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Theory of Linear Operations

Stefan Banach, translated by F. Jellett
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The Basic Library List Committee strongly recommends this book for acquisition by undergraduate mathematics libraries.

[Reviewed by
Allen Stenger
, on

This is a historically important and influential book that was the first text on functional analysis, comprising mostly new material that had never been published before. The present volume is a 2008 reprint of the 1987 translation of the 1932 book that was published in Warsaw in French. (The original is still in print from AMS/Chelsea.) The translation helpfully uses modern terminology for everything; for example, the main spaces of interest are called Banach spaces instead of spaces of type (B). The body of the book is only 160 pages; it bound with a 70 page report, “Some aspects of the present theory of Banach spaces” by A. Pelczynski and Cz. Bessaga, that details modern developments in many of the topics covered in the body. The report is not dated but appears to have been written around 1978 or 1979.

Most of the material deals with Lebesgue-Stieltjes integration, abstract spaces of various types, and with what today is called functional analysis. Applications include integral equations and Fourier series.

Beyond its historical importance, is the book still useful today? Probably not, unless you follow Abel’s advice to “study the masters and not the pupils”. Although Banach’s book has a surprisingly modern approach to the subject, it is still the first book and the subject has been streamlined a lot since then, and many new areas have opened up that are covered very briefly or not at all in the present book. Even a short introductory text such as Saxe’s Beginning Functional Analysis has more extensive coverage of the subject that Banach’s book. In addition Banach’s book has no exercises, and has a poor physical appearance: it is typewritten (not typeset) and the type is very small.

Allen Stenger is a math hobbyist, library propagandist, and retired computer programmer. He volunteers in his spare time at, a math help site that fosters inquiry learning. His mathematical interests are number theory and classical analysis.



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


Index of terminology

Some aspects of the present theory of Banach spaces
by A. Pelczynski and Cz. Bessaga


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


Additional Bibliography