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Chapter Ⅰ．Linear spaces 1

1．Linear spaces and linear dependence 1

2．Linear functions and conjugate spaces 4

3．The Hahn-Banach extension theorem 8

4．Linear topological spaces 11

5．Conjugate spaces 17

6．Cones，wedges，order relations 20

Chapter Ⅱ．Normed Linear spaces 24

1．Elementary definitions and properties 24

2．Examples of normed spaces；constructions of new spaces from old 28

3．Category proofs 33

4．Geometry and approximation 38

5．Comparison of topologies in a normed space 39

Chapter Ⅲ．Completeness，compactness，and reflexivity 44

1．Completeness in a linear topological space 44

2．Compactness 47

3．Completely continuous linear operators 53

4．Reflexivity 56

Chapter Ⅳ．Unconditional convergence and bases 58

1．Series and unconditional convergence 58

2．Tensor products of locally convex spaces 63

3．Schauder bases in separable spaces 67

4．Unconditional bases 73

Chapter Ⅴ．Compact convex sets and continuous function spaces 77

1．Extreme points of compact convex sets 77

2．The fixed-point theorem 82

3．Some properties of continuous function spaces 84

4．Characterizations of continuous function spaces among Banach spaces 87

Chapter Ⅵ．Norm and order 96

1．Vector lattices and normed lattices 96

2．Linear sublattices of continuous function spaces 101

3．Monotone projections and extensions 104

4．Special properties of(AL)-spaces 107

Chapter Ⅶ．Metric geometry in normed spaces 110

1．Isometry and the linear structure 110

2．Rotundity and smoothness 111

3．Characterizations of inner-product spaces 115

Chapter Ⅷ．Reader＇s guide 121

Bibliography 124

Index of symbols 132

Index 135