Book:Frigyes Riesz/Functional Analysis

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