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

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