A Course of Higher Mathematics

Contents

Introduction

Preface

Chapter I 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. Holly's Theorem


13. Selection Principle


14. Space of Continuous Functions


15. General Form of the Functional in Space G


16. Linear Operators in C


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 Integral

Chapter 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 Classing- Convergence in The Mean


57. Hilbert Function Space. Orthogonal Systems of Functions


59. The Space L2


60. Lineals In


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 Functions

Chapter 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 Integral

Chapter 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 C, Lp and lp


103. Weak Convergence in 0, 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 C, Lp and lp


109. Generalized Derivatives


110. Generalized Derivatives (Continued)


111. The Case of A Star-Shaped Domain


112. Spaces and 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(1)(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 Selfconjugate 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 Symetric 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 Function

Index

Volumes Published in This Series