an introduction to functional analysis - vitali milman

29 2.3.1 Linear functionals in a general linear space.. 29 2.3.2 Bounded linear functionals in normed spaces.. Chapter 1Linear spaces; normed spaces; first examples 1.1 Linear spaces I N

Vitali Milman

An Introduction To Functional Analysis

WORLD 1999

2

dedications

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1 Linear spaces; normed spaces; first examples 9

1.1 Linear spaces 9

1.2 Normed spaces; first examples 11

1.2.1 H¨older inequality 12

1.2.2 Minkowski inequality 13

1.3 Completeness; completion 16

1.3.1 Construction of completion 17

1.4 Exercises 18

2 Hilbert spaces 21 2.1 Basic notions; first examples 21

2.1.1 Cauchy-Schwartz inequality 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: the geometric 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 of a non-separable Hilbert space: 32 2.4 Exercises 33

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6 CONTENTS

3 The dual space

39 3.1 Hahn-Banach theorem and its first consequences 39

3.2 Dual Spaces 41

3.3 Exercises: 42

4 Bounded linear operators 43 4.1 Completeness of the space of bounded linear opera-tors 43

4.2 Examples of linear operators 44

4.3 Compact operators 45

4.3.1 Compact sets 46

4.3.2 The space of compact operators 48

4.4 Dual Operators 48

4.5 Different convergences in the space   of bounded operators 50

4.6 Invertible Operators 52

4.7 Exercises 52

5 Spectral theory 57 5.1 Classification of spectrum 57

5.2 Fredholm Theory of compact operators 58

5.3 Exercises 63

6 Self adjoint compact operators 65 6.1 General Properties 65

6.2 Exercises 72

7 Self-adjoint bounded operators 73 7.1 Order in the space of symmetric operators 73

7.1.1 Properties 73

7.2 Projections (projection operators) 77

7.2.1 Some properties of projections in linear spaces 77

8 Functions of operators 79 8.1 Properties of this correspondence (   ) 80

8.2 The main inequality 82

8.3 Simple spectrum 85

9 Spectral theory of unitary operators 87 9.1 Spectral properties 87

CONTENTS 7

10.1 The open mapping theorem 92

10.2 The Closed Graph Theorem 94

10.3 The Banach-Steinhaus Theorem 95

10.4 Bases In Banach Spaces 99

10.5 Hahn-Banach Theorem Linear functionals 100

10.6 Extremal points; The Krein-Milman Theorem 108

11 Banach algebras 111 11.1 Analytic functions 114

11.2 Radicals 118

11.3 Involutions 120

12 Unbounded self-adjoint and symmetric operators in 127 12.1 More Properties Of Operators 131

12.2 The Spectrum
 132

12.3 Elements Of The “Graph Method” 133

12.4 Reduction Of Operator 134

12.5 Cayley Transform 136

8 CONTENTS

Chapter 1

Linear spaces; normed

spaces; first examples

1.1 Linear spaces

I N THIS course we study linear spaces over the field of real

or complex numbers
or  The simplest examples of linearspaces studied in a course of Linear Algebra are those of the

wehave that





 





10CHAPTER 1 LINEAR SPACES; NORMED SPACES; FIRST EXAMPLES

one and onto linear map and consequently it is invertible We write



for its inverse

Examples of linear spaces

1

is the set of finite support sequences; that is, the quences with all but finite zero elements It is a linear space withrespect to addition of sequences and obviously isomorphic to thespace of all polynomials

se-2 The set  of sequences tending to zero

3 The set of all convergent sequences

and the linear structure of restricted

on gives the linear structure of We will write  

not all of them zero, so that

  

We define the linear span of a subset of a linear space to bethe intersection of all subspaces of containing That is,

is invariant and it is called the dimension

of the space We writedim 

form a basis of

1.2 NORMED SPACES; FIRST EXAMPLES 11

Next we introduce the notion of quotient spaces For a subspace

of we define a new linear space called the quotient space of

with respect to in the following way First we consider the

is a zero of the new space  ... proceed to define the notion of “distance” in a linear space

This is necessary if one wants to analysis and study convergence