now gets us
324 00:08:28,029 –> 00:08:29,920 to a similar result as before
325 00:08:30,809 –> 00:08:32,750 namely the norm of X and
326 00:08:32,760 –> 00:08:33,619 Y squared.
327 00:08:33,630 –> 00:08:34,799 And in the middle, we get
328 00:08:34,808 –> 00:08:36,479 two times the real part of
329 00:08:36,489 –> 00:08:37,239 the product.
330 00:08:37,890 –> 00:08:39,210 Now, of course, if we use
331 00:08:39,219 –> 00:08:40,409 the absolute value instead
332 00:08:40,419 –> 00:08:42,020 of the real part, it gets
333 00:08:42,030 –> 00:08:42,489 bigger.
334 00:08:43,369 –> 00:08:44,710 And of course, here we now
335 00:08:44,719 –> 00:08:46,599 can use our Cauchy Schwarz
336 00:08:46,609 –> 00:08:47,510 inequality.
337 00:08:48,080 –> 00:08:49,630 And now knowing the binomial
338 00:08:49,640 –> 00:08:51,109 theorem, you see this is
339 00:08:51,119 –> 00:08:52,669 just the square of the sum
340 00:08:52,679 –> 00:08:53,729 of the two norms
341 00:08:54,320 –> 00:08:55,419 taking the square root on
342 00:08:55,429 –> 00:08:56,109 both sides.
343 00:08:56,119 –> 00:08:57,640 You see this is simply the
344 00:08:57,650 –> 00:08:59,130 triangle inequality for the
345 00:08:59,140 –> 00:09:00,820 norm we defined by the inner
346 00:09:00,830 –> 00:09:01,349 product.
347 00:09:02,169 –> 00:09:03,210 The other two properties
348 00:09:03,219 –> 00:09:04,609 for norm shouldn’t be a problem
349 00:09:04,619 –> 00:09:05,250 to show.
350 00:09:05,299 –> 00:09:06,789 And therefore now it makes
351 00:09:06,799 –> 00:09:08,650 sense to use this symbol
352 00:09:08,659 –> 00:09:09,770 and call it a norm in an
353 00:09:09,780 –> 00:09:11,229 inner product space.
354 00:09:11,770 –> 00:09:12,229 OK?
355 00:09:12,239 –> 00:09:13,190 I think that’s good enough
356 00:09:13,200 –> 00:09:13,909 for today.
357 00:09:13,929 –> 00:09:15,169 In the next videos, we will
358 00:09:15,179 –> 00:09:16,590 talk a little bit more about
359 00:09:16,599 –> 00:09:17,669 the geometry.
360 00:09:17,679 –> 00:09:18,989 An inner product gives the
361 00:09:19,000 –> 00:09:20,780 space and I
362 00:09:20,789 –> 00:09:22,280 already told you if you need
363 00:09:22,289 –> 00:09:23,650 it, you can download the
364 00:09:23,659 –> 00:09:25,169 PDF version of this video
365 00:09:25,179 –> 00:09:27,159 in the description and of
366 00:09:27,169 –> 00:09:28,679 course use the comments if
367 00:09:28,690 –> 00:09:29,599 you have questions.
368 00:09:29,719 –> 00:09:30,979 Otherwise I see you in the
369 00:09:30,989 –> 00:09:31,609 next video.
370 00:09:32,590 –> 00:09:33,609 Have a nice day.
371 00:09:33,700 –> 00:09:34,289 Bye.