# Book:Frigyes Riesz/Functional Analysis

## Frigyes Riesz and Béla Sz.-Nagy: Functional Analysis

Published $\text {1990}$, Dover

ISBN 978-0486662893 (translated by Leo F. Boron)

### Subject Matter

Functional Analysis

### Contents

Part I: 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 Integrbility of a Limit Function
21. The Schwarsz, Hölder and Minkowski Inequalities
22. Measurable Sets and Measurable Functions
Chapter III: The Stieltjes integral and its generalizations
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. Weal 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 Rectange Function. Parallel Displacement of the Net
42. Rectangle Functions of Bounded Variation. Conjugate Nets
43. Additive Set Functions. Sets Measurable $\paren B$
Other Definitions of the Lebesgue Integral
44. Sets Measurable $\paren L$
45. Functions Measurable $\paren L$ and the Integral $\paren L$
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

Part II: Integral equations. Linear transforms

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. Approximations 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
Another Method, Based on Complete Continuity
76. Complete Continuity
77. Subspaces ${\mathfrak M}_n$ and ${\mathfrak R}_n$
78. The Cases $\nu = 0$ and $\nu \ge 1$. 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
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 $j - \lambda A f = g$
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
Applications to the Vibrating-String Problem and to Almost 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 Transformation
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
116. Permutability. Reduction
117. The Graph of a Transformation
118. The Transformation $B = \paren {I + T^*T}^{-1}$ and $X = T \paren{I + T^*T}^{-1}$
119. Symmetric and Self-Adjoined Transformations. Definitions and Examples
120. Spectral Decomposition of a Self-Adjoint Transformation
121. Von Neumann's Method. Cayley Transforms
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 Transformation
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 Trasformations
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 by Symmetric, Unitary and Normal Transformations by Their Spectral Sets

Bibliography

Appendix

Index

Notation & symbols