Banach, Fréchet, Hilbert and Neumann Spaces - Simon, Jacques

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Résumé :

Introduction xi

Familiarization with Semi-normed Spaces xv

Notations xvii

Chapter 1 Prerequisites 1

1.1 Sets, mappings, orders 1

1.2 Countability 3

1.3 Construction of R 4

1.4 Properties of R 5

Part 1 Semi-normed Spaces 9

Chapter 2 Semi-normed Spaces 11

2.1 Definition of semi-normed spaces 11

2.2 Convergent sequences 15

2.3 Bounded, open and closed sets 17

2.4 Interior, closure, balls and semi-balls 21

2.5 Density, separability 23

2.6 Compact sets 25

2.7 Connected and convex sets 30

Chapter 3 Comparison of Semi-normed Spaces 33

3.1 Equivalent families of semi-norms 33

3.2 Topological equalities and inclusions 34

3.3 Topological subspaces 39

3.4 Filtering families of semi-norms 43

3.5 Sums of sets 46

Chapter 4 Banach, Fr?chet and Neumann Spaces 49

4.1 Metrizable spaces 49

4.2 Properties of sets in metrizable spaces 51

4.3 Banach, Fr?chet and Neumann spaces 55

4.4 Compacts sets in Fr?chet spaces 57

4.5 Properties of R 58

4.6 Convergent sequences 60

4.7 Sequential completion of a semi-normed space 62

Chapter 5 Hilbert Spaces 65

5.1 Hilbert spaces 65

5.2 Projection in a Hilbert space 68

5.3 The space Rd 70

Chapter 6 Product, Intersection, Sum and Quotient of Spaces 73

6.1 Product of semi-normed spaces 73

6.2 Product of a semi-normed space by itself 78

6.3 Intersection of semi-normed spaces 80

6.4 Sum of semi-normed spaces 83

6.5 Direct sum of semi-normed spaces 89

6.6 Quotient space 93

Part 2 Continuous Mappings 95

Chapter 7 Continuous Mappings 97

7.1 Continuous mappings 97

7.2 Continuity and change of topology or restriction 100

7.3 Continuity of composite mappings 102

7.4 Continuous semi-norms 102

7.5 Continuous linear mappings 104

7.6 Continuous multilinear mappings 107

7.7 Some continuous mappings 111

Chapter 8 Images of Sets Under Continuous Mappings 115

8.1 Images of open and closed sets 115

8.2 Images of dense, separable and connected sets 117

8.3 Images of compact sets 119

8.4 Images under continuous linear mappings 121

8.5 Continuous mappings in compact sets 123

8.6 Continuous real mappings 124

8.7 Compacting mappings 125

Chapter 9 Properties of Mappings in Metrizable Spaces 129

9.1 Continuous mappings in metrizable spaces 129

9.2 Banach's fixed point theorem 133

9.3 Baire's theorem 134

9.4 Open mapping theorem 136

9.5 Banach-Schauder's continuity theorem 138

9.6 Closed graph theorem 139

Chapter 10 Extension of Mappings, Equicontinuity 141

10.1 Extension of equalities by continuity 141

10.2 Continuous extension of mappings 142

10.3 Equicontinuous families of mappings 146

10.4 Banach-Steinhaus equicontinuity theorem 148

Chapter 11 Compactness in Mapping Spaces 153

11.1 The spaces F(X...

Biographie:
F)-pt 153

11.2 Zorn's lemma 154

11.3 Compactness in F(X...

Sommaire:
F) and C(X...