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, L^{r}, c, l^{r}, m and in the subspaces of m

5. Closed and complete sequences in the spaces C, L^{r}, c and l^{r}

6. Approximation of functions belonging to C and L^{r} 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 L^{r}, c and l^{r}

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, L^{p}, c and l^{p}

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 L^{p} and l^{p} 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 L^{2} and l^{2}

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 l^{p}, for p ≥ 1

3. Linear dimension of the spaces L^{p} and l^{p} 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 L^{p} spaces

12. The isomorphic structure of the spaces L^{p}(μ)

Chapter VI.

13. The topological structure of linear metric spaces

14. Added in proof

Bibliography

Additional Bibliography