Preface
Introduction
A. The Lebesgue-Stieltjes Integral
1. Some theorems in the theory of the Lebesgue integral
2. Some inequalities for pth-power summable functions
3. Asymptotic convergence
4. Mean convergence
S. The Stieltjes Integral
6. Lebesgue’s Theorem
B. (B)-Measurable sets and operators in metric spaces
7. Metric spaces
8. Sets in metric spaces
9. Mappings in metric spaces
Chapter I. Groups
1. Definition of G-spaces
2. Properties of sub-groups
3. Additive and linear operators
4. A theorem on the condensation of singularities
Chapter II. General vector spaces
1. Definition and elementary properties of vector spaces
2. Extension of additive homogeneous functional s
3. Applications: generalisation of the notions of integral, of measure and of limit
Chapter III. F-spaces
1. Definitions and preliminaries
2. Homogeneous operators
3. Series of elements. Inversion of linear operators
4. Continuous non-differentiable functions
5. The continuity of solutions of partial differential equations
6. Systems of linear equations in infinitely many unknowns
7. The space s
Chapter IV. Normed spaces
1. Definition of normed vector spaces and of Banach spaces
2. Properties of linear operators. Extension of linear functionals
3. Fundamental sets and total sets
4. The general form of bounded linear functionals in the spaces C, Lr, c, lr, m and in the subspaces of m
5. Closed and complete sequences in the spaces C, Lr, c and lr
6. Approximation of functions belonging to C and Lr by linear combinations of functions
7. The problem of moments
8. Condition for the existence of solutions of certain systems of equations in infinitely many unknowns
Chapter V. Banach spaces
1. Linear operators in Banach spaces
2. The principle of condensation of singularities
3. Compactness in Banach spaces
4. A property of the spaces Lr, c and lr
5. Banach spaces of measurable functions
6. Examples of bounded linear operators in some special Banach spaces
7. Some theorems on summation methods
Chapter VI. Compact operators
1. Compact operators
2. Examples of compact operators in some special spaces
3. Adjoint (conjugate) operators
4. Applications. Examples of adjoint operators in some special spaces
Chapter VII. Biorthogonal sequences
1. Definition and general properties
2. Biorthogonal sequences in some special spaces
3 Bases in Banach spaces
4. Some applications to the theory of orthogonal expansions
Chapter VIII. Linear functionals
1. Preliminaries
2. Regularly closed linear spaces of linear functionals
3. Transfinitely closed sets of bounded linear functionals
4. Weak convergence of bounded linear functionals
5. Weakly closed sets of bounded linear functionals in separable Banach spaces
6. Conditions for the weak convergence of bounded linear functionals on the spaces C, Lp, c and lp
7. Weak compactness of bounded sets in certain spaces
8. Weakly continuous linear functionals defined on the space of bounded linear functionals
Chapter IX. Weakly convergent sequences
1. Definition. Conditions for the weak convergence of sequences of elements
2. Weak convergence of sequences
3. The relationship between weak and strong (norm) convergence in the spaces Lp and lp for p > 1
4. Weakly complete spaces
5. A theorem on weak convergence
Chapter X. Linear functional equations
1. Relations between bounded linear operators and their adjoints
2. Riesz’ theory of linear equations associated with compact linear operators
3. Regular values and proper values in linear equations
4. Theorems of Fredholm in the theory of compact operators
S. Fredholm integral equations
6. Volterra integral equations
7. Symmetric integral equations
Chapter XI. Isometry , equivalence, isomorphism
1. Isometry
2. The spaces L2 and l2
3. Isometric transformations of normed vector spaces
4. Spaces of continuous real-valued functions
5. Rotations
6. Isomorphism and equivalence
7. Products of Banach spaces
8. The space C as the universal space
9. Dual spaces
Chapter XII. Linear dimension
1. Definitions
2. Linear dimension of the spaces c and lp, for p ≥ 1
3. Linear dimension of the spaces Lp and lp for p> 1
Appendix. Weak convergence in Banach spaces
1. The weak derived sets of sets of bounded linear functionals
2. Weak convergence of elements
Remarks
Index of terminology
Some aspects of the present theory of Banach spaces
by A. Pelczynski and Cz. Bessaga
Introduction
Notation and terminology
Chapter I.
1. Reflexive and weakly compactly generated Banach spaces. Related counter examples
Chapter II. Local properties of Banach spaces
2. The Banach-Mazur distance and projection constants
3. Local representability of Banach spaces
4. The moduli of convexity and smoothness; super-reflexive Banach spaces. Unconditionally convergent series
Chapter III. The approximation property and bases
5. The approximation property
6. The bounded approximation property
7. Bases and their relation to the approximation property
8. Unconditional bases
Chapter IV.
9. Characterizations of Hilbert spaces in the class of Banach spaces
Chapter V. Classical Banach spaces
10. The isometric theory of classical Banach spaces
11. The isomorphic theory of Lp spaces
12. The isomorphic structure of the spaces Lp(μ)
Chapter VI.
13. The topological structure of linear metric spaces
14. Added in proof
Bibliography
Additional Bibliography