Contents 8
Preface 6
Introduction A. The Lebesgue-Stieltjes Integral 12
1. Some theorems in the theory of the Lebesgue integral 12
2. Some inequalities for pth-power summable functions 12
3. Asymptotic convergence 13
4. Mean convergence 13
5.The Stieltjes Integral 14
6. Lebesgue's Theorem 16
B. (B)-Measurable sets and operators in metric spaces 16
7. Metric spaces 16
8. Sets in metric spaces 18
9. Mappings in metric spaces 20
Chapter I. Groups 24
1. Definition of G-spaces 24
2. Properties of sub-groups 24
3. Additive and linear operators 25
4. A theorem on the condensation of singularities 26
Chapter II. General vector spaces 28
1. Definition and elementary properties of vector spaces 28
2. Extension of additive homogeneous functionals 28
3. Applications: generalisation of the notions of integral of measure and of limit 30
Chapter III. F-spaces 34
1. Definitions and preliminaries 34
2. Homogeneous operators 35
3. Series of elements. Inversion of linear operators 35
4. Continuous non-differentiable functions 38
5. The continuity of solutions of partial differential equations 39
6. Systems of linear equations in infinitely many unknowns 40
7. The space 8 42
Chapter IV. Normed spaces 44
1. Definition of normed vector spaces and of Banach spaces 44
2. Properties of linear operators. Extension of linear functionals 44
3. Fundamental sets and total sets 46
4. The general form of bounded linear functionals in the spaces C,Lr,c,lr,m and in the subspaces of rn 47
5. Closed and complete sequences in the spaces cC,Lr, c and lr 56
6. Approximation of functions belonging to C and Lr by linear combinations of functions 56
7. The problem of moments 57
8. Condition for the existence of solutions of certain systems of equations in infinitely many unknowns 58
Chapter V. Banach spaces 60
1. Linear operators in Banach spaces 60
2. The principle of condensation of singularities 61
3. Compactness in Banach spaces 63
4. A property of the spaces Lr, c and lr 63
5. Banach spaces of measurable functions 64
6. Examples of bounded linear operators in some special Banach spaces 65
7. Some theorems on summation methods 66
Chpter VI. Compact operators 70
1. Compact operators 70
2. Examples of compact operators in some special spaces 70
3. Adjoint (conjugate) operators 72
4. Applications . Examples of adjoint operators in some special spaces 73
Chapter VII. Biorthogonal sequences 76
1. Definition and general properties 76
2. Biorthogonal sequences in some special spaces 77
3. Bases in Banach spaces 78
4. Some applications to the theory of orthogonal expansions 79
Chapter VIII. Linear functionals 82
1. Preliminaries 82
2. Regularly closed linear spaces of linear functionals 77
3. Transfinitely Closed sets of bounded linear functionals 83
4. Weak convergence of bounded linear functionals 86
5. Weakly closed sets of bounded linear functionals in separable Banach spaces 87
6. Conditions for the weak convergence of bounded linear functionals on the spaces C, Lp,c and zp 88
7. Weak compactness of bounded sets in certain spaces 90
8. Weakly continuous linear functionals defined on the space of bounded linear functionals 91
Chapter IX . Weakly convergent sequences 92
1. Definition. Conditions for the weak convergence of sequences of elements 92
2. Weak convergence of sequences in the spaces 92
3. The relationship between weak and strong (norm) convergence in the spaces LP and zp for p> I 95
4. Weakly complete spaces 96
5. A theorem on weak convergence 98
Chapter X. Linear functional equations 100
1. Relations between bounded linear operators and their adjoints 100
2. Riesz' theory of linear equations associated with compact linear operators 103
3. Regular values and proper values in linear equations 106
4. Theorems of Fredholm in the theory of compact operators 108
5. Fredholm integral equations 109
6. Volterra integral equations 109
7. Symmetric integral equations 110
Chapter XI. Isometry, equivalence, isomorphism 112
1. Isometry 112
2. The spaces and L2 and Z2 112
3. Isometric transformations of normed vector spaces 112
4. Spaces of continuous real-valued functions 113
5. Rotations 116
6. Isomorphism and equivalence 120
7. Products of Banach spaces 121
8. The space C as the universal space 123
9. Dual spaces 124
Chapter XII. Linear dimension 128
l. Definitions 128
2. Linear dimension of the spaces c and zp, for p>= 1 128
3. Linear dimension of the spaces Lp and Zp for p>1 130
Appendix. Weak convergence in Banach spaces 138
1. The weak derived sets of sets of bounded linear functionals 138
2. Weak convergence of elements 143
Remarks 148
Index of terminology 168
Some aspects of the present theory of Banach spaces 172
Introduotion 174
Notation and tenninoZogy 174
Chapter I. 176
1. Reflexive and weakly compactly generated Banach spaces. Related counter examples 176
Chapter II . Local properties of Banach spaces 180
2. The Banach-Mazur distance and projection constants 180
3. Local representability of Banach spaces 182
4. The moduli of convexity and smoothness; super-reflexive Banach spaces . Unconditionally convergent series 185
Chapter III. The approximation property and bases 190
5. The approximation property 190
6. The bounded approximation property 192
7. Bases and their relation to the approximation property 194
8. Unconditional bases 196
Chapter IV. 200
9. Characterizations of Hilbert spaces in the class of Banach spaces. 200
Chapter V. Classical Banach spaces 206
10. The isometric theory of classical Banach spaces 206
11. The isomorphic theory of Lp spaces 210
12. The isomorphic structure of the spaces Lp(渭) 215
Chapter VI. 220
13. The topological structure of linear metric spaces 220
14. Added in proof 224
Bibliography 228
Additional Bibliography 244