Frigyes Riesz was a dangerous man with whom to collaborate in writing a paper or a book. He was constantly having new ideas on how to proceed, and the latest brain child was the favourite. This would lead to disconcerting results for the collaborator, who was perpetually out of step. An example was told me by Tibor Rado, his ex-assistant. During the academic year, Riesz would lecture on measure theory and functional analysis. Rado would take copious notes. When summer arrived, Riesz would depart for a cooler spot (Győr). Rado would sweat it out for three months, writing up at Riesz's request all the material, to be in publishable form in the fall. At the end of September Riesz would put in his first day at the Institute, and Rado would come to the library to greet his superior, proudly carrying a stack of eight hundred pages, which he placed in Riesz' lap with great satisfaction. Riesz glanced at the bundle, recognised what it was, and raised his eyes with a mixture of kindness and thankfulness, and at the same time with a spark of merriment, as if he had pulled off a fast one. "Oh, very good, very good. Yes, this is very nice, really nice. But let me tell you. During the summer I had an idea. We will do it all another way. You will see as I give the course. You will like it." This took place many years in a row. The book was not written until Riesz, probably under the pressure of advancing age, wrote the book in collaboration with Béla Szőkefalvi-Nagy some 18 years later. As we all know, the book, Leçons d'analyse fonctionnelle, was an international best seller for decades.

2.1. Preface (translated from the French).

This book has developed from courses entitled "Real Functions," "Integral Equations," "Hilbert Space," etc., which the two authors have taught for several years at the Universities of Szeged and Budapest. As the printing was delayed by technical difficulties we have in the meantime added some paragraphs dealing with recent results.

The first part, on modern theories of differentiation and integration, serves as introduction to the second, which treats integral equations and linear functionals and transformations. This division into two parts corresponds to the division of the work by the two authors; although they have worked together, the first part was written principally by the first, and the second by the second author.

The two parts form an organic unit centred about the concept of linear operator. This concept is reflected in the method by which we have constructed the Lebesgue integral; this method, which seems to us to be simpler and clearer than that based on the theory of measure, has been used by the first author in his lectures for more than twenty years although it has not been published in definitive form.

The first part begins with a direct proof of the Lebesgue theorem on the differentiation of monotonic functions and its application to the study of the relations between the derivatives and the integrals of interval functions. After this we construct the theory of the Lebesgue integral and study the spaces $L_{2}$ and $L_{p}$ and their linear functionals. The Stieltjes integral and its generalisations are introduced in terms of linear operations on the space of continuous functions.

The second part begins with a chapter on integral equations, the subject which prepared the way for the general theory of linear transformations. We present several methods for arriving at the Fredholm alternative, and in the succeeding chapter we apply them to completely continuous functional equations of general type on either a Hilbert space or a Banach space. Symmetric completely continuous linear transformations are studied in a separate chapter.

We then develop the spectral theory of self-adjoint transformations, either bounded or unbounded, of Hilbert space. We also consider the problem of the extensions of unbounded symmetric transformations. A special chapter is devoted to functions of a self-adjoint transformation, as well as to the study of the spectrum and its perturbations. Stone's theorem on groups of unitary transformations and some related theorems, as well as certain ergodic theorems, are the subject of another chapter.

The last chapter surveys the beginnings, as yet fragmentary, of the spectral theory of linear transformations which are not necessarily normal; we present the method based on the calculus of residues, and we also include a study of the very recent results of J von Neumann on spectral sets.

In the exposition we have not attempted to study in detail all possible generalisations; rather we have sought to present the principal problems and the methods for handling them. At times we have presented several methods for attaining the same goal and have compared them and discussed their scopes.

We wish to express here our profound gratitude to the Hungarian Academy of Science for publishing our book in French, thus assuring for it an international public. Our hearty thanks are also due A Császár, who read the manuscript and whose critical remarks helped us to improve the text. We also acknowledge our gratitude to H Grenet and K Tandori for the care they gave to the correction of the proofs.

Budapest and Szeged, February 1952

2.2. Review by: J L B Copper.
The Mathematical Gazette 37 (320) (1953), 157-158.

Functional Analysis, and particularly the theory of linear spaces with which this book deals, has a large and growing literature; but in spite of the growing importance and diversity of its applications, it has lacked readable accounts, suitable for student as well as specialist, of the subject as a whole. For a major part, the part of most frequent application, this need is met brilliantly by the book under review.

The authors' names are sufficient guarantee of the excellence of the work. Both have made major contributions to the subject and have written important and, what is nowadays rare, lucid books on it. The elder, Riesz, is known to all analysts; his work began in the days when Lebesgue's integral and Hilbert's work on Integral Equations transformed analysis: he contributed greatly to both theories, simplifying their formulation and acting as a pioneer in giving them their modern, more abstract, form. A search for the general ideas under-lying theories and for a classical elegance of exposition have been characteristic of his work. These tendencies have done much to form the present book.

It begins with a discussion of the differentiation of monotonic functions, developed with no use of the theory of integration, and then gives an account of the Lebesgue integral by a definition based on extension from step functions to wider classes of functions: a definition which is frequently preferred to that of Lebesgue and is due to Riesz. The $L^{p}$ spaces are discussed as metric linear spaces, and other definitions of the Lebesgue integral, including that of Lebesgue, are given. The Stieltjes integral is then defined, directly, and as a functional on the space of continuous functions; its extension to the Lebesgue-Stieltjes integral is discussed. An account of the Daniell abstract integral closes the first third of the book, which forms a self-contained account of real variable theory.

The second part begins by discussing integral equations, first of all by more or less classical methods and then by means of the abstract theory of completely continuous operators on L². Abstract Hilbert Space and Banach Spaces are defined in the next chapter: most of the arguments are carried through for Hilbert Space, but the possible generalisations to Banach Spaces are pointed out. Four chapters are devoted to the theory of symmetric and normal operators in Hilbert Space, going with increasing generality from completely continuous to bounded and then to unbounded functions. The last of these chapters concerns itself with the functional calculus for these operators and with perturbation theory. The major applications to ordinary analysis, the boundary problems of potential theory, other problems of differential equations, almost periodic functions, are dealt with quite fully. The next chapter discusses the theory of groups and semigroups of transformations dealing mainly with the unitary and self-adjoint representations of the group of real numbers, but indicating the theory for other groups and it concludes with an account of ergodic theory. The final chapter deals with the spectral theory of general bounded operators. It commences with an account of the modern developments of the use of complex integration for the problem, which includes a very interesting application to Wiener's theorem concerning absolutely convergent Fourier Series, and proceeds to some recent contributions of von Neumann.

The book is so written that the ordinary honours course in analysis will suffice for its reading. Its method of exposition is to prepare the way for abstract concepts by well chosen examples of concrete cases. Generality for its own sake has been avoided; greater generality might sometimes have increased the value of the work as a book of reference, but the authors have made a good compromise between this and the very great pedagogic advantages of teaching the general methods by their application to the concrete cases. As a work of exposition this deserves to rank with the great French classics. It can be recommended strongly to all students of analysis; and its readers will share the authors' gratitude to the Hungarian Academy of Sciences for making it available to an international public. The book is well and accurately produced, and closes with a comprehensive bibliography and a useful index.

2.3. Review by: Mahlon Marsh Day.
Mathematical Reviews MR0050159 (14,286d).

This book is divided into two parts, modern theories of differentiation and integration, and spectral theory of linear transformations in Hilbert space; a long chapter on integral equations serves as a bridge between these two topics.

Chapter I defines zero measure, and works from the Lebesgue theorem that every monotone function has a derivative almost everywhere, to a study of interval functions, used to discuss Riemann integrals and bounded variation.

Chapter II defines the Lebesgue integral in terms of limits of monotone sequences of step functions; the whole presentation is made to depend on two lemmas in such a way that the generalisation to Lebesgue-Stieltjes integrals and to n dimensions can be made by a simple transition principle. Measurable functions and measure are defined; absolute continuity of the integral is proved along with the Lebesgue decomposition of a function of bounded variation. $L^{p}$ spaces, mean and weak convergence, and the Riesz-Fischer theorem are followed by a comparison of this approach with the original definition by way of measure.

Chapter III discusses Stieltjes integrals and the representation of linear functionals on the space $C$ of continuous functions, describes the Lebesgue-Stieltjes integral in $n$ dimensions and the Daniell integral, and gives one variant of the Radon-Nikodým theorem.

Chapter IV treats integral equations by two methods, with an application to potential theory. Chapter V defines Hilbert and Banach spaces and gives a condition for complete continuity of a linear transformation of $C$ into itself. Chapter VI studies symmetric, completely continuous transformations in Hilbert space, with application to the vibrating string and to almost periodic functions.

Chapter VII begins with two proofs of the spectral decomposition of a bounded symmetric operator in Hilbert space, then extends these results to unitary and normal transformations. Chapter VIII gives the corresponding decomposition for unbounded, self-adjoint transformations and discusses extensions of symmetric transformations. Chapter IX describes a functional calculus for self-adjoint transformations and discusses the behaviour of the spectrum under perturbation.

Chapter X gives Stone's theorem on groups of unitary operators, discusses groups and semigroups of operators, and concludes with proofs of mean ergodic theorems in Hilbert space. Chapter XI defines a functional calculus for not-necessarily-normal transformations by means of line integrals in the resolvent set, and closes with a discussion of spectral sets of such a transformation.

The book ends with a ten-page bibliography, an index, a list of notations, and a page of errata. ...

This book shows the authors' skill in choosing for each result the proof best suited to give a student sufficient understanding to make further applications and extensions appear easy and natural. Careful references are made throughout, both for the material in the text and for related topics. This book covers so much material so well that it should be noted that it does not discuss rings of operators in Hilbert space nor spectral multiplicity. But a student interested in integration or spectral theory will find in this book much pertinent background material as well as a clear, concise discussion of these major topics.

3.1. Preface (translated from the French).

The favourable reception which this book has received has necessitated a new edition. In it we have tried to eliminate the misprints of the first edition and to improve some passages. There are major changes in Chapters X and XI, mainly in the sections dealing with semi-groups of general type, the relations between the spectrum of a linear transformation and the norms of the iterated transformations, and the spectral sets of von Neumann.

We wish to express our thanks to all those, and particularly to A Császár, who by their criticisms have facilitated our task of improving the text.

Budapest and Szeged, May 1953
F. R. and B. Sz.-N.

3.2. Review by: Editors.
Mathematical Reviews MR0056821(15,132d).

This edition differs from the first only in correction of known misprints and errors, rewriting of parts of certain sections, an added new section in chapter X, and additions to the bibliography.

4.1. Contents.

PREMIÈRE PARTIE

THÉORIES MODERNES DE LA DÉRIVATION ET DE L'INTÉGRATION

I -DÉRIVATION
Théorème de Lebesgue sur la dérivée d'une fonction monotone.
- Exemple d'une fonction continue non dérivable.
- Théorème de Lebesgue sur la dérivée d'une fonction monotone. Ensembles de mesure nulle.
- Démonstration du théorème de Lebesgue.
- Fonctions à variation bornée.
Quelques conséquences immédiates du théorème de Lebesgue.
- Théorème de Fubini sur la dérivation des séries à termes monotones.
- Points de densité des ensembles linéaires.
- Fonctions des sauts.
- Fonctions à variation bornée quelconques.
- Théorème de Denjoy-Young-Saks sur les nombres dérivés des fonctions les plus générales.
Fonctions d'intervalle.
- Préliminaires.
- Premier théorème fondamental.
- Second théorème fondamental.
- Les intégrales de Darboux et celle de Riemann.
- Théorème de Darboux.
- Fonctions à variation bornée et rectification des courbes.

II -INTÉGRALE DE LEBESGUE
Définition et propriétés fondamentales.
- Intégrale des fonctions en escalier. Deux lemmes.
- Intégrale des fonctions sommables.
- Intégration terme à terme d'une suite croissante (théorème de Beppo Levi).
- Intégration terme à terme d'une suite majorée (théorème de Lebesgue).
- Théorèmes affirmant l'intégrabilité d'une fonction limite.
- Inégalités de Schwarz, de Hölder et de Minkowski.
- Ensembles et fonctions mesurables.
Intégrales indéfinies; fonctions absolument continues.
- Variation totale et dérivée de l'intégrale indéfinie.
- Exemple d'une fonction monotone continue dont la dérivée s'annule presque partout.
- Fonctions absolument continues. Décomposition canonique des fonctions monotones.
- Intégration par parties et intégration par substitution.
- L'intégrale comme fonction d'ensemble.
L'espace $L^{2}$ et ses fonctionnelles linéaires. Les espaces $L^{p}$.
- L'espace $L^{2}$; convergence en moyenne ; théorème de Riesz-Fischer.
- Convergence faible.
- Fonctionnelles linéaires.
- Suites de fonctionnelles linéaires ; un théorème d'Osgood.
- Séparabilité de $L^{2}$. Théorème du choix.
- Systèmes orthonormaux.
- Sous-espaces de $L^{2}$. Théorème de décomposition.
- Une autre démonstration du théorème de choix; prolongement des fonctionnelles.
- L'espace $L^{p}$ et ses fonctionnelles linéaires.
- Un théorème sur la convergence en moyenne.
- Un théorème de Banach et Saks.
Fonctions de plusieurs variables.
- Définitions. Principe de transition.
- Intégrations successives; théorème de Fubini.
- Dérivées, sur un réseau, d'une fonction additive non-négative de rectangle. Déplacement parallèle du réseau.
- Fonctions de rectangle à variation bornée. Réseaux conjugués.
- Fonctions additives d'ensemble. Ensembles mesurables ($B$).
Autres définitions de l'intégrale de Lebesgue.
- Ensembles mesurables ($L$).
- Fonctions mesurables ($L$) et intégrale ($L$).
- Autres définitions. Théorème d'Egoroff.
- Démonstration élémentaire des théorèmes d'Arzelà et d'Osgood.
- L'intégrale de Lebesgue considérée comme opération inverse de la dérivation.

III - INTÉGRALE DE STIELTJES ET SES GÉNÉRALISATIONS
Fonctionnelles linéaires dans l'espace des fonctions continues.
- L'intégrale de Stieltjes.
- Fonctionnelles linéaires dans l'espace $C$.
- Unicité de la fonction génératrice.
- Prolongement d'une fonctionnelle linéaire.
- Théorème d'approximation. Problèmes des moments.
- Intégration par parties. Second théorème de la moyenne.
- Suites de fonctionnelles.
Généralisations de l'intégrale de Stieltjes.
- Intégrales de Stieltjes-Riemann et de Stieltjes-Lebesgue.
- Réduction de l'intégrale de Stieltjes-Lebesgue à celle de Lebesgue.
- Relations entre deux intégrales de Stieltjes-Lebesgue.
- Fonctions de plusieurs variables. Définition directe.
- Définition moyennant le principe de transition.
Intégrale de Daniell.
- Fonctionnelles linéaires positives.
- Fonctionnelles de signe variable.
- Dérivée d'une fonctionnelle linéaire par rapport à une autre.

SECONDE PARTIE

ÉQUATIONS INTÉGRALES. TRANSFORMATIONS LINÉAIRES

IV - ÉQUATIONS INTÉGRALES
Méthode des approximations successives.
- Idée d'une équation intégrale.
- Noyaux bornées.
- Noyaux de carré sommable. Transformations linéaires de l'espace $L^{2}$.
- Transformation inverse. Valeurs régulières et singulières.
- Noyaux itérés, noyaux résolvants.
- Approximation d'un noyau quelconque par des noyaux de rang fini.
Alternative de Fredholm.
- Équations intégrales à noyau de rang fini.
- Équations intégrales à noyau de type général.
- Décomposition correspondant à une valeur singulière.
- L'alternative de Fredholm pour des noyaux généraux.
Déterminants de Fredholm.
- La méthode de Fredholm.
- Intégrale de Hadamard.
Autre méthode, fondée sur la continuité complète.
- Continuité complète.
- Les sous-espaces $M_{n}$ et $R_{n}$.
- Théorème de décomposition.
- Répartition des valeurs singulières.
- Décomposition canonique correspondant à une valeur singulière.
Applications à la théorie du potentiel.
- Problèmes de Dirichlet et de Neumann ; solution par la méthode de Fredholm.

V - ESPACES DE HILBERT ET DE BANACH
Espaces de Hilbert.
- Espaces de Hilbert des coordonnées.
- Espace de Hilbert abstrait.
- Transformations linéaires de l'espace de Hilbert. Notions fondamentales.
- Transformations linéaires complètement continues.
- Suites biorthogonales. Un théorème de Paley et Wiener.
Espaces de Banach.
- Espaces de Banach et leurs espaces conjugués.
- Transformations linéaires et leurs adjointes.
- Équations fonctionnelles.
- Transformations de l'espace des fonctions continues.
- Retour à la théorie du potentiel.

VI - TRANSFORMATIONS SYMÉTRIQUES COMPLÈTEMENT CONTINUES DE L'ESPACE DE HILBERT
Existence d'éléments propres. Théorème du développement.
- Valeurs propres et éléments propres. Premières propriétés des transformations symétriques.
- Transformations symétriques complètement continues.
- Détermination directe de la $n$-ième valeur propre de signe donné.
- Autre méthode de construire les valeurs propres et les éléments propres.
Transformations à noyau symétrique.
- Théorèmes de Hilbert et de Schmidt.
- Théorème de Mercer.
Applications au problème de la corde vibrante et aux fonctions presque-périodiques.
- Problème de la corde vibrante. Espaces $D$ et $H$.
- Problème de la corde vibrante. Vibrations propres.
- L'espace des fonctions presque-périodiques.
- Démonstration du théorème fondamental sur les fonctions presque-périodiques.
- Transformations isométriques d'un espace de dimension finie.

VII - TRANSFORMATIONS SYMÉTRIQUES, UNITAIRES ET NORMALES BORNÉES DE L'ESPACE DE HILBERT
Transformations symétriques.
- Quelques propriétés fondamentales.
- Projections.
- Fonctions d'une transformation symétrique bornée.
- Décomposition spectrale d'une transformation symétrique bornée.
- Parties positive et négative d'une transformation symétrique. Autre démonstration de la décomposition spectrale.
Transformations unitaires et normales.
- Transformations unitaires.
- Transformations normales. Factorisations.
- Décomposition spectrale des transformations normales. Fonctions de plusieurs transformations.
Transformations unitaires de l'espace $L^{2}$.
- Un théorème de Bochner.
- Transformations de Fourier-Plancherel et de Watson.

VIII - TRANSFORMATIONS LINÉAIRES NON BORNÉES DE L'ESPACE DE HILBERT
Généralisation de l'idée de transformation linéaire.
- Un théorème de Hellinger et Toeplitz. Extension de la notion de transformation linéaire.
- Transformations adjointes.
- Permutabilité. Réduction.
- Le graphique d'une transformation
- Les transformations $B$ et $C$.
Transformations autoadjointes. Décomposition spectrale.
- Transformations symétriques et autoadjointes. Définitions et exemples.
- Décomposition spectrale d'une transformation autoadjointe.
- Méthode de von Neumann. Transformées cayleyennes.
- Transformations autoadjointes semi-bornées.
Prolongement des transformations symétriques.
- Transformées cayleyennes. Indices de défaut.
- Transformations symétriques semi-bornées. Méthode de Friedrichs.
- Méthode de Krein.

IX - TRANSFORMATIONS AUTOADJOINTES : CALCUL FONCTIONNEL, SPECTRE, PERTURBATIONS
Calcul fonctionnel.
- Fonctions bornées.
- Fonctions non bornées. Définitions.
- Fonctions non bornées. Règles de calcul.
- Propriétés caractéristiques des fonctions d'une transformation autoadjointe.
- Ensembles finis ou dénombrables de transformations autoadjointes permutables.
- Ensembles quelconques de transformations autoadjointes permutables.
Le spectre d'une transformation autoadjointe et ses perturbations.
- Le spectre d'une transformation autoadjointe. Décomposition suivant le spectre ponctuel et le spectre continu.
- Points-limite du spectre.
- Perturbations du spectre par addition d'une transformation complètement continue.
- Perturbations continues.
- Perturbations analytiques.

X - GROUPES ET SEMI-GROUPES DE TRANSFORMATIONS
Transformations unitaires.
- Théorème de Stone.
- Autre démonstration, fondée sur un théorème de Bochner.
- Quelques applications du théorème de Stone.
- Représentations unitaires de groupes plus généraux.
Transformations non unitaires.
- Groupes et semi-groupes de transformations autoadjointes.
- Semi-groupes de transformations de type général. - Formules exponentielles.
Théorèmes ergodiques.
- Premières méthodes.
- Méthodes reposant sur des raisonnements de convexité.
- Semi-groupes de contractions non permutables.

XI - THÉORIES SPECTRALES DE TRANSFORMATIONS LINÉAIRES DE TYPE GÉNÉRAL
Applications des méthodes de la théorie des fonctions.
- Le spectre; intégrales curvilignes.
- Théorème de décomposition.
- Relations entre le spectre et les normes des transformations itérées.
- Application aux séries trigonométriques absolument convergentes.
- Éléments d'un calcul fonctionnel.
Ensembles spectraux d'après John von Neumann.
- Théorèmes principaux.
- Ensembles spectraux.
- Caractérisation des transformations symétriques, unitaires et normales par leurs ensembles spectraux.

APPENDICE
Prolongements des transformations de l'espace de Hilbert qui sortent de cet espace, par Béla SZ.-NAGY.
- Introduction.
- Familles spectrales au sens large. Théorème de Neumark.
- Suites de moments.
- Contractions de l'espace de Hilbert.
- Prolongements normaux.
- Théorème principal.
- Démonstration du théorème de Neumark.
- Démonstration du théorème sur les suites de moments.
- Démonstration des théorèmes sur les contractions.
- Démonstration du théorème sur les prolongements normaux.

4.2. Review by: Editors.
Mathematical Reviews MR0068139 (16,837b).

Aside from minor changes, this edition differs from the last in the addition of the Appendix reviewed above.

5.1. Contents.

Part One

MODERN THEORIES OF DIFFERENTIATION AND INTEGRATION

CHAPTER I: DIFFERENTIATION

Lebesgue's Theorem on the Derivative of a Monotonic Function

1. Example of a Nondifferentiable Continuous Function

2. Lebesgue's Theorem on the Differentiation of a Monotonic Function. Sets of Measure Zero

3. Proof of Lebesgue's Theorem

4. Functions of Bounded Variation

Some Immediate Consequences of Lebesgue's Theorem

5. Fubini's Theorem on the Differentiation of Series with Monotonic Terms

6. Density Points of Linear Sets

7. Saltus Functions

8. Arbitrary Functions of Bounded Variation

9. The Denjoy-Young-Saks Theorem on the Derived Numbers of Arbitrary Functions

Interval Functions

10. Preliminaries

11. First Fundamental Theorem

12. Second Fundamental Theorem

13. The Darboux Integrals and the Riemann Integral

14. Darboux's Theorem

15. Functions of Bounded Variation and Rectification of Curves

CHAPTER II: THE LEBESGUE INTEGRAL

Definition and Fundamental Properties

16. The Integral for Step Functions. Two Lemmas

17. The Integral for Summable Functions

18. Term-by-Term integration of an Increasing Sequence (Beppo Levi's Theorem)

19. Term-by-Term Integration of a Majorized Sequence (Lebesgue's Theorem)

20. Theorems Affirming the Integrability of a Limit Function

21. The Schwarz, Hölder, and Minkowski Inequalities

22. Measurable Sets and Measurable Functions

Indefinite Integrals. Absolutely Continuous Functions

23. The Total Variation and the Derivative of the Indefinite Integral

24. Example of a Monotonic Continuous Function Whose Derivative is Zero Almost Everywhere

25. Absolutely Continuous Functions. Canonical Decomposition of Monotonic Functions

26. Integration by Parts and Integration by Substitution

27. The Integral as a Set Function

The Space $L^{2}$ and its Linear Functionals. $L^{p}$ Spaces

28. The Space $L^{2}$; Convergence in the Mean; the Riesz-Fischer Theorem

29. Weak Convergence

30. Linear Functionals

31. Sequence of Linear Functionals; a Theorem of Osgood

32. Separability of $L^{2}$. The Theorem of Choice

33. Orthonormal Systems

34. Subspaces of $L^{2}$. The Decomposition Theorem

35. Another Proof of the Theorem of Choice. Extension of Functionals

36. The Space $L^{p}$ and Its Linear Functionals

37. A Theorem on Mean Convergence

38. A Theorem of Banach and Saks

Functions of Several Variables

39. Definitions. Principle of Transition

40. Successive Integrations. Fubini's Theorem

41. The Derivative Over a Net of a Non-negative, Additive Rectangle Function. Parallel Displacement of the Net

42. Rectangle Functions of Bounded Variation. Conjugate Nets

43. Additive Set Functions. Sets Measurable ($B$)

Other Definitions of the Lebesgue Integral

44. Sets Measurable ($L$)

45. Functions Measurable ($L$) and the Integral ($L$)

46. Other Definitions. Egoroff's Theorem

47. Elementary Proof of the Theorems of Arzelà and Osgood

48. The Lebesgue Integral Considered as the Inverse Operation of Differentiation

CHAPTER III: THE STIELTJES INTEGRAL AND ITS GENERALIZATIONS

Linear Functionals on the Space of Continuous Functions

49. The Stieltjes Integral

50. Linear Functionals on the Space $C$

51. Uniqueness of the Generating Function

52. Extension of a Linear Functional

53. The Approximation Theorem. Moment Problems

54. Integration by Parts. The Second Theorem of the Mean

55. Sequences of Functionals

Generalization of the Stieltjes Integral

56. The Riemann-Stieltjes and Lebesgue-Stieltjes Integrals

57. Reduction of the Lebesgue-Stieltjes Integral to that of Lebesgue

58. Relations Between Two Lebesgue-Stieltjes Integrals

59. Functions of Several Variables. Direct Definition

60. Definition by Means of the Principle of Transition

The Daniel Integral

61. Positive Linear Functionals

62. Functionals of Variable Sign

63. The Derivative of One Linear Functional With Respect to Another

Part Two

INTEGRAL EQUATIONS. LINEAR TRANSFORMATIONS

CHAPTER IV: INTEGRAL EQUATIONS

The Method of Successive Approximations

64. The Concept of an Integral Equation

65. Bounded Kernels

66. Square-Summable Kernels. Linear Transformations of the Space $L^{2}$

67. Inverse Transformations. Regular and Singular Values

68. Iterated Kernels. Resolvent Kernels

69. Approximation of an Arbitrary Kernel by Means of Kernels of Finite Rank

The Fredholm Alternative

70. Integral Equations With Kernels of Finite Rank

71. Integral Equations With Kernels of General Type

72. Decomposition Corresponding to a Singular Value

73. The Fredholm Alternative for General Kernels

Fredholm Determinants

74. The Method of Fredholm

75. Hadamard's Inequality

Another Method, Based on Complete Continuity

76. Complete Continuity

77. The Subspaces $\mathfrak {M}_n$ and $\mathfrak {N}_n$

78. The Cases $\nu = 0$ and $\nu ≥ 1$. The Decomposition Theorem

79. The Distribution of the Singular Values

80. The Canonical Decomposition Corresponding to a Singular Value

Applications to Potential Theory

81. The Dirichlet and Neumann Problems. Solution by Fredholm's Method

CHAPTER V: HILBERT AND BANACH SPACES

Hilbert Space

82. Hilbert Coordinate Space

83. Abstract Hilbert Space

84. Linear Transformations of Hilbert Space. Fundamental Concepts

85. Completely Continuous Linear Transformations

86. Biorthogonal Sequences. A Theorem of Paley and Wiener

Banach Spaces

87. Banach Spaces and Their Conjugate Spaces

88. Linear Transformations and Their Adjoints

89. Functional Equations

90. Transformations of the Space of Continuous Functions

91. A Return to Potential Theory

CHAPTER VI: COMPLETELY CONTINUOUS SYMMETRIC TRANSFORMATIONS OF HILBERT SPACE

Existence of Characteristic Elements. Theorem on Series Development

92. Characteristic Values and Characteristic Elements. Fundamental Properties of Symmetric Transformations

93. Completely Continuous Symmetric Transformations

94. Solution of the Functional Equation $j - \lambda Af = g$

95. Direct Determination of the $n$-th Characteristic Value of Given Sign

96. Another Method of Constructing Characteristic Values and Characteristic Elements

Transformations with Symmetric Kernel

97. Theorems of Hilbert and Schmidt

98. Mercer's Theorem

Periodic Functions

99. The Vibrating-String Problem. The Spaces $D$ and $H$

100. The Vibrating-String Problem. Characteristic Vibrations

101. Space of Almost Periodic Functions

102. Proof of the Fundamental Theorem on Almost Periodic Functions

103. Isometric Transformations of a Finite-Dimensional Space

CHAPTER VII: BOUNDED SYMMETRIC, UNITARY, AND NORMAL TRANSFORMATIONS OF HILBERT SPACE

Symmetric Transformations

104. Some Fundamental Properties

105. Projections

106. Functions of a Bounded Symmetric Transformation

107. Spectral Decomposition of a Bounded Symmetric Transformation

108. Positive and Negative Parts of a Symmetric Transformation. Another Proof of the Spectral Decomposition

Unitary and Normal Transformations

109. Unitary Transformations

110. Normal Transformations. Factorizations

111. The Spectral Decomposition of Normal Transformations. Functions of Several Transformations

Unitary Transformations of the Space $L^{2}$

112. A Theorem of Bochner

113. Fourier-Plancherel and Watson Transformations

CHAPTER VIII: UNBOUNDED LINEAR TRANSFORMATIONS OF HILBERT SPACE

Generalization of the Concept of Linear Transformation

114. A Theorem of Hellinger and Toeplitz. Extension of the Concept of Linear Transformation

115. Adjoint Transformations

116. Permutability. Reduction

117. The Graph of a Transformation

118. The Transformation $B = (I + T^{*}T)^{-1}$ and $C = T(I + T^{*}T)^{-1}$

Self-Adjoint Transformations. Spectral Decomposition

119. Symmetric and Self-Adjoined Transformations. Definitions and Examples

120. Spectral Decomposition of a Self-Adjoint Transformation

121. Von Neumann's Method. Cayley Transforms

122. Semi-Bounded Self-Adjoint Transformations

Extensions of Symmetric Transformations

123. Cayley Transforms. Deficiency Indices

124. Semi-Bounded Symmetric Transformations. The Method of Friedrichs

125. Krein's Method

CHAPTER IX: SELF-ADJOINT TRANSFORMATIONS. FUNCTIONAL CALCULUS, SPECTRUM, PERTURBATIONS

Functional Calculus

126. Bounded Functions

127. Unbounded Functions. Definitions

128. Unbounded Functions. Rules of Calculation

129. Characteristic Properties of Functions of a Self-Adjoint Transformation

130. Finite or Denumerable Sets of Permutable Self-Adjoint Transformations

131. Arbitrary Sets of Permutable Self-Adjoint Transformations

The Spectrum of a Self-Adjoint Transformation and Its Perturbations

132. The Spectrum of a Self-Adjoint Transformation. Decomposition in Terms of the Point Spectrum and the Continuous Spectrum

133. Limit Points of the Spectrum

134. Perturbation of the Spectrum by the Addition of a Completely Continuous Transformation

135. Continuous Perturbations

136. Analytic Perturbations

CHAPTER X: GROUPS AND SEMIGROUPS OF TRANSFORMATIONS

Unitary Transformations

137. Stone's Theorem

138. Another Proof. Based on a Theorem of Bochner

139. Some Applications of Stone's Theorem

140. Unitary Representations of More General Groups

Non-Unitary Transformations

141. Groups and Semigroups of Self-Adjoint Transformations

142. Infinitesimal Transformation of a Semigroup of Transformations of General Type

143. Exponential Formulas

Ergodic Theorems

144. Fundamental Methods

145. Methods Based on Convexity Arguments

146. Semigroups of Nonpermutable Contractions

CHAPTER XI: SPECTRAL THEORIES FOR LINEAR TRANSFORMATIONS OF GENERAL TYPE

Applications of Methods from the Theory of Functions

147. The Spectrum. Curvilinear Integrals

148. Decomposition Theorem

149. Relations Between the Spectrum and the Norms of Iterated Transformations

150. Application to Absolutely Convergent Trigonometric Series

151. Elements of a Functional Calculus

152. Two Examples

Von Neumann's Theory of Spectral Sets

153. Principal Theorems

154. Spectral Sets

155. Characterization of Symmetric, Unitary, and Normal Transformations by Their Spectral Sets

Bibliography

Appendix

Extensions of Linear Transformations in Hilbert Space

Which Extend Beyond This Space

Index

Notation & Symbols

5.2. Review by: L S Bosanquet.
Science Progress 45 (178) (1957), 340-341.

The late Frederic Riesz, brother of Marcel Riesz, devoted more than fifty years to research and was one of the pioneers of Functional Analysis. His pupil Béla Sz.-Nagy has been researching for some twenty years. The book is a brilliant piece of collaboration by these two distinguished Hungarian mathematicians. The first edition was published in 1952 in French by the Hungarian Academy of Science under the title Leçons d'analyse fonctionnelle.

The first part contains an account of the Lebesgue integral and the Stieltjes integral and its extensions. The Lebesgue theory is established without the theory of measure (except for sets of measure zero), integrable functions being defined in terms of limits of monotonic sequences of step-functions. The choice of theorems and proofs is guided by the requirements of the more general theory discussed later. Thus the classes $L^{p}$ and $C$ of functions of integrable $p$-th power and continuous functions serve to introduce Banach spaces. Here the generalised scalar product $(f, g)$ plays a fundamental rôle, every functional $F(f)$ being expressible in the form $(f, g)$ for a suitable $g$. In the space $L^{2}$, $(f, f)^{1/2}$ coincides with the norm $|| f ||$, which is a characteristic property of a Hilbert space. A theorem of Osgood on sequences of continuous functions serves to introduce the Banach-Steinhaus theorem, while the Hahn-Banach extension theorem is first met (as it was historically) in the spaces $L^{2}$ and $C$. All this is of interest in itself, but becomes even more fascinating when read again in the light of results given later.

The second part begins with an account of the Fredholm integral equation

          $f(x) - \int K(x, y) f(y) dy = g(x)$

where $K$ is of integrable square over a rectangle. This is an introduction to later theory in which the integral transformation is replaced by a general transformation (or operator). Next the authors give the general definitions of Banach and Hilbert spaces, and discuss the equation $T(f) = \mu f$ in the case where $T$ is a completely continuous bounded symmetric (or self-adjoint) transformation. Here the non-trivial solutions are the characteristic elements (eigenelements) corresponding to the characteristic values (eigenvalues) permitted for $\mu$. This takes us past the middle of the book, which continues to build on itself as the subject matter becomes more comprehensive. Later topics include: spectral decomposition of bounded symmetric transformations; unbounded linear and self-adjoint transformations; spectral theories for transformations of general type.

Although the reader will be impressed by the wide and deep learning of the authors, he will be even more struck by the charm and simplicity of their style, and the trouble they have taken to give him insight by discussing concrete examples of abstract ideas. Finally, a debt of gratitude is owing to the translator for the skill and accuracy with which he has carried out his share of the work.

5.3. Review by: Editors.
Mathematical Reviews MR0071727 (17,175).

Translation of the authors' Leçons d'analyse fonctionnelle, 2d ed. [1953].

5.4. Review by: T A A Broadbent.
The Mathematical Gazette 41 (335) (1957), 68.

Leçons d'analyse fonctionnelle appeared first in 1952, with second and third editions in 1953 and 1955; this rapid production endorses the opinion of experts that the work is outstanding among treatises on linear analysis. It is in two parts: the first, by Riesz, deals with real variable and integration, the Lebesgue integral being defined by Riesz's own process; the second part, by Nagy, starts with integral equations and proceeds to discuss the main problems of linear spaces and operators. For further details, we may refer to the review of the first edition in the Gazette, XXXVII, pp. 157-8. It is difficult to believe that any competent reader would find trouble with the limpid clarity of the French original; but for those who need a translation, this appears to be accurate and lucid, and the volume has been beautifully printed and produced (though I think there should be a reference to Nikolsky on p. 217).