00:00 Introduction

02:25 Inner product definition

06:25 Norm and Hilbert space

1 00:00:00,469 –> 00:00:02,210 Hello and welcome back to

2 00:00:02,220 –> 00:00:03,390 functional analysis.

3 00:00:03,500 –> 00:00:04,610 And as always, I want to

4 00:00:04,619 –> 00:00:06,090 thank all the nice people

5 00:00:06,099 –> 00:00:07,780 that support my channel on

6 00:00:07,789 –> 00:00:08,920 Steady or paypal.

7 00:00:09,739 –> 00:00:11,229 In today’s part eight, we

8 00:00:11,239 –> 00:00:12,819 introduce inner products

9 00:00:12,829 –> 00:00:14,130 and Hilbert spaces.

10 00:00:15,250 –> 00:00:16,819 You will see such an inner

11 00:00:16,829 –> 00:00:18,399 product gives the space more

12 00:00:18,409 –> 00:00:19,149 structure.

13 00:00:19,309 –> 00:00:20,950 So maybe let’s recall which

14 00:00:20,959 –> 00:00:22,309 structures we already know.

15 00:00:22,840 –> 00:00:24,040 First, we discussed

16 00:00:24,049 –> 00:00:25,870 metrics and we learned

17 00:00:25,879 –> 00:00:27,270 that a metric measures

18 00:00:27,280 –> 00:00:28,700 distances in the space.

19 00:00:29,739 –> 00:00:31,280 Afterwards, we introduced

20 00:00:31,290 –> 00:00:33,240 norms for vector spaces.

21 00:00:34,029 –> 00:00:35,229 And there we learned that

22 00:00:35,240 –> 00:00:36,569 a norm measures distances

23 00:00:36,580 –> 00:00:38,250 as well, but it also

24 00:00:38,259 –> 00:00:39,750 measures lengths of

25 00:00:39,759 –> 00:00:40,369 vectors.

26 00:00:41,099 –> 00:00:42,669 And now an inner product

27 00:00:42,680 –> 00:00:43,990 should do even more in a

28 00:00:44,000 –> 00:00:44,869 vector space.

29 00:00:45,549 –> 00:00:47,270 Besides measuring distances

30 00:00:47,279 –> 00:00:49,209 and lengths, it is also able

31 00:00:49,220 –> 00:00:51,119 to measure angles between

32 00:00:51,130 –> 00:00:51,880 two vectors.

33 00:00:52,580 –> 00:00:54,119 So you see an inner product

34 00:00:54,130 –> 00:00:56,020 gives you a geometry like

35 00:00:56,029 –> 00:00:57,279 you naturally have it on

36 00:00:57,290 –> 00:00:58,599 the plane or the surrounding

37 00:00:58,610 –> 00:00:59,080 space.

38 00:00:59,779 –> 00:01:01,349 The user notation one uses

39 00:01:01,360 –> 00:01:03,009 for an inner product is given

40 00:01:03,020 –> 00:01:04,319 by these brackets.

41 00:01:05,080 –> 00:01:06,779 So what we have is one vector

42 00:01:06,790 –> 00:01:08,349 X on the left hand side and

43 00:01:08,360 –> 00:01:09,970 another vector Y on the right

44 00:01:09,980 –> 00:01:11,809 hand side, a nice

45 00:01:11,819 –> 00:01:13,290 visualization for this would

46 00:01:13,300 –> 00:01:15,169 be to have the vector X on

47 00:01:15,180 –> 00:01:16,360 the horizontal line

48 00:01:17,169 –> 00:01:18,930 and the vector Y in another

49 00:01:18,940 –> 00:01:20,349 direction as an arrow,

50 00:01:21,099 –> 00:01:22,849 which means in this picture,

51 00:01:22,860 –> 00:01:24,610 we find an angle which we

52 00:01:24,620 –> 00:01:25,599 could call alpha.

53 00:01:25,610 –> 00:01:26,330 For example.

54 00:01:27,120 –> 00:01:28,629 Now the inner product should

55 00:01:28,639 –> 00:01:30,160 be a multiplication of these

56 00:01:30,169 –> 00:01:31,889 two vectors where only the

57 00:01:31,900 –> 00:01:33,279 component of Y

58 00:01:33,300 –> 00:01:34,889 interdiction of X is

59 00:01:34,900 –> 00:01:36,459 needed in the

60 00:01:36,470 –> 00:01:37,050 picture.

61 00:01:37,059 –> 00:01:38,400 This would mean that we have

62 00:01:38,410 –> 00:01:40,099 here a wide angle and

63 00:01:40,110 –> 00:01:41,900 only look at this arrow here.

64 00:01:42,470 –> 00:01:43,709 Now, since you’re good with

65 00:01:43,720 –> 00:01:45,059 trigonometric functions,

66 00:01:45,069 –> 00:01:46,419 you can easily calculate

67 00:01:46,430 –> 00:01:48,059 the length of this yellow

68 00:01:48,069 –> 00:01:48,580 arrow.

69 00:01:49,230 –> 00:01:50,889 It is simply the length of

70 00:01:50,900 –> 00:01:52,389 Y times the

71 00:01:52,800 –> 00:01:53,900 cosine of alpha.

72 00:01:54,949 –> 00:01:56,430 And this length now should

73 00:01:56,440 –> 00:01:58,209 be multiplied with the length

74 00:01:58,220 –> 00:01:59,519 of the AX.

75 00:02:00,389 –> 00:02:02,209 And this formula now explains

76 00:02:02,220 –> 00:02:03,529 what we want from an inner

77 00:02:03,540 –> 00:02:04,169 product.

78 00:02:04,760 –> 00:02:06,089 However, it does not define

79 00:02:06,099 –> 00:02:06,769 anything yet.

80 00:02:06,779 –> 00:02:07,910 It does not explain what

81 00:02:07,919 –> 00:02:09,258 the norm should mean here

82 00:02:09,369 –> 00:02:10,820 and what the angle alpha

83 00:02:10,830 –> 00:02:12,369 is in a general vector space.

84 00:02:13,169 –> 00:02:14,360 Of course, the correct logic

85 00:02:14,369 –> 00:02:15,759 should be that we start with

86 00:02:15,770 –> 00:02:17,110 an inner product, then we

87 00:02:17,119 –> 00:02:18,630 define norm and then we

88 00:02:18,639 –> 00:02:20,429 define the angle such that

89 00:02:20,440 –> 00:02:21,759 we get a formula in this

90 00:02:21,770 –> 00:02:22,149 sense.

91 00:02:23,179 –> 00:02:24,490 In order to do that, we now

92 00:02:24,500 –> 00:02:26,289 finally state the definition

93 00:02:27,240 –> 00:02:28,720 as usual, we use the letter

94 00:02:28,729 –> 00:02:30,539 F to denote either R or

95 00:02:30,550 –> 00:02:32,490 C and then X

96 00:02:32,500 –> 00:02:34,199 should be just an F vector

97 00:02:34,210 –> 00:02:34,660 space.

98 00:02:35,679 –> 00:02:36,979 Now, the inner product is

99 00:02:36,990 –> 00:02:38,979 just a map where we use these

100 00:02:38,990 –> 00:02:39,839 angle brackets.

101 00:02:39,850 –> 00:02:41,520 Again, the

102 00:02:41,529 –> 00:02:43,389 input is two vectors such

103 00:02:43,399 –> 00:02:44,639 that the domain should be

104 00:02:44,649 –> 00:02:46,029 X times X.

105 00:02:46,750 –> 00:02:48,199 We’ve already discussed that

106 00:02:48,210 –> 00:02:49,619 the outcome should be just

107 00:02:49,630 –> 00:02:50,259 a number.

108 00:02:50,270 –> 00:02:51,850 So is scalar in F

109 00:02:52,500 –> 00:02:53,970 and there we already see

110 00:02:53,979 –> 00:02:55,809 that if we are in the complex

111 00:02:55,820 –> 00:02:57,309 numbers in a complex vector

112 00:02:57,320 –> 00:02:58,699 space, this formula from

113 00:02:58,710 –> 00:03:00,130 above can’t be completely

114 00:03:00,139 –> 00:03:00,529 true.

115 00:03:01,520 –> 00:03:02,850 So let’s use some quotation

116 00:03:02,860 –> 00:03:04,619 marks to say that this was

117 00:03:04,630 –> 00:03:06,449 just an informal explanation

118 00:03:06,460 –> 00:03:07,240 at the beginning.

119 00:03:07,750 –> 00:03:09,229 Nevertheless, it can still

120 00:03:09,240 –> 00:03:10,789 guide us through the definition

121 00:03:10,800 –> 00:03:11,110 here.

122 00:03:11,850 –> 00:03:12,220 OK.

123 00:03:12,229 –> 00:03:13,589 Now, such a map is called

124 00:03:13,600 –> 00:03:14,380 an inner product.

125 00:03:14,389 –> 00:03:16,360 If it fulfills three properties.

126 00:03:16,919 –> 00:03:18,110 The first property tells

127 00:03:18,119 –> 00:03:19,250 us that the inner product

128 00:03:19,259 –> 00:03:20,600 should be able to measure

129 00:03:20,610 –> 00:03:21,139 lengths.

130 00:03:21,979 –> 00:03:23,770 This means that if we put

131 00:03:23,779 –> 00:03:25,729 the same vector X into both

132 00:03:25,740 –> 00:03:27,289 sides, then the

133 00:03:27,300 –> 00:03:28,539 outcome should be a nonne

134 00:03:29,300 –> 00:03:30,970 real number no matter if

135 00:03:30,979 –> 00:03:31,949 we are in the real or the

136 00:03:31,960 –> 00:03:32,809 complex case.

137 00:03:33,639 –> 00:03:35,169 On the other hand, you already

138 00:03:35,179 –> 00:03:36,970 know length zero should be

139 00:03:36,979 –> 00:03:38,660 only possible for the zero

140 00:03:38,669 –> 00:03:39,199 vector.

141 00:03:40,050 –> 00:03:41,779 In other words, XX is

142 00:03:41,789 –> 00:03:43,720 zero if and only if

143 00:03:43,729 –> 00:03:45,100 X is the zero vector.

144 00:03:45,830 –> 00:03:47,110 Of course, you recognize

145 00:03:47,119 –> 00:03:48,820 this property, it’s called

146 00:03:48,830 –> 00:03:50,059 positive definite.

147 00:03:50,710 –> 00:03:51,779 Indeed, you see it’s the

148 00:03:51,789 –> 00:03:53,460 same first property as we

149 00:03:53,470 –> 00:03:54,490 had it for the norm.

150 00:03:55,320 –> 00:03:56,630 Now, the second property

151 00:03:56,639 –> 00:03:58,350 is about what changes when

152 00:03:58,360 –> 00:04:00,009 we exchange the two vectors

153 00:04:00,020 –> 00:04:01,020 in the inner product

154 00:04:01,929 –> 00:04:03,130 having a picture from above

155 00:04:03,139 –> 00:04:04,830 in mind, you see we still

156 00:04:04,839 –> 00:04:05,960 have the same angle.

157 00:04:05,970 –> 00:04:06,940 So it shouldn’t make any

158 00:04:06,949 –> 00:04:08,270 difference when we exchange

159 00:04:08,279 –> 00:04:09,039 the two vectors.

160 00:04:10,160 –> 00:04:11,539 So this should be

161 00:04:11,820 –> 00:04:13,380 YNX.

162 00:04:14,649 –> 00:04:16,178 However, this only makes

163 00:04:16,190 –> 00:04:17,709 sense in the case that we

164 00:04:17,720 –> 00:04:19,399 are in a real vector space.

165 00:04:19,410 –> 00:04:21,200 So our F is indeed

166 00:04:21,209 –> 00:04:21,678 R.

167 00:04:22,859 –> 00:04:24,600 Now from complex vector space,

168 00:04:24,609 –> 00:04:26,279 we’ve already seen that this

169 00:04:26,290 –> 00:04:27,799 equation here can’t be the

170 00:04:27,809 –> 00:04:28,739 complete truth.

171 00:04:29,429 –> 00:04:31,230 Indeed, the complex conjugation

172 00:04:31,239 –> 00:04:32,600 has to be involved here.

173 00:04:33,329 –> 00:04:34,589 Please keep in mind that

174 00:04:34,600 –> 00:04:36,230 in the first property also,

175 00:04:36,239 –> 00:04:37,970 in a complex case, we want

176 00:04:37,980 –> 00:04:39,709 to get out a real number

177 00:04:39,720 –> 00:04:41,029 when we put in the same vector

178 00:04:41,040 –> 00:04:41,540 twice.

179 00:04:42,420 –> 00:04:43,929 So therefore, if we exchange

180 00:04:43,940 –> 00:04:45,640 it to vector here, we also

181 00:04:45,649 –> 00:04:47,609 have to add a complex conjugation.

182 00:04:48,559 –> 00:04:49,959 For this reason, we call

183 00:04:49,970 –> 00:04:51,459 this property conjugate

184 00:04:51,470 –> 00:04:52,779 symmetry or just

185 00:04:52,790 –> 00:04:54,450 symmetry if we are in a real

186 00:04:54,459 –> 00:04:54,899 case.

187 00:04:55,559 –> 00:04:56,989 And now the third and last

188 00:04:57,000 –> 00:04:58,970 property should finally be

189 00:04:58,980 –> 00:04:59,899 the linearity.

190 00:05:00,700 –> 00:05:02,089 The linearity makes sense.

191 00:05:02,100 –> 00:05:03,070 If you look at the picture

192 00:05:03,079 –> 00:05:04,720 again, if you scale

193 00:05:04,730 –> 00:05:06,230 Y or if you add another

194 00:05:06,239 –> 00:05:08,209 vector, then you could just

195 00:05:08,220 –> 00:05:09,790 do the same with the outcome

196 00:05:10,690 –> 00:05:11,910 for the formula, we then

197 00:05:11,920 –> 00:05:13,049 need two vectors.

198 00:05:13,059 –> 00:05:14,640 So let’s call them why one

199 00:05:14,649 –> 00:05:16,309 and why two and add them

200 00:05:17,589 –> 00:05:18,529 the inner product should

201 00:05:18,540 –> 00:05:19,670 be additive in the second

202 00:05:19,679 –> 00:05:21,089 component, which means we

203 00:05:21,100 –> 00:05:22,570 can pull out the addition

204 00:05:22,579 –> 00:05:22,890 here.

205 00:05:23,339 –> 00:05:24,429 Of course, it should also

206 00:05:24,440 –> 00:05:26,109 be homogeneous, which means

207 00:05:26,119 –> 00:05:27,600 we can pull out a scalar.

208 00:05:28,299 –> 00:05:29,720 So let’s call that scalar

209 00:05:29,730 –> 00:05:30,359 lambda.

210 00:05:30,369 –> 00:05:31,549 And then the equation looks

211 00:05:31,559 –> 00:05:33,510 like this, which means

212 00:05:33,519 –> 00:05:34,769 that on the right hand side,

213 00:05:34,779 –> 00:05:36,459 we just have the multiplication

214 00:05:36,470 –> 00:05:38,070 of two numbers in F.

215 00:05:38,820 –> 00:05:40,190 And indeed, that’s the part

216 00:05:40,200 –> 00:05:41,640 that only works in the second

217 00:05:41,649 –> 00:05:43,269 component because if you

218 00:05:43,279 –> 00:05:44,640 want to do that in the first

219 00:05:44,649 –> 00:05:46,269 component as well, you have

220 00:05:46,279 –> 00:05:47,600 to use property two.

221 00:05:48,299 –> 00:05:49,619 And that one tells you you

222 00:05:49,630 –> 00:05:50,799 don’t have any problem in

223 00:05:50,809 –> 00:05:51,670 the real case.

224 00:05:51,679 –> 00:05:53,239 But in the complex case,

225 00:05:53,250 –> 00:05:55,109 you get out lambda bar

226 00:05:55,119 –> 00:05:55,920 instead of lambda.

227 00:05:56,690 –> 00:05:58,079 So please keep that in mind.

228 00:05:58,089 –> 00:05:59,880 Both sides are not the same.

229 00:06:00,709 –> 00:06:02,059 And here we have chosen the

230 00:06:02,070 –> 00:06:03,700 second part to be the linear

231 00:06:03,709 –> 00:06:04,059 one.

232 00:06:04,700 –> 00:06:06,399 I emphasize that because

233 00:06:06,410 –> 00:06:07,950 there are other people that

234 00:06:07,959 –> 00:06:09,359 choose the first part to

235 00:06:09,369 –> 00:06:10,339 be the linear one.

236 00:06:10,890 –> 00:06:12,440 Therefore, please be careful

237 00:06:12,450 –> 00:06:12,720 there.

238 00:06:13,570 –> 00:06:14,000 OK.

239 00:06:14,010 –> 00:06:15,670 So that’s the whole definition

240 00:06:15,679 –> 00:06:16,829 of an inner product.

241 00:06:16,950 –> 00:06:18,350 And the vector space X

242 00:06:18,359 –> 00:06:20,059 together with an inner product,

243 00:06:20,070 –> 00:06:21,940 we just call an inner product

244 00:06:21,950 –> 00:06:22,309 space.

245 00:06:23,149 –> 00:06:24,609 Now, an important part we’ve

246 00:06:24,619 –> 00:06:26,269 already mentioned is that

247 00:06:26,279 –> 00:06:27,730 in an inner product space,

248 00:06:27,739 –> 00:06:29,279 we can measure lengths

249 00:06:29,799 –> 00:06:31,329 and we can do that simply

250 00:06:31,339 –> 00:06:32,929 by defining a norm.

251 00:06:32,940 –> 00:06:34,850 And we do this by using the

252 00:06:34,859 –> 00:06:36,010 formula from above.

253 00:06:36,019 –> 00:06:37,989 So we set the norm of X

254 00:06:38,000 –> 00:06:39,320 to be the square root of

255 00:06:39,329 –> 00:06:40,209 the inner product

256 00:06:41,100 –> 00:06:43,079 by using these three properties.

257 00:06:43,089 –> 00:06:44,929 It’s indeed easy to show

258 00:06:44,940 –> 00:06:46,559 that this defines a norm.

259 00:06:47,250 –> 00:06:48,910 And if we need to be careful

260 00:06:48,920 –> 00:06:50,420 which norm we talk about,

261 00:06:50,450 –> 00:06:52,429 we can set the inner product

262 00:06:52,440 –> 00:06:53,510 as an index here.

263 00:06:54,350 –> 00:06:55,809 Now, to close this video,

264 00:06:55,820 –> 00:06:57,540 I tell you now what a Hilbert

265 00:06:57,549 –> 00:06:59,399 space is an

266 00:06:59,410 –> 00:07:00,799 inner product space, which

267 00:07:00,809 –> 00:07:02,500 means X together with an

268 00:07:02,510 –> 00:07:04,489 inner product, it’s

269 00:07:04,500 –> 00:07:06,209 called a Hilbert space

270 00:07:07,260 –> 00:07:08,730 if X together with the

271 00:07:08,739 –> 00:07:10,619 corresponding norm is a ban

272 00:07:10,630 –> 00:07:11,380 of space.

273 00:07:12,040 –> 00:07:13,920 So this means we have a vector

274 00:07:13,929 –> 00:07:15,339 space where we can measure

275 00:07:15,350 –> 00:07:16,950 lengths and angles and it’s

276 00:07:16,959 –> 00:07:18,739 also a complete metric space.

277 00:07:19,779 –> 00:07:20,239 OK?

278 00:07:20,250 –> 00:07:21,149 I think that’s good enough

279 00:07:21,160 –> 00:07:21,859 for today.

280 00:07:21,869 –> 00:07:23,660 I see you next time when

281 00:07:23,670 –> 00:07:25,250 we discuss examples of Hilbert

282 00:07:25,260 –> 00:07:25,829 spaces.

283 00:07:26,540 –> 00:07:28,119 So have a nice day and see

284 00:07:28,130 –> 00:07:28,640 you then.

285 00:07:28,649 –> 00:07:29,279 Bye.

Q1: Let $\langle \cdot, \cdot \rangle$ be an inner product on $\mathbb{C}^n$. What is not correct in general?

A1: $\langle x, y \rangle = 0$ implies $x = y$

A2: $\langle x, x \rangle = 0$ implies $x = 0$

A3: $\langle x, 0 \rangle = 0$ for all $x \in \mathbb{C}^n$

A4: $\langle x, i y\rangle = - \langle i x, y\rangle$ for all $x,y \in \mathbb{C}^n$

Q2: Is $\mathbb{R}^n$ together with the standard inner product a Hilbert space?

A1: Yes!

A2: No!

A3: One needs more information.

Q3: Is $\mathbb{C}^n$ together with the standard inner product a Hilbert space?

A1: Yes!

A2: No!

A3: One needs more information.