00:00 Introduction

00:33 Definition (norm)

04:17 Normed space

04:50 Connection to metrics

06:00 Banach space

1 00:00:00,420 –> 00:00:06,780 hello and welcome back to functional analysis and  as always I want to thank all the nice people that

2 00:00:06,780 –> 00:00:14,760 support the channel on Steady or PayPal today’s  part 6 finally is about Banach spaces. We will

3 00:00:14,760 –> 00:00:20,340 later see that a so-called Banach space is indeed  one of the most important objects in functional

4 00:00:20,340 –> 00:00:28,140 analysis. In its core it’s just a vector space but  with more analytical structure on it. In order to

5 00:00:28,140 –> 00:00:34,980 understand such a Banach space we first have to  define what a norm is. For this, I will use the

6 00:00:34,980 –> 00:00:42,420 letter F to denote a field of numbers. However here  it will always be the real or the complex numbers.

7 00:00:43,500 –> 00:00:48,660 Of course, this makes our life easier because we  don’t have to write down the definition two times.

8 00:00:49,740 –> 00:00:55,800 Now we have a set X that also carries some  operations such that it is a vector space.

9 00:00:57,000 –> 00:01:04,140 F Vector space always means that the scaling  of vectors is done with numbers form F. This

10 00:01:04,140 –> 00:01:10,800 means if F is R you can scale with real numbers  and in the case that F is C you can scale even

11 00:01:10,800 –> 00:01:19,320 with complex numbers. The usual visualization for  points in a vector space is given by arrows. It’s

12 00:01:19,320 –> 00:01:24,720 a good idea because you see the two operations  scaling and adding two vectors immediately.

13 00:01:25,560 –> 00:01:31,680 However from the analytical point of view it would  be also nice to know how long such an arrow is.

14 00:01:32,520 –> 00:01:36,660 Measuring such a length is what we  call in the abstract sense and norm.

15 00:01:37,500 –> 00:01:43,140 The symbol one uses for a norm are just two  lines on the left and two lines on the right.

16 00:01:43,980 –> 00:01:49,380 And because we are measuring a length, the only  possible values should be non-negative numbers.

17 00:01:50,340 –> 00:01:56,160 This means that each Vector in X gets  a number which is positive or zero,

18 00:01:56,160 –> 00:02:00,840 and this map we get we call a norm  if it fulfills three properties.

19 00:02:01,920 –> 00:02:06,540 Since you already know metrics you might  easily guess some of the details here.

20 00:02:07,500 –> 00:02:13,920 For example in (a) we find the positive definite  part which means if we have the length 0,

21 00:02:15,480 –> 00:02:21,240 then this is equivalent for having the  zero vector, or in other words the zero

22 00:02:21,240 –> 00:02:26,400 vector has length 0 (which makes sense) but it’s also the only one with length 0.

23 00:02:27,000 –> 00:02:34,140 Now Part (b) explains what happens to the length  when we scale a vector. Scaling here means we

24 00:02:34,140 –> 00:02:41,760 have a vector x and multiply it from the  left with a scalar so a number in F. In the

25 00:02:41,760 –> 00:02:47,460 picture you always visualize that with the arrow  getting longer or smaller depending on Lambda.

26 00:02:48,540 –> 00:02:55,800 Hence the norm has to satisfy this which means the  length gets also multiplied by this Factor Lambda.

27 00:02:57,480 –> 00:03:03,240 However Lambda could be a negative number or  even a complex one such that this only makes

28 00:03:03,240 –> 00:03:10,380 sense if we consider the absolute value of Lambda. Now depending what F is, R or C, we have here the

29 00:03:10,380 –> 00:03:15,900 absolute value in the real numbers or in the  complex numbers. Therefore this part (b) now tells

30 00:03:15,900 –> 00:03:23,100 you that the norm is always absolutely homogeneous. So you can always pull out scalars but outside of

31 00:03:23,100 –> 00:03:29,520 the norm they always get absolute values. Okay so  we are in a vector space we have two operations

32 00:03:29,520 –> 00:03:34,920 the scalar multiplication we had in (b) and now in (c)  we have to explain what happens under the addition.

33 00:03:35,700 –> 00:03:41,880 So let’s look at two vectors x and  y. You already know how to visualize

34 00:03:41,880 –> 00:03:46,980 the addition in a vector space: you just put  the two arrows together to get out the sum.

35 00:03:48,000 –> 00:03:53,820 Then you see the triangle here and you think  that the length the norm should fulfill the

36 00:03:53,820 –> 00:04:00,420 normal geometry in this sense this means  that it satisfies the triangle inequality.

37 00:04:01,860 –> 00:04:06,960 It looks similar what we have for the metrics but  keep in mind here’s with respect to the vector

38 00:04:06,960 –> 00:04:14,160 addition. Therefore the norm is not a linear map in  general because you can pull out the addition but

39 00:04:14,160 –> 00:04:20,820 what you get is only an inequality. Now you might  already guess that a vector space together with a

40 00:04:20,820 –> 00:04:27,600 chosen norm gets a special name and you are right. This pair is what we simply call a normed space.

41 00:04:28,680 –> 00:04:34,560 So what we have is a real or complex vector space  where we can measure the length in a meaningful

42 00:04:34,560 –> 00:04:41,040 way. Now you might ask what is the connection  to the metric spaces we defined at the beginning s 43 00:04:41,040 –> 00:04:47,460 of this video series. Since the property (a)  and (c) look very similar, the norm could be

44 00:04:47,460 –> 00:04:52,860 a special case of a metric. Indeed this one is  an important fact you always should remember.

45 00:04:53,940 –> 00:05:01,020 If you have a norm for the vector space X, you can  immediately define a metric. So maybe let’s put the

46 00:05:01,020 –> 00:05:06,960 norm in the index of the metric and then we can  define the distance between two points x and y.

47 00:05:07,860 –> 00:05:13,740 If we look at the elements of the set X, we  should think of the end points of the arrows,

48 00:05:14,640 –> 00:05:19,620 and then the distance between the two points  should be given by the connection vector which

49 00:05:19,620 –> 00:05:28,140 is x minus y and then we take the length of it. In fact, this then defines a metric for the set X.

50 00:05:29,220 –> 00:05:32,640 This one is not hard to show. I advise you to try it out.

51 00:05:33,300 –> 00:05:38,460 So proving the three properties of the metric  by just using the three properties of the norm.

52 00:05:39,660 –> 00:05:45,780 Now the most important thing to remember here is  because we always have this definition in mind and

53 00:05:45,780 –> 00:05:52,200 a normed space is indeed a special case of a metric  space. In particular all the definitions we have

54 00:05:52,200 –> 00:06:00,840 for metric spaces also work for normed spaces. So for  example closedness, openness and so on. And with this

55 00:06:00,840 –> 00:06:06,900 in mind we can now eventually define what a Banach space is. So it could happen that our new

56 00:06:06,900 –> 00:06:15,000 metric space here (X,d) is a complete one. This one  is a very nice property which we discussed in the

57 00:06:15,000 –> 00:06:23,460 last video. All the Cauchy sequences converge. We  don’t have holes in this metric space. In this case

58 00:06:23,460 –> 00:06:31,620 then the original normed space, X with its norm, gets a special name: we call it a Banach space.

59 00:06:32,640 –> 00:06:38,580 Okay so now you know the definition of one of  the most important objects in functional analysis.

60 00:06:39,300 –> 00:06:46,200 So you see we have ingredients from the algebraic  side and also from the analytical side. On the one

61 00:06:46,200 –> 00:06:52,800 hand we have a real or complex vector space and  on the other hand we have a complete metric space.

62 00:06:54,120 –> 00:06:58,740 And the thing that connects both  sides very smoothly is our norm.

63 00:06:59,700 –> 00:07:04,620 So please keep that in mind we don’t have any  metric here we have the metric that comes from

64 00:07:04,620 –> 00:07:11,700 this norm and by definition the norm can  deal with the algebraic operations. So this

65 00:07:11,700 –> 00:07:18,480 is how I see a Banach space in the abstract  sense to get more concrete we will look at a

66 00:07:18,480 –> 00:07:25,404 lot of examples in the next video so thanks for  listening and I hope I see you there bye [Music]

Q1: What is not a property of a norm?

A1: positive definite

A2: absolutely homogenous

A3: linear

A4: triangle inequality

Q2: Let $X = \mathbb{R}$ the vector space of real numbers. Is the absolute value $|\cdot|$ a norm on $X$?

A1: Yes!

A2: No!

Q3: What is the correct definition of a Banach space?

A1: A normed vector space that is also complex.

A2: A normed vector space that is also real.

A3: A normed vector space that is also complete.

A4: A normed vector space that is also a metric space.