Title: Metric Space
Series: Functional Analysis
YouTube-Title: Functional Analysis 1 | Metric Space - How to Measure Distances?
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Timestamps00:00 Introduction into functional analysis 01:49 Metric space (introduction) 02:25 Metric (definition) 03:20 Metric (definiteness) 03:36 Metric (symmetry) 04:07 Metric (triangle inequality) 05:31 Credits
Subtitle in English
1 00:00:00,560 –> 00:00:03,200 Hello and welcome back to a new video.
2 00:00:03,200 –> 00:00:07,760 And as always I want to thank all the nice people that support this channel on steady.
3 00:00:08,560 –> 00:00:14,480 The topic for today is functional analysis because this is the one most of you voted for.
4 00:00:15,680 –> 00:00:20,800 Therefore now I have a whole serious planned to get into the real interesting stuff concerning
5 00:00:20,800 –> 00:00:27,280 functional analysis. What you should know is that functional analysis is a really wide field.
6 00:00:28,160 –> 00:00:32,720 However to get an idea what it is about you can just look at these two blocks.
7 00:00:33,760 –> 00:00:39,120 On the one side you have the linear algebra and on the other side you have the real or complex
8 00:00:39,120 –> 00:00:46,000 analysis. In a rough sense if you look at infinite dimensional spaces in the linear algebra sense,
9 00:00:47,120 –> 00:00:52,960 and if you want to apply some analysis for these spaces, then you are on the realm of functional
10 00:00:52,960 –> 00:00:59,840 analysis. Of course there’s much more to it but if you have some knowledge in linear algebra and
11 00:00:59,840 –> 00:01:07,280 analysis now you know what to expect. For example here we consider spaces consisting of functions
12 00:01:07,280 –> 00:01:12,880 or of sequences and of course also linear maps between such spaces.
13 00:01:14,000 –> 00:01:19,280 Because in this field we always combine algebraic structures with analytic properties,
14 00:01:19,280 –> 00:01:25,200 one often summarizes functional analysis just by saying: it’s the study of topological algebraic
15 00:01:25,200 –> 00:01:32,560 structures. Don’t worry what this exactly means we will clear up in the next videos. However what
16 00:01:32,560 –> 00:01:38,880 you should see here is that topology plays a role here and indeed it’s not a bad idea to start with
17 00:01:38,880 –> 00:01:44,560 topology when dealing with functional analysis. However that’s not what I want to do here
18 00:01:44,560 –> 00:01:50,080 because I want to start with something that is closer to a normal analysis course. And
19 00:01:50,080 –> 00:01:57,440 that is the structure we call a metric space. A so-called metric space is not at all complicated:
20 00:01:58,560 –> 00:02:06,720 we start with a set X and we visualize that as a collection of points. For example, in the set X
21 00:02:06,720 –> 00:02:14,720 we have a lowercase point x, so just an element in our set and another element we can call y.
22 00:02:16,160 –> 00:02:20,640 If we just consider a set, we can’t say anything about these points other than:
23 00:02:20,640 –> 00:02:25,600 they are equal or not. So there are different points but we don’t know anything more than that.
24 00:02:26,880 –> 00:02:30,240 Now this is what we now want to change: we want to give the set
25 00:02:30,240 –> 00:02:37,440 X more structure. In this case, we want to know what the distance between two chosen points is.
26 00:02:38,720 –> 00:02:43,360 Now if we want such a distance then of course it should fulfill some reasonable properties.
27 00:02:44,560 –> 00:02:50,400 For example, if we choose two distinct points, then the distance should be always positive.
28 00:02:51,680 –> 00:02:57,520 Of course the distance could also be zero but only in the case when x and y are the same point.
29 00:02:58,720 –> 00:03:06,400 In order to formalize that, we have to write down a map. This map is called a metric and it measures
30 00:03:06,400 –> 00:03:13,120 distances and now it gets two points, so it lives on the Cartesian product of X with X. And it
31 00:03:13,120 –> 00:03:22,000 maps into the non-negative real numbers, so zero infinity, where zero is included. Now such a map
32 00:03:22,000 –> 00:03:28,720 we call a metric if it fulfills three properties. And the first one we already know. If the distance
33 00:03:28,720 –> 00:03:36,640 between two points is exactly zero, then this is only if and only the case if x is equal to y.
34 00:03:37,760 –> 00:03:43,600 The second property is also very descriptive. So if we look again at two points x and y.
35 00:03:44,720 –> 00:03:51,440 And now we measure the distance from x to y. And of course this should be the same as starting by y
36 00:03:52,000 –> 00:03:58,320 and going to x. So the distance from y to x should be the same as the distance from x to y.
37 00:03:59,440 –> 00:04:04,640 This means for the map it is symmetric: it does not matter which is the first and which is the
38 00:04:04,640 –> 00:04:12,560 second element. You get the same result. And now the last property is very important it is called
39 00:04:12,560 –> 00:04:19,680 the triangle inequality. The name already tells you now we consider three points instead of two,
40 00:04:19,680 –> 00:04:27,840 so there’s a point z. Now we have a distance from x to z, and also distance from z to y. Now the
41 00:04:27,840 –> 00:04:35,760 triangle inequality tells you if you go a detour, then your distance gets longer. In other words if
42 00:04:35,760 –> 00:04:43,760 you add the gray length, then it gets longer than the blue line here. In a formal way this means:
43 00:04:43,760 –> 00:04:55,840 d(x, y), the blue line, is less or equal than distance x to z plus distance z to y.
44 00:04:57,040 –> 00:05:01,440 Now with such a map, we can now measure all the distances in the set X.
45 00:05:01,440 –> 00:05:08,720 Therefore we call the set X with the metric d a metric space. And you already know
46 00:05:08,720 –> 00:05:15,120 if we can measure distances, we can do a lot of analytical stuff. And this is what we will do
47 00:05:15,120 –> 00:05:21,120 in the next videos. So please use the comments to tell me what you want to see in a functional
48 00:05:21,120 –> 00:05:37,840 analysis course. So thanks for listening and hopefully, I see you in the next videos. Bye!
49 00:05:58,400 –> 00:05:58,900 you