A Course of Higher Mathematics

A Course of Higher Mathematics

International Series of Monographs in Pure and Applied Mathematics, Volume 62: A Course of Higher Mathematics, V: Integration and Functional Analysis

1st Edition - January 1, 1964

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  • Author: V. I. Smirnov
  • eBook ISBN: 9781483139371

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International Series of Monographs in Pure and Applied Mathematics, Volume 62: A Course of Higher Mathematics, V: Integration and Functional Analysis focuses on the theory of functions. The book first discusses the Stieltjes integral. Concerns include sets and their powers, Darboux sums, improper Stieltjes integral, jump functions, Helly’s theorem, and selection principles. The text then takes a look at set functions and the Lebesgue integral. Operations on sets, measurable sets, properties of closed and open sets, criteria for measurability, and exterior measure and its properties are discussed. The text also examines set functions, absolute continuity, and generalization of the integral. Absolutely continuous set functions; absolutely continuous functions of several variables; supplementary propositions; and the properties of the Hellinger integral are presented. The text also focuses on metric and normed spaces. Separability, compactness, linear functionals, conjugate spaces, and operators in normed spaces are underscored. The book also discusses Hilbert space. Linear functionals, projections, axioms of the space, sequences of operators, and weak convergence are described. The text is a valuable source of information for students and mathematicians interested in studying the theory of functions.

Table of Contents

  • Contents



    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


    Volumes Published in This Series

Product details

  • No. of pages: 652
  • Language: English
  • Copyright: © Pergamon 2013
  • Published: January 1, 1964
  • Imprint: Pergamon
  • eBook ISBN: 9781483139371

About the Author

V. I. Smirnov

About the Editors

I. N. Sneddon

M. Stark

S. Ulam

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