00:00 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

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

Q1: In a metric space, we measure distances between points. Let us consider the set $X = { 2, 6, 7 }$ and a metric $d$. Which of the following claims is definitely false?

A1: $d(2,6) = 7$

A2: $d(6,7) = 0$

A3: $d(2,2) = 0$

A4: $d(7,6) = 2$

Q2: A metric on a set $X$ is a map $d: X \rightarrow [0,\infty)$ with three certain properties.

A1: True

A2: False

Q3: What is the correct triangle inequality?

A1: $d(x,y) < d(x,z) + d(z,y)$

A2: $d(x,y) > d(x,z) + d(z,y)$

A3: $d(x,y) \leq d(x,z) + d(z,y)$

A4: $d(x,y) \geq d(x,z) + d(z,y)$

Q4: Consider the set $\mathbb{R}$. Is the map $d(x,y) := \frac{1}{2} |x-y|$ 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.