# Book:Frigyes Riesz/Functional Analysis

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## Frigyes Riesz and Béla Sz.-Nagy:

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

Published $1990$, **Dover**

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

### Subject Matter

### 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

- 75. Hadamard's Inequality

**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

- 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 $j - \lambda A f = g$

- 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

- 115. Adjoint Transformations

- 116. Permutability. Reduction

- 117. The Graph of a Transformation

- 118. The Transformation $B = \paren {I + T^*T}^{-1}$ and $X = T \paren{I + T^*T}^{-1}$

**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 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**