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Title: Examples of Inner Products and Hilbert Spaces
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Series: Functional Analysis
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Chapter: Banach and Hilbert Spaces
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YouTube-Title: Functional Analysis 9 | Examples of Inner Products and Hilbert Spaces
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Forum: Ask a question in Mattermost
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Quiz: Test your knowledge
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Exercise Download PDF sheets
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Subtitle on GitHub: fa09_sub_eng.srt
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Timestamps
00:00 Introduction
00:28 Examples
02:39 Checking properties for $\ell^2$
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Subtitle in English
1 00:00:00.575 –> 00:00:03.245 Hello and welcome back to Functional Analysis
2 00:00:03.585 –> 00:00:05.325 and many thanks to all the people
3 00:00:05.325 –> 00:00:07.605 that support this channel on Steady or PayPal.
4 00:00:08.815 –> 00:00:10.405 We’ve already reached part nine
5 00:00:10.745 –> 00:00:13.765 and we will talk about examples of Hilbert spaces today.
6 00:00:14.545 –> 00:00:17.325 Please recall that the Hilbert space is a real
7 00:00:17.345 –> 00:00:20.965 or complex vector space within a product such
8 00:00:20.965 –> 00:00:23.845 that the associated metric space is complete.
9 00:00:24.865 –> 00:00:27.645 Of course, most of you already know a typical example,
10 00:00:28.065 –> 00:00:31.845 namely RN or CN, with the standard in a product.
11 00:00:32.865 –> 00:00:34.325 And this one is given by the sum
12 00:00:34.495 –> 00:00:36.325 where you multiply the components.
13 00:00:37.345 –> 00:00:39.725 And the important thing in CN would be
14 00:00:39.725 –> 00:00:42.605 that you have the complex conjugate in the first component.
15 00:00:43.665 –> 00:00:45.565 The next example is a generalization
16 00:00:45.705 –> 00:00:47.085 for an infinite dimension.
17 00:00:48.035 –> 00:00:50.765 It’s the l two space we already had in part
18 00:00:50.895 –> 00:00:52.005 seven of this series.
19 00:00:53.265 –> 00:00:54.785 And there you might already guess
20 00:00:54.935 –> 00:00:56.225 what the inner product should be.
21 00:00:56.405 –> 00:00:57.785 It should be like this one,
22 00:00:57.965 –> 00:01:00.545 but you go to infinity there later.
23 00:01:00.645 –> 00:01:02.625 In this video I want to discuss
24 00:01:02.625 –> 00:01:05.425 with you why this is indeed an inner product
25 00:01:05.685 –> 00:01:07.105 for vector space l two.
26 00:01:07.785 –> 00:01:08.795 However, first,
27 00:01:08.935 –> 00:01:11.435 let me show you another infinite dimensional example.
28 00:01:12.425 –> 00:01:15.155 It’s about continuous functions defined
29 00:01:15.295 –> 00:01:16.435 on the unit interval.
30 00:01:17.105 –> 00:01:19.795 Therefore, I write zero one for the domain
31 00:01:20.295 –> 00:01:21.795 and F for the codomain.
32 00:01:23.345 –> 00:01:26.045 Now you should know all these continuous functions together
33 00:01:26.435 –> 00:01:29.565 form an F vector space with respect to the natural addition
34 00:01:29.585 –> 00:01:30.925 and scalar multiplication.
35 00:01:32.025 –> 00:01:33.725 And now for two functions, f
36 00:01:33.905 –> 00:01:38.085 and g, we can define an inner product, simply
37 00:01:38.225 –> 00:01:40.605 by looking at the integral from zero to one
38 00:01:40.775 –> 00:01:44.165 where we put in the function f and the function g.
39 00:01:45.305 –> 00:01:47.005 And of course, in the complex case,
40 00:01:47.185 –> 00:01:49.805 we need the complex conjugation for the first function.
41 00:01:50.835 –> 00:01:53.205 Okay, I would say these three are one
42 00:01:53.205 –> 00:01:55.245 of the most important examples at the
43 00:01:55.245 –> 00:01:56.285 beginning of such a course.
44 00:01:57.385 –> 00:01:59.165 The first one gets us the normal,
45 00:01:59.385 –> 00:02:01.765 the Euclidean geometry in RN
46 00:02:01.865 –> 00:02:05.325 or CN, part (b) then generalizes that
47 00:02:05.345 –> 00:02:06.405 to an infinite dimension.
48 00:02:07.025 –> 00:02:10.605 And part (c) gives us a geometry for continuous functions.
49 00:02:11.435 –> 00:02:12.765 However, in spite
50 00:02:12.765 –> 00:02:14.845 of having an inner product here on the right,
51 00:02:15.225 –> 00:02:17.565 we don’t get out a hilbert space in (c).
52 00:02:18.185 –> 00:02:19.645 So please keep that in mind.
53 00:02:20.025 –> 00:02:23.285 We have an inner product, but the completeness fails here.
54 00:02:24.435 –> 00:02:25.925 Okay, we can talk about this later.
55 00:02:25.935 –> 00:02:27.805 First, I want to show you in part (b)
56 00:02:27.995 –> 00:02:30.765 that we have a Hilbert space there.
57 00:02:30.785 –> 00:02:32.325 You already know the completeness.
58 00:02:32.585 –> 00:02:34.765 So let’s discuss the inner product part here.
59 00:02:35.935 –> 00:02:38.385 This means that we have to check all the properties.
60 00:02:39.485 –> 00:02:41.705 And the first thing should always be showing
61 00:02:41.735 –> 00:02:45.105 that this one is a well-defined map from l two times
62 00:02:45.305 –> 00:02:46.385 l two to F.
63 00:02:47.775 –> 00:02:51.625 This means that this limit as a series should always exist.
64 00:02:52.375 –> 00:02:55.865 However, for this we need some technical details I just want
65 00:02:55.865 –> 00:02:57.185 to do later in the series.
66 00:02:58.285 –> 00:03:00.555 So don’t worry. There will be a video about that.
67 00:03:00.785 –> 00:03:03.515 Here we focus on the three properties of an inner product,
68 00:03:04.415 –> 00:03:07.675 and the first part is showing that it is positive definite,
69 00:03:08.445 –> 00:03:11.395 which means when putting in the same vector x, we want
70 00:03:11.395 –> 00:03:15.115 to get out a non-negative number, which is easy to see
71 00:03:15.115 –> 00:03:17.315 because we have xi times xi.
72 00:03:17.335 –> 00:03:19.835 And the first one is the complex conjugate one
73 00:03:19.855 –> 00:03:20.875 in the complex case.
74 00:03:21.295 –> 00:03:25.395 So in other words, it’s the absolute value squared,
75 00:03:26.755 –> 00:03:29.185 which is clearly non-negative.
76 00:03:30.725 –> 00:03:34.345 And the other part would be looking at the case when the
77 00:03:34.415 –> 00:03:39.005 outcome is zero, which means by the calculation above
78 00:03:39.665 –> 00:03:44.485 all the xi squared have to be zero, which then
79 00:03:44.505 –> 00:03:47.445 of course means all the xi have to be zero.
80 00:03:48.705 –> 00:03:52.045 And in conclusion, this is of course the zero vector itself,
81 00:03:53.425 –> 00:03:56.365 and now we know it’s positive definite.
82 00:03:58.185 –> 00:04:00.845 Now going to the second property, which was
83 00:04:01.075 –> 00:04:03.965 that the inner product is conjugate symmetric.
84 00:04:04.945 –> 00:04:07.125 Of course, this is now very simple to show.
85 00:04:07.275 –> 00:04:09.725 Just look at the inner product <y,x>
86 00:04:09.725 –> 00:04:12.125 where we look at the complex conjugation.
87 00:04:13.225 –> 00:04:14.765 So let’s mark that in green
88 00:04:15.185 –> 00:04:17.405 and we have it then over the whole series,
89 00:04:18.905 –> 00:04:21.565 but of course we can pull that inside.
90 00:04:23.145 –> 00:04:25.285 And then the normal calculation rules tell us
91 00:04:25.285 –> 00:04:29.445 that we have y_i x_i complex conjugation,
92 00:04:30.395 –> 00:04:35.065 which is then of course <x, y> in this order.
93 00:04:36.485 –> 00:04:39.465 And now the last part, the third part is the linearity
94 00:04:40.005 –> 00:04:41.305 in the second argument.
95 00:04:42.245 –> 00:04:44.305 So maybe that’s already easy to see,
96 00:04:44.725 –> 00:04:46.145 but still, let’s write it down.
97 00:04:46.875 –> 00:04:49.665 Since I don’t want to get conflicts with the indices here.
98 00:04:50.065 –> 00:04:53.905 I use y and z as the two vectors in the second component.
99 00:04:54.805 –> 00:04:56.985 Now by definition, this is the inner product
100 00:04:57.125 –> 00:04:59.025 and we can write it as to series.
101 00:04:59.925 –> 00:05:03.185 And as you can see, this is simply the inner product with x
102 00:05:03.285 –> 00:05:05.025 and y and x and z.
103 00:05:05.765 –> 00:05:08.345 And now we can do the same for the homogeneous part.
104 00:05:08.525 –> 00:05:11.585 So we look at the inner product x, lambda y,
105 00:05:12.315 –> 00:05:15.905 which is the series x_i bar Lambda, y_i,
106 00:05:16.685 –> 00:05:19.385 and there we can simply pull out the Lambda factor,
107 00:05:20.155 –> 00:05:23.225 which is then lambda times the inner product.
108 00:05:24.165 –> 00:05:26.105 And indeed that’s the linearity.
109 00:05:27.125 –> 00:05:28.745 Now, what we have learned here is
110 00:05:28.745 –> 00:05:32.425 that checking all three properties is often not hard at all
111 00:05:33.215 –> 00:05:35.625 Here, it was just a matter of writing it down.
112 00:05:36.375 –> 00:05:37.425 However, showing
113 00:05:37.425 –> 00:05:39.785 that the map itself is well defined
114 00:05:39.915 –> 00:05:41.265 could be a much harder problem.
115 00:05:42.275 –> 00:05:46.025 Hence, in this case, we will do that in another long video.
116 00:05:46.905 –> 00:05:50.345 Nevertheless, combining this with the three properties
117 00:05:50.725 –> 00:05:53.985 and the fact that this corresponding norm we find here
118 00:05:54.875 –> 00:05:57.805 makes l two two Banach space, tells us
119 00:05:58.075 –> 00:06:00.365 that the whole thing is a Hilbert space.
120 00:06:01.335 –> 00:06:04.605 Don’t forget the completeness we’ve already discussed in
121 00:06:04.605 –> 00:06:07.285 part seven, in the upcoming video,
122 00:06:07.505 –> 00:06:10.125 we don’t talk about the technical details yet,
123 00:06:10.785 –> 00:06:13.965 but I want to show you all the nice properties a general
124 00:06:13.975 –> 00:06:15.245 inner product has.
125 00:06:16.205 –> 00:06:17.535 Okay, I hope I see you’re there.
126 00:06:17.675 –> 00:06:21.055 And I can also tell you that I put a link to a PDF version
127 00:06:21.235 –> 00:06:23.015 of this video in the description.
128 00:06:23.895 –> 00:06:26.915 And indeed, I want to do this for all upcoming videos.
129 00:06:27.735 –> 00:06:29.955 So please enjoy it, use it when you need it,
130 00:06:30.735 –> 00:06:32.805 and with this, thanks for listening
131 00:06:32.945 –> 00:06:34.325 and see you in the next video.
132 00:06:34.865 –> 00:06:35.085 Bye.
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Quiz Content
Q1: Which of following inner product spaces is not a Hilbert space?
A1: $\mathbb{C}^n$ with standard inner product
A2: $\ell^2(\mathbb{N}, \mathbb{F})$ with inner product $\langle x, y \rangle = \sum_{j=1}^\infty \overline{x_j} y_j$
A3: $C([0,1], \mathbb{F})$ with inner product $\langle f, g \rangle = \int_{0}^1 \overline{f(t)} g(t) , dt$
A4: $\mathbb{R}^n$ with standard inner product
Q2: Which of following maps does define an inner product on $\ell^2(\mathbb{N}, \mathbb{C})$?
A1: $$\ell^2 \times \ell^2 \to \mathbb{C} $$ $$(x,y) \mapsto \sum_{i = 1}^\infty \overline{x_i} y_i$$
A2: $$\ell^2 \times \ell^2 \to \mathbb{C} $$ $$ (x,y) \mapsto \sum_{i = 10}^\infty \overline{x_i} y_i$$
A3: $$\ell^2 \times \ell^2 \to \mathbb{C} $$ $$ (x,y) \mapsto \sum_{i = 1}^\infty x_i y_i$$
A4: $$\ell^2 \times \ell^2 \to \mathbb{C} $$ $$ (x,y) \mapsto - \sum_{i = 1}^\infty \overline{x_i} y_i$$
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Last update: 2025-09
