A Course of Higher Mathematics

IntroductionPrefaceChapter The Stieltjes Integral 1. Sets and their Powers 2. The Stieltjes Integral and its Basic Properties 3. Darboux Sums 4. The Stieltjes Integral of a Continuous Function 5. The Improper Stieltjes Integral 6. Jump Functions 7. Physical Interpretation 8. Functions of Bounded Variation 9. An Integrating Function of Bounded Variation 10. Existence of the Stieltjes Integral 11. Passage to the Limit in the Stieltjes Integral 12. Helly's Theorem 13. Selection Principle 14. Space of Continuous Functions 15. General Form of the Functional in Space G 16. Linear Operators in G 17. Functions of an Interval 18. The General Stieltjes Integral 19. Properties of the (General) Stieltjes Integral 20. The Existence of the General Stieltjes Integral 21. Functions of a Two-Dimensional Interval 22. Passage to Point Functions 23. The Stieltjes Integral on a Plane 24. Functions of Bounded Variation on the Plane 25. The Space of Continuous Functions of Several Variables 26. the Fourier-Stieltjes Integral 27. Inversion Formula 28. Convolution Theorem 29. The Cauchy-Stieltjes IntegralChapter II Set Functions and the Lebesgue Integral § 1. Set Functions and the Theory of Measure 30. Operations on Sets 31. Point Sets 32. Properties of Closed and Open Sets 33. Elementary Figures 34. Exterior Measure and its Properties 35. Measurable Sets 36. Measurable Sets (Continued) 37. Criteria for Measurability 38. Field of Sets 39. Independence of the Choice of Axes 40. The B Field 41. The Case of a Single Variable § 2. Measurable Functions 42. Definition of Measurable Function 43. Properties of Measurable Functions 44. the Limit of a Measurable Function 45. the G Property 46. Piecewise Constant Functions 47. Class B § 3. The Lebesgue Integral 48. The Integral of a Bounded Function 49. Properties of the Integral 50. The Integral of a Non-Negative Unbounded Function 51. Properties of the Integral 52. Functions of Any Sign 53. Complex Summable Functions 54. Passage to the Limit Under the Integral Sign 55. The Class L2 56. Convergence in the Mean 57. Hubert Function Space. Orthogonal Systems of Functions 59. The Space U 60. Lineals in 58. L2 61. Examples of Closed Systems 62. The Holder and Minkovskii Inequalities 63. Integral Over a Set of Infinite Measure 64. The Class L2 on a Set of Infinite Measure 65. An Integrating Function of Bounded Variation 66. The Reduction of Multiple Integrals 67. The Case of the Characteristic Function 68. Fubini's Theorem 69. Change of the Order of Integration 70. Continuity in the Mean 71. Mean FunctionsChapter III Set Functions. Absolute Continuity. Generalization of the Integral 72. Additive Set Functions 73. Singular Function 74. The Case of One Variable 75. Absolutely Continuous Set Functions 76. Example 77. Absolutely Continuous Functions of Several Variables 78. Supplementary Propositions 79. Supplementary Propositions (Continued) 80. Fundamental Theorem 81. Hellinger's Integral 82. The Case of a Single Variable 83. Properties of the Hellinger IntegralChapter IV Metric and Normed Spaces 84. Metric Space 85. The Completion of a Metric Space 86. Operators and Functionals. The Principle of Compressed Mappings 87. Examples 88. Examples of Applications of the Principle of Compressed Mappings 89. Compactness 90. Compactness in C 91. Compactness in Lp 92. Compactness In Lp 93. Functionals on Mutually Compact Sets 94. Separability 95. Linear Normed Spaces 96. Examples of Normed Spaces 97. Operators in Normed Spaces 98. Linear Functionals 99. Conjugate Spaces 100. Weak Convergence of Functionals 101. The Weak Convergence of Elements 102. Linear Functionals In G, Lp and lp 103. Weak Convergence In C, Lp and lp 104. the Space of Linear Operators and the Convergence of Sequences of Operators 105. Conjugate Operators 106. Completely Continuous Operators 107. Operator Equations 108. Completely Continuous Operators in C,Lp and lp 109. Generalized Derivatives 110. Generalized Derivatives (Continued) 111. The Case of a Star-Shaped Domain 112. Spaces W(l)p and W(l)p 113. Properties of Functions of Space W(l)p(E) 114. Embedding Theorems 115. Integral Operators with a Polar Kernel 116. Sobolev's Integral Forms 117. Embedding Theorems 118. Domains of a More General Type 119. Space C(l)(D)Chapter V Hilbert Space § 1. The Theory of Bounded Operators 120. Axioms of the Space 121. Orthogonality and Orthogonal Systems of Elements 122. Projections 123. Linear Functionals 124. Linear Operators 125. Bilinear and Quadratic Functionals 126. Bounds of a Self-Conjugate Operator 127. The Inverse Operator 128. Spectrum of an Operator 129. The Spectrum of a Self-Conjugate Operator 130. The Resolvent 131. Sequences of Operators 132. Weak Convergence 133. Completely Continuous Operators 134. Spaces H and l2 135. Linear Equations in Completely Continuous Operators 136. Completely Continuous Self-Conjugate Operators 137. Unitary Operators 138. The Absolute Norm of an Operator 139. Operations on Subspaces 140. Projection Operators 141. The Resolution of the Identity. The Stieltjes Integral 142. The Spectral Function of a Self-Conjugate Operator 143. Continuous Functions of a Self-Conjugate Operator 144. A Formula for the Resolvent and a Characteristic of Regular Values of λ 145. Eigenvalues and Eigenelements 146. Purely Point Spectra 147. A Continuous Simple Spectrum 148. Invariant Subspaces 149. The General Case of a Continuous Spectrum 150. The Case of a Mixed Spectrum 151. Differential Solutions 152. The Operation of Multiplication by the Independent Variable 153. The Unitary Equivalence of Self-Conjugate Operators 154. The Spectral Resolution of Unitary Operators 155. Functions of a Self-Conjugate Operator 156. Commuting Operators 157. Perturbations of the Spectrum of a Self-Conjugate Operator 158. Normal Operators 159. Auxiliary Propositions 160. Power Series of Operators 161. The Spectral Function § 2. Spaces l2 and L2 162. Linear Operators In l2 163. Bounded Operators 164. Unitary Matrices and Projection Matrices 165. Self-Conjugate Matrices 166. The Case of a Continuous Spectrum 167. Jacobian Matrices 168. Differential Solutions 169. Examples 170. Weak Convergence in L2 171. Completely Continuous Operators in l2 172. Integral Operators in L2 173. The Conjugate Operator 174. Completely Continuous Operators 175. Spectral Functions 176. The Spectral Function (Continued) 177. Unitary Transformations in L2 178. Fourier Transformations 179. Fourier Transformations and Hermitian Functions 180. The Operation of Multiplication 181. Kernels that Depend on a Difference 182. Weak Convergence 183. Other Concrete Forms of Space H § 3. Unbounded Operators 184. Closed Operators 185. Conjugate Operators 186. The Graph of an Operator 187. Symmetric and Self-Conjugate Operators 188. Examples of Unbounded Operators 189. The Spectrum of a Self-Conjugate Operator 190. The Case of a Point Spectrum 191. Invariant Subspaces and the Reducibility of an Operator 192. Resolutions of the Indentity the Stieltjes Integral 193. Continuous Functions of a Self-Conjugate Operator 194. The Resolvent 195. Eigenvalues 196. The Case of a Mixed Spectrum 197. Functions of a Self-Conjugate Operator 198. Small Perturbations of the Spectrum 199. The Operator of Multiplication 200. Integral Operators 201. The Extension of a Closed Symmetric Operator 202. Deficiency Indices 203. The Conjugate Operator 204. Maximal Operators 205. Extension of Symmetric Semi-Bounded Operators 206. The Comparison of Semi-Bounded Operators 207. Examples on the Theory of Extensions 208. The Spectrum of a Symmetric Operator 209. Some Theorems on Extensions and their Spectra 210. The Independence of the Deficiency Indices on λ 211. The Invariance of the Continuous Part of the Spectral Kernel in the Case of Symmetric Extensions 212. The Spectra of Self-Conjugate Extensions 213. Examples 214. Infinite Matrices 215. Jacobian Matrices 216. Matrices and Operators 217. The Unitary Equivalence of C-Matrices 218. The Existence of the Spectral FunctionIndex