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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
Author: V. I. Smirnov
Editors: I. N. Sneddon, M. Stark, S. Ulam
Language: English
eBook ISBN:9781483139371
9 7 8 - 1 - 4 8 3 1 - 3 9 3 7 - 1
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…Read more
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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.
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