00:00 Introduction

00:16 Metric space

00:45 Examples

04:07 Discrete metric space

1 00:00:00,600 –> 00:00:02,246 Hello and welcome back to

2 00:00:02,278 –> 00:00:03,358 functional analysis.

3 00:00:03,454 –> 00:00:04,862 First, big thanks to all

4 00:00:04,886 –> 00:00:06,502 the nice people who support

5 00:00:06,606 –> 00:00:08,182 my channel on Steady or

6 00:00:08,206 –> 00:00:08,970 Paypal.

7 00:00:09,590 –> 00:00:10,942 Now, this is part two of

8 00:00:10,966 –> 00:00:12,566 the series, and today we

9 00:00:12,598 –> 00:00:14,006 go deeper into the topic

10 00:00:14,038 –> 00:00:15,690 of metric spaces.

11 00:00:16,390 –> 00:00:17,886 We’ve already learned if

12 00:00:17,918 –> 00:00:19,662 we have a set X and a

13 00:00:19,686 –> 00:00:21,286 metric for this set, then

14 00:00:21,318 –> 00:00:22,878 we call this a metric

15 00:00:22,934 –> 00:00:23,530 space.

16 00:00:24,270 –> 00:00:25,894 In a metric space, one can

17 00:00:25,942 –> 00:00:27,478 now generalize a lot of

18 00:00:27,534 –> 00:00:28,982 notions you may know from

19 00:00:29,006 –> 00:00:30,530 a real analysis course.

20 00:00:31,160 –> 00:00:32,592 For example, we can define

21 00:00:32,656 –> 00:00:34,080 sequences, limits,

22 00:00:34,160 –> 00:00:35,456 accumulation points, and

23 00:00:35,488 –> 00:00:36,300 so on.

24 00:00:36,760 –> 00:00:38,088 Of course, this is what we

25 00:00:38,104 –> 00:00:39,980 will use a lot in this series.

26 00:00:41,000 –> 00:00:42,584 However, before we do that,

27 00:00:42,672 –> 00:00:43,824 let’s talk about some

28 00:00:43,872 –> 00:00:44,700 examples.

29 00:00:45,560 –> 00:00:47,144 So let’s start with a simple

30 00:00:47,192 –> 00:00:48,912 one, which is also one of

31 00:00:48,936 –> 00:00:50,540 the most important ones.

32 00:00:51,400 –> 00:00:52,888 The set is just the complex

33 00:00:52,944 –> 00:00:54,880 numbers, and the metric should

34 00:00:54,920 –> 00:00:56,624 be the usual notion of

35 00:00:56,672 –> 00:00:58,232 measuring distances in the

36 00:00:58,256 –> 00:00:59,870 complex numbers, which means

37 00:00:59,950 –> 00:01:01,486 its the absolute value where

38 00:01:01,518 –> 00:01:02,566 we look at the difference

39 00:01:02,638 –> 00:01:03,770 of both points.

40 00:01:04,750 –> 00:01:06,198 Now, the visualization in

41 00:01:06,214 –> 00:01:07,926 the complex plane gives us

42 00:01:07,958 –> 00:01:09,798 indeed the normal geometry

43 00:01:09,894 –> 00:01:11,330 we have in the plane.

44 00:01:11,870 –> 00:01:13,654 If we pick two complex numbers,

45 00:01:13,702 –> 00:01:15,406 x and y, we know we can

46 00:01:15,438 –> 00:01:16,622 calculate the difference,

47 00:01:16,726 –> 00:01:18,530 which is a new complex number.

48 00:01:19,070 –> 00:01:20,486 And the length of this new

49 00:01:20,518 –> 00:01:22,326 complex number is exactly

50 00:01:22,358 –> 00:01:23,730 the distance we want.

51 00:01:24,840 –> 00:01:26,360 Now, you know this normal

52 00:01:26,400 –> 00:01:28,024 geometry we have for measuring

53 00:01:28,072 –> 00:01:29,824 distances you can also do

54 00:01:29,912 –> 00:01:31,260 in higher dimensions.

55 00:01:31,840 –> 00:01:33,336 For example, if the set is

56 00:01:33,368 –> 00:01:35,184 Rn, we would have a standard

57 00:01:35,232 –> 00:01:36,992 metric we can choose, and

58 00:01:37,016 –> 00:01:38,456 this is usually called the

59 00:01:38,488 –> 00:01:39,952 Euclidean metric or the

60 00:01:39,976 –> 00:01:41,380 euclidean distance.

61 00:01:41,920 –> 00:01:43,584 And because you know Pythagora’s

62 00:01:43,632 –> 00:01:45,352 theorem, you also know the

63 00:01:45,376 –> 00:01:46,700 definition of d.

64 00:01:47,200 –> 00:01:48,632 It’s just the first component

65 00:01:48,656 –> 00:01:50,432 of x minus the first component

66 00:01:50,456 –> 00:01:52,106 of y squared,

67 00:01:52,258 –> 00:01:53,562 plus the second

68 00:01:53,626 –> 00:01:55,466 component, and so

69 00:01:55,498 –> 00:01:57,130 on and so on, until you reach

70 00:01:57,210 –> 00:01:58,402 the last component, which

71 00:01:58,426 –> 00:02:00,026 is xn minus

72 00:02:00,098 –> 00:02:01,950 yn squared.

73 00:02:02,690 –> 00:02:04,306 And then, of course, we need

74 00:02:04,338 –> 00:02:05,390 the square root.

75 00:02:06,170 –> 00:02:07,930 Okay, so this is the euclidean

76 00:02:07,970 –> 00:02:09,810 metric, the common, but

77 00:02:09,850 –> 00:02:11,482 only a possible choice of

78 00:02:11,506 –> 00:02:12,910 a metric in Rn.

79 00:02:13,370 –> 00:02:14,738 So please remember, for a

80 00:02:14,754 –> 00:02:16,050 metric, we only need the

81 00:02:16,090 –> 00:02:16,962 three properties.

82 00:02:17,066 –> 00:02:18,926 It should be positive definite,

83 00:02:18,948 –> 00:02:20,642 symmetric, and fulfill the

84 00:02:20,666 –> 00:02:22,190 triangle inequality.

85 00:02:22,690 –> 00:02:24,034 We already know that these

86 00:02:24,082 –> 00:02:25,914 two fulfill the three properties.

87 00:02:26,002 –> 00:02:27,698 So there are metrics on

88 00:02:27,754 –> 00:02:29,630 Rn or C respectively.

89 00:02:30,290 –> 00:02:31,506 So let’s look at another

90 00:02:31,578 –> 00:02:32,602 distance function we could

91 00:02:32,626 –> 00:02:33,954 define for Rn.

92 00:02:34,042 –> 00:02:35,378 This would be the

93 00:02:35,434 –> 00:02:36,390 maximum,

94 00:02:37,530 –> 00:02:38,954 and then we look at the difference

95 00:02:39,042 –> 00:02:40,842 x one minus y

96 00:02:40,906 –> 00:02:42,650 one, and use the normal

97 00:02:42,690 –> 00:02:44,400 absolute value, or modulus

98 00:02:44,530 –> 00:02:45,480 in r.

99 00:02:46,860 –> 00:02:48,060 So let’s write it down for

100 00:02:48,100 –> 00:02:49,796 all components, which means

101 00:02:49,908 –> 00:02:51,652 we want the maximum of

102 00:02:51,716 –> 00:02:53,120 all these differences.

103 00:02:53,980 –> 00:02:55,260 In fact, this is of course,

104 00:02:55,300 –> 00:02:56,316 a metric as well.

105 00:02:56,388 –> 00:02:57,588 It’s not hard to check that

106 00:02:57,604 –> 00:02:59,348 it fulfills our three properties.

107 00:02:59,484 –> 00:03:00,860 And most of the time, only

108 00:03:00,900 –> 00:03:02,612 the triangle inequality

109 00:03:02,756 –> 00:03:04,080 needs some work.

110 00:03:04,700 –> 00:03:05,892 However, we should first

111 00:03:05,956 –> 00:03:07,116 visualize the whole thing,

112 00:03:07,188 –> 00:03:08,840 maybe again in the plane.

113 00:03:09,220 –> 00:03:10,572 So here we see two points,

114 00:03:10,636 –> 00:03:11,348 x and y.

115 00:03:11,444 –> 00:03:12,540 And what we now have to do

116 00:03:12,580 –> 00:03:13,676 is calculate the distances

117 00:03:13,748 –> 00:03:15,400 of the components here.

118 00:03:15,910 –> 00:03:17,246 Now in blue we have the distance

119 00:03:17,278 –> 00:03:18,558 in the first component and

120 00:03:18,574 –> 00:03:20,190 in green in the second component.

121 00:03:20,270 –> 00:03:21,886 And as you can see, the blue

122 00:03:21,918 –> 00:03:22,918 one is bigger.

123 00:03:23,014 –> 00:03:24,542 So the maximum is indeed

124 00:03:24,606 –> 00:03:25,850 this number here,

125 00:03:26,430 –> 00:03:27,894 which means using this

126 00:03:27,942 –> 00:03:29,702 metric, this is exactly the

127 00:03:29,726 –> 00:03:31,358 distance between x and y.

128 00:03:31,494 –> 00:03:33,142 Or in other words, if you

129 00:03:33,166 –> 00:03:34,862 choose a point z here, for

130 00:03:34,886 –> 00:03:36,686 example, it has the same

131 00:03:36,758 –> 00:03:38,486 distance from x than the

132 00:03:38,518 –> 00:03:39,310 distance x to

133 00:03:39,350 –> 00:03:40,940 y.

134 00:03:41,010 –> 00:03:42,056 Of course this shouldn’t

135 00:03:42,088 –> 00:03:43,008 be so surprising.

136 00:03:43,104 –> 00:03:44,224 Of course you could have

137 00:03:44,272 –> 00:03:45,472 different points that have

138 00:03:45,496 –> 00:03:46,728 the same distance from a

139 00:03:46,744 –> 00:03:47,980 chosen point x.

140 00:03:48,560 –> 00:03:50,032 In our Euclidean metric from

141 00:03:50,056 –> 00:03:51,168 above or in a picture in

142 00:03:51,184 –> 00:03:52,472 a complex plane, this would

143 00:03:52,496 –> 00:03:54,080 be just a circle

144 00:03:54,160 –> 00:03:55,180 around x.

145 00:03:56,240 –> 00:03:57,920 However, with another metric,

146 00:03:58,000 –> 00:03:59,320 such a thing we could call

147 00:03:59,360 –> 00:04:01,160 a circle might look completely

148 00:04:01,200 –> 00:04:01,780 different.

149 00:04:02,440 –> 00:04:02,936 Okay,

150 00:04:02,968 –> 00:04:04,328 So please keep that in mind

151 00:04:04,384 –> 00:04:05,872 whenever we use circles to

152 00:04:05,896 –> 00:04:07,020 visualize things.

153 00:04:07,840 –> 00:04:09,216 For our last example here,

154 00:04:09,248 –> 00:04:10,576 lets choose a more abstract

155 00:04:10,648 –> 00:04:12,168 one, just any set

156 00:04:12,224 –> 00:04:13,728 X which should not be the

157 00:04:13,744 –> 00:04:14,540 empty set.

158 00:04:15,360 –> 00:04:16,992 Then we define a metric

159 00:04:17,096 –> 00:04:18,504 by distinguishing two

160 00:04:18,552 –> 00:04:19,140 cases.

161 00:04:20,000 –> 00:04:21,720 The first case is just X

162 00:04:21,760 –> 00:04:23,656 is equal to y and the

163 00:04:23,688 –> 00:04:25,592 second, of course x is not

164 00:04:25,656 –> 00:04:26,860 equal to y.

165 00:04:27,480 –> 00:04:28,776 By the first property of

166 00:04:28,808 –> 00:04:30,536 the metric we already know

167 00:04:30,648 –> 00:04:31,980 here we need a zero.

168 00:04:32,930 –> 00:04:34,266 However, we are not allowed

169 00:04:34,298 –> 00:04:35,938 to have a zero here, so we

170 00:04:35,954 –> 00:04:37,250 can choose whatever we want.

171 00:04:37,330 –> 00:04:38,594 And most of the time we just

172 00:04:38,642 –> 00:04:39,710 choose one.

173 00:04:40,610 –> 00:04:42,090 Ok, so let’s check together

174 00:04:42,170 –> 00:04:44,110 that this is indeed a metric.

175 00:04:44,650 –> 00:04:46,010 As it happens very often

176 00:04:46,090 –> 00:04:47,442 the first two properties

177 00:04:47,546 –> 00:04:49,150 are not a problem at all.

178 00:04:49,570 –> 00:04:51,290 Its positive definite by

179 00:04:51,330 –> 00:04:52,290 our construction.

180 00:04:52,370 –> 00:04:53,870 So no problem here.

181 00:04:54,570 –> 00:04:56,026 The second property is the

182 00:04:56,058 –> 00:04:56,762 symmetry.

183 00:04:56,906 –> 00:04:58,266 And here you can see the

184 00:04:58,298 –> 00:04:59,974 whole definition is symmetric.

185 00:05:00,082 –> 00:05:02,010 So no problem here at all.

186 00:05:02,470 –> 00:05:04,086 The third property is the

187 00:05:04,118 –> 00:05:05,662 triangle inequality.

188 00:05:05,806 –> 00:05:07,606 And this one I want to

189 00:05:07,638 –> 00:05:09,598 write down in

190 00:05:09,614 –> 00:05:10,934 order to show that we need

191 00:05:10,982 –> 00:05:12,862 three points, so x,

192 00:05:12,886 –> 00:05:14,686 y and z out of the set.

193 00:05:14,838 –> 00:05:16,650 And then we look at the inequality.

194 00:05:17,150 –> 00:05:18,342 This means that we go the

195 00:05:18,366 –> 00:05:19,926 detour over the point z.

196 00:05:20,038 –> 00:05:21,766 Or in other words, we add

197 00:05:21,798 –> 00:05:23,662 the distances on the right.

198 00:05:23,846 –> 00:05:25,286 Okay, so this is what we

199 00:05:25,318 –> 00:05:26,526 have to show in general.

200 00:05:26,638 –> 00:05:28,190 So let’s first look at two

201 00:05:28,270 –> 00:05:29,210 different cases.

202 00:05:29,750 –> 00:05:30,966 Now the first case would

203 00:05:30,998 –> 00:05:32,510 be x is equal to

204 00:05:32,550 –> 00:05:34,454 y because then we know

205 00:05:34,542 –> 00:05:36,254 the distance of x to y

206 00:05:36,342 –> 00:05:37,730 is exactly zero.

207 00:05:38,350 –> 00:05:39,742 Okay, so let’s write that

208 00:05:39,806 –> 00:05:40,410 here.

209 00:05:40,750 –> 00:05:41,942 This is zero.

210 00:05:42,126 –> 00:05:42,774 Okay.

211 00:05:42,862 –> 00:05:44,670 But then we know this inequality

212 00:05:44,710 –> 00:05:46,102 is correct because on the

213 00:05:46,126 –> 00:05:47,798 right hand side we just have

214 00:05:47,854 –> 00:05:49,678 the metric or the distances

215 00:05:49,814 –> 00:05:51,222 and we already know they

216 00:05:51,246 –> 00:05:52,662 are positive or zero.

217 00:05:52,726 –> 00:05:54,720 In the worst case, this

218 00:05:54,760 –> 00:05:56,064 means that the triangle

219 00:05:56,112 –> 00:05:57,744 inequality in this first

220 00:05:57,792 –> 00:05:59,500 case is satisfied

221 00:06:00,360 –> 00:06:01,912 now, the second case of course,

222 00:06:01,976 –> 00:06:03,016 should be the opposite.

223 00:06:03,128 –> 00:06:04,816 So we have x is not equal

224 00:06:04,848 –> 00:06:06,832 to y, which means the distance

225 00:06:06,896 –> 00:06:08,760 is by definition exactly

226 00:06:08,800 –> 00:06:09,380 one.

227 00:06:10,000 –> 00:06:11,616 Now, because we know that

228 00:06:11,648 –> 00:06:13,312 the only allowed distances

229 00:06:13,416 –> 00:06:15,272 are zero or one, there

230 00:06:15,296 –> 00:06:16,448 is nothing in between.

231 00:06:16,624 –> 00:06:18,376 We know that xz or

232 00:06:18,408 –> 00:06:20,204 zy have also

233 00:06:20,292 –> 00:06:21,440 one or zero.

234 00:06:22,180 –> 00:06:23,900 Or to put it in another way,

235 00:06:24,020 –> 00:06:24,708 we know that

236 00:06:24,764 –> 00:06:26,280 d(x,z)

237 00:06:26,820 –> 00:06:28,284 or the other one.

238 00:06:28,372 –> 00:06:30,300 So zy is

239 00:06:30,340 –> 00:06:31,880 also exactly one.

240 00:06:32,540 –> 00:06:34,012 At least one of them has

241 00:06:34,036 –> 00:06:35,756 to be one, otherwise both

242 00:06:35,788 –> 00:06:37,356 of them would be zero, which

243 00:06:37,388 –> 00:06:38,700 means that z is equal to

244 00:06:38,740 –> 00:06:40,372 x and z is equal to y.

245 00:06:40,476 –> 00:06:42,076 But then this would be not

246 00:06:42,108 –> 00:06:43,160 a second case.

247 00:06:44,100 –> 00:06:45,580 However, this means now,

248 00:06:45,660 –> 00:06:47,180 if we add both of these

249 00:06:47,220 –> 00:06:49,156 distances, we will get

250 00:06:49,228 –> 00:06:49,676 one.

251 00:06:49,788 –> 00:06:50,916 Or in the worst case, when

252 00:06:50,948 –> 00:06:51,972 both of them are one, we

253 00:06:51,996 –> 00:06:52,996 get out two.

254 00:06:53,108 –> 00:06:54,960 So we have the inequality.

255 00:06:56,140 –> 00:06:57,060 So very good.

256 00:06:57,140 –> 00:06:58,244 Also in the second case,

257 00:06:58,332 –> 00:07:00,236 the triangle inequality is

258 00:07:00,268 –> 00:07:01,120 fulfilled.

259 00:07:01,820 –> 00:07:03,620 Well now, to close this example,

260 00:07:03,660 –> 00:07:05,636 let’s recall that this definition

261 00:07:05,708 –> 00:07:06,828 defines a metric.

262 00:07:06,924 –> 00:07:08,748 It works on any set, and

263 00:07:08,764 –> 00:07:10,116 it’s called the discrete

264 00:07:10,148 –> 00:07:10,840 metric.

265 00:07:11,460 –> 00:07:13,228 So you might try to visualize

266 00:07:13,284 –> 00:07:14,454 this metric space.

267 00:07:14,612 –> 00:07:15,874 It’s a little bit strange

268 00:07:15,962 –> 00:07:17,570 because there are no neighbors

269 00:07:17,610 –> 00:07:19,290 around a given point, simply

270 00:07:19,330 –> 00:07:20,754 because theres a fixed

271 00:07:20,802 –> 00:07:22,618 distance from one point to

272 00:07:22,674 –> 00:07:23,990 all the other ones.

273 00:07:24,490 –> 00:07:26,138 This means that all the points

274 00:07:26,234 –> 00:07:27,710 are isolated points.

275 00:07:28,410 –> 00:07:29,954 Ok, so now we have a few

276 00:07:30,002 –> 00:07:31,530 examples of metric spaces

277 00:07:31,570 –> 00:07:32,482 we can work with.

278 00:07:32,586 –> 00:07:34,346 And in the next video I will

279 00:07:34,378 –> 00:07:35,930 talk about other objects

280 00:07:35,970 –> 00:07:37,950 we find in metric spaces.

281 00:07:38,610 –> 00:07:40,386 So thanks for listening and

282 00:07:40,418 –> 00:07:41,570 I hope I see you in the next

283 00:07:41,610 –> 00:07:42,074 video.

284 00:07:42,202 –> 00:07:42,990 Bye.

Q1: Consider the set $\mathbb{C}$. Is the map $d(x,y) := \frac{1}{4} |x-y|$ a metric?

A1: Yes, all properties are satisfied.

A2: No, the triangle inequality is not satisfied.

A3: No, because $d(x,y) = 0 \Rightarrow x = y$ is not satisfied.

A4: No, because $d(x,y) = d(y,x)$ is not satisfied.

Q2: For any set $X$, the map $d: X \times X \rightarrow [0,\infty)$ with $d(x,y) = 1$ for $x \neq y$ and $d(x,y) = 0$ for $x = y$ is a metric

A1: True

A2: False

Q3: Consider the set $\mathbb{R}^n$. Are there many different metrics on this set?

A1: Yes, there is the euclidean metric and the discrete metric, for example.

A2: No, there is only one metric, the euclidean metric.