Calculo

Vitali Milman
An Introduction To
Functional Analysis

W ORLD 1999

2

3

dedications

4

Contents
1 Linear spaces; normed spaces; first examples
1.1
Linear spaces . . . . . . . . . . . . . . . . .
1.2
Normed spaces; first examples . . . . . . .
1.2.1 H¨ lder inequality. . . . . . . . . . .
o
1.2.2 Minkowski inequality . . . . . . . .
1.3
Completeness; completion . . . .. . . . .
1.3.1 Construction of completion . . . .
1.4
Exercises . . . . . . . . . . . . . . . . . . .
2

.
.
.
.
.
.
.

.
.
.
.
.
.
.

.
.
.
.
.
.
.

.
.
.
.
.
.
.

.
.
.
.
.
.
.

.
.
.
.
.
.
.

9
9
11
12
13
16
17
18

Hilbert spaces
21
2.1
Basic notions; first examples . . . . . . . . . . . . . .
21
2.1.1 Cauchy-Schwartzinequality . . . . . . . . . .
22
2.1.2 Bessel’s inequality . . . . . . . . . . . . . . .
23
2.1.3 Gram-Schmidt orthogonalization procedure .
24
2.1.4 Parseval’s equality . . . . . . . . . . . . . . .
25
2.2
Projections; decompositions . . . . . . . . . . . . . .
27
2.2.1 Separable case . . . . . . . . . . . . . . . . . .
27
2.2.2 Uniqueness of the distance from a point to a
convex set: thegeometric meaning . . . . . .
27
2.2.3 Orthogonal decomposition . . . . . . . . . . .
28
2.3
Linear functionals . . . . . . . . . . . . . . . . . . . .
29
2.3.1 Linear functionals in a general linear space .
29
2.3.2 Bounded linear functionals in normed spaces.
The norm of a functional . . . . . . . . . . . .
31
2.3.3 Bounded linear functionals in a Hilbert space 32
2.3.4 An Example ofa non-separable Hilbert space: 32
2.4
Exercises . . . . . . . . . . . . . . . . . . . . . . . . .
33
5

CONTENTS

6

¡ 

3 The dual space
3.1
Hahn-Banach theorem and its first consequences .
3.2
Dual Spaces . . . . . . . . . . . . . . . . . . . . . . .
3.3
Exercises: . . . . . . . . . . . . . . . . . . . . . . . . .

39
39
41
42

4 Bounded linear operators
4.1Completeness of the space of bounded
tors . . . . . . . . . . . . . . . . . . . .
4.2
Examples of linear operators . . . . . .
4.3
Compact operators . . . . . . . . . . .
4.3.1 Compact sets . . . . . . . . . .
4.3.2 The space of compact operators
4.4
Dual Operators . . . . . . . . . . . . . .
4.5
Different convergences in the space
of bounded operators . . . . . . .
4.6
Invertible Operators .. . . . . . . . .
4.7
Exercises . . . . . . . . . . . . . . . . .

43
linear
....
....
....
....
....
....

opera....
....
....
....
....
....

43
44
45
46
48
48

........
........
........

50
52
52

5 Spectral theory
5.1
Classification of spectrum . . . . . . . . . . . . . . .
5.2
Fredholm Theory of compact operators . . . . . . . .
5.3
Exercises . . . .. . . . . . . . . . . . . . . . . . . . .

57
57
58
63

6 Self adjoint compact operators
6.1
General Properties . . . . . . . . . . . . . . . . . . .
6.2
Exercises . . . . . . . . . . . . . . . . . . . . . . . . .

65
65
72

7 Self-adjoint bounded operators
7.1
Order in the space of symmetric operators . . .
7.1.1 Properties . . . . . . . . . . . . . . . . .
7.2
Projections(projection operators) . . . . . . . .
7.2.1 Some properties of projections in linear
spaces . . . . . . . . . . . . . . . . . . .

...
...
...

73
73
73
77

...

77

8 Functions of operators
8.1
Properties of this correspondence (
)......
8.2
The main inequality . . . . . . . . . . . . . . . . . . .
8.3
Simple spectrum . . . . . . . . . . . . . . . . . . . . .

79
80
8285

9 Spectral theory of unitary operators
9.1
Spectral properties . . . . . . . . . . . . . . . . . . .

87
87

¥ ¤£ ¢

© ¨ §¦

CONTENTS

7

10 The Fundamental Theorems.
10.1 The open mapping theorem . . . .
10.2 The Closed Graph Theorem . . . .
10.3 The Banach-Steinhaus Theorem .
10.4 Bases In Banach Spaces . . . . . .
10.5 Hahn-Banach Theorem.
Linear functionals . ....