A Course of Higher Mathematics - 1st Edition - ISBN: 9781483197470, 9781483222509

A Course of Higher Mathematics

1st Edition

Integration and Functional Analysis

Authors: V. I. Smirnov
Editors: A. J. Lohwater
eBook ISBN: 9781483222509
Imprint: Pergamon
Published Date: 1st January 1964
Page Count: 650
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A Course of Higher Mathematics, Volume V focuses on the theory of integration and elements of functional analysis.

This book is organized into five chapters. Chapter I discusses the theory of the classical Stieltjes integral and space C of continuous functions, while Chapter II deals with the foundations of the metric theory of functions of a real variable and Lebesgue-Stieltjes integral. The theory of completely additive set functions and case of the one-dimensional Hellinger integral are analyzed in Chapter III. Chapter IV contains an exposition of the foundations of the general theory of metric and normed spaces. The general theory of Hilbert space is covered in Chapter V.

This volume is suitable for engineers, physicists, and students of pure mathematics.

Table of Contents



Chapter 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 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 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 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 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 Function



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© Pergamon 1964
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About the Author

V. I. Smirnov

About the Editor

A. J. Lohwater