Smirnov / Sneddon / Stark 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
1. Auflage 2014
ISBN: 978-1-4831-3937-1
Verlag: Elsevier Science & Techn.
Format: PDF
Kopierschutz: 1 - PDF Watermark

E-Book, Englisch, 652 Seiten, Web PDF

Reihe: International Series in Pure and Applied Mathematics

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.

Autoren/Hrsg.

Smirnov, V. I.

Sneddon, I. N.

Stark, M.

Ulam, S.

Inhaltsverzeichnis

1;Front Cover;1
2;Higher Mathematics;4
3;Copyright Page ;5
4;Table of Contents;6
5;INTRODUCTION;10
6;PREFACE;12
7;CHAPTER 1. THE STIELTJES INTEGRAL;16
7.1;1. Sets and their powers;16
7.2;2. The Stieltjes integral and its basic properties;19
7.3;3. Darboux sums;23
7.4;4. The Stieltjes integral of a continuous function;28
7.5;5· The improper Stieltjes integral;31
7.6;6. Jump functions;33
7.7;7. Physical interpretation;37
7.8;8. Functions of bounded variation;39
7.9;9· An integrating function of bounded variation;45
7.10;10. Existence of the Stieltjes integral;47
7.11;11. Passage to the limit in the Stieltjes integral;48
7.12;12. Helly's theorem;50
7.13;13. Selection principle;54
7.14;14. Space of continuoue fonctione;55
7.15;15. General form of the functional in space C;58
7.16;16. Linear operators in C;62
7.17;17· Functions of an interval;63
7.18;18. The general Stieltjes integral;65
7.19;19. Properties of the (general) Stieltjes integral;68
7.20;20. The existence of the general Stieltjes integral;71
7.21;21· Functions of a two-dimensional interval;73
7.22;22. Passage to point functions;76
7.23;23. The Stieltjes integral on a plane;79
7.24;24. Functions of bounded variation on the plane;81
7.25;25. The space of continuous functions of several variables;84
7.26;26. The Fourier-Stieltjes integral;85
7.27;27. Inversion formula;88
7.28;28. ConvoIution theorem;90
7.29;29. The Cauchy–Stieltjes integral;92
8;CHAPTER 2. SET FUNCTIONS AND THE LEBESGUE INTEGRAL;96
8.1;§ 1· Set functions and the theory of measure;96
8.1.1;30. Operations on sets;96
8.1.2;31· Point sets;99
8.1.3;32· Properties of closed and open sets;101
8.1.4;33. Elementary figures;104
8.1.5;34. Exterior measure and its properties;107
8.1.6;35. Measurable sets;110
8.1.7;36. Measurable sets (continued);118
8.1.8;37. Criteria for measurability;120
8.1.9;38. Field of sets;121
8.1.10;39. Independence of the choice of axes;124
8.1.11;40. The . field;125
8.1.12;41. The case of a single variable;127
8.2;§ 2· Measurable functions;128
8.2.1;42. Definition of measurable function;128
8.2.2;43. Properties of measurable functions;132
8.2.3;44. The limit of a measurable function;133
8.2.4;45. The C properly;138
8.2.5;46. Piecewise constant functions;138
8.2.6;47. Class B ;141
8.3;§ 3. The Lebesgue integral;142
8.3.1;48. The integral of a bounded function;142
8.3.2;49. Properties of the integral;145
8.3.3;50· The integral of a non-negative unbounded function;150
8.3.4;51. Properties of the integral;153
8.3.5;52· Functions of any sign;156
8.3.6;53. Complex summable functions;161
8.3.7;54. Passage to the limit under the integral sign;162
8.3.8;55· The class L2;167
8.3.9;56. Convergence in the mean;169
8.3.10;57. Hilbert function space;172
8.3.11;58· Orthogonal systems of functions;175
8.3.12;59. The space l2;180
8.3.13;60. Lineals in L2;183
8.3.14;61. Examples of closed systems;186
8.3.15;62. The Holder and Minhkoskii inequalities;188
8.3.16;63. Integral over a set of infinite measure;193
8.3.17;64. The class L2 on a set of infinite measure;197
8.3.18;65· An integrating function of bounded variation;200
8.3.19;66. The reduction of multiple integrals;202
8.3.20;67· The case of the characteristic function;206
8.3.21;68· Fubini's theorem;209
8.3.22;69. Change of the order of integration;213
8.3.23;70. Continuity in the mean;215
8.3.24;71. Mean functions;216
9;CHAPTER 3. SET FUNCTIONS. ABSOLUTE CONTINUITY GENERALIZATION OF THE INTEGRAL;224
9.1;72. Additive set functions;224
9.2;73. Siogular function;228
9.3;74· The case of one variable;231
9.4;75. Absolutely continuous set functions;236
9.5;76. Example;243
9.6;77. Absolutely contínuous functions of seyeral variables;245
9.7;78· Supplementary propositions;247
9.8;79. Supplementary propositions (continued);252
9.9;80. Fundamental theorem;256
9.10;81. Hellinger'e iutegrals;260
9.11;82. The case of a single variable;263
9.12;83. Properties of the Hellinger integral;268
10;CHAPTER 4. METRIC AND NORMED SPACES;272
10.1;84. Metric space;272
10.2;85. The completion of a metric space;274
10.3;86· Operators and functionals. The principle of compressed mappings;279
10.4;87. Examples;281
10.5;88. Examples of applications of the principle of compressed mappings;283
10.6;89. Compactness;286
10.7;90· Compactness in C;288
10.8;91. Compactness in Lp;289
10.9;92. Compactness in lp;292
10.10;93. Functionals on mutually compact sets;294
10.11;94. Separability;295
10.12;95. Linear normed spaces;296
10.13;96. Examples of normed spaces;300
10.14;97. Operators in normed spaces;300
10.15;98. Linear functionals;304
10.16;99. Conjugate spaces;308
10.17;100. Weak convergence of functionals;310
10.18;101. The weak convergence of elements;313
10.19;102. Linear functionals in C, Lp, and lp;317
10.20;103. Weak convergence in C, Lp, and lp;324
10.21;104· The space of linear operators and the convergence of sequences of operators;326
10.22;105. Conjugate operators;328
10.23;106. Completely continuous operators;329
10.24;107· Operator equations;330
10.25;108. Completely continuous operators in C, Lp, and lp;332
10.26;109· Generalized derivatives;335
10.27;110. Generalized derivatives (continued);340
10.28;111. The case of a star-shaped domain;343
10.29;112. Spaces W(1)p and W(1)p;344
10.30;113. Properties of functions of space W(1)p (D);347
10.31;114. Embedding theorems;354
10.32;115. Integral operators with a polar kernel;358
10.33;116. Soboley'e integral forms;364
10.34;117. Embedding theorems;367
10.35;118. Domains of a more general type;370
10.36;119· Space C(1)D;371
11;CHAPTER 5. HILBERT SPACE;382
11.1;§ 1. The theory of bounded operators;382
11.1.1;120. Axioms of the space;382
11.1.2;121· Orthogonality and orthogonal systems of elements;384
11.1.3;122. Projections;389
11.1.4;123. Linear functionals;391
11.1.5;124. Linear operators;393
11.1.6;125· Bilinear and quadratic functionals;396
11.1.7;126. Bounds of a self-conjugate operator;398
11.1.8;127. The inverse operator;400
11.1.9;128. Spectrum of an operator;404
11.1.10;129. The spectrum of a self-conjugate operator;407
11.1.11;130. The resolvent;411
11.1.12;131. Sequences of operators;412
11.1.13;132. Weak convergence;413
11.1.14;134. Spaces . and l2;417
11.1.15;135· Linear equations in completely continuous operators;420
11.1.16;136. Completely continuous self-conjugate operators;425
11.1.17;137· Unitary operators;430
11.1.18;138. The absolute norm of an operator;433
11.1.19;140. Projection operators;438
11.1.20;141. The resolution of the identity. The Stieltjes integral;443
11.1.21;142. The spectral function of a self-conjugate operator;448
11.1.22;143. Continuous functions of a self-conjugate operator;451
11.1.23;144. A formula for the resolvent and a characteristic of regular values of .;453
11.1.24;145· Eigenvalues and eigenelements;456
11.1.25;146. Purely point spectra;458
11.1.26;147. A continuous simple spectrum;459
11.1.27;148. Invariant subspaces;465
11.1.28;149. The general case of a continuous spectrum;468
11.1.29;150. The case of a mixed spectrum;470
11.1.30;151. Differential solutions;471
11.1.31;152. The operation of multiplication by the independent variable;475
11.1.32;153. The unitary equivalence of self-conjugate operators;478
11.1.33;154. The spectral resolution of unitary operators;479
11.1.34;155· Functions of a self-conjugate operator;480
11.1.35;156. Commuting operators;484
11.1.36;157· Perturbations of the spectrum of a self-conjugate operator;486
11.1.37;158. Normal operators;488
11.1.38;159. Auxiliary propositions;490
11.1.39;160. Power series of operators;493
11.1.40;161. The spectral function;495
11.2;§ 2. Spaces l2 and L2;498
11.2.1;162· Linear operators in l2;498
11.2.2;163· Bounded operators;500
11.2.3;164. Unitary matrices and projection matrices;505
11.2.4;165. Self-conjugate matrices;507
11.2.5;166. The case of a continuous spectrum;509
11.2.6;167. Jacobian matrices;514
11.2.7;168. Differential solutions;517
11.2.8;169· Examples;519
11.2.9;170. Weak convergence in l2;522
11.2.10;171· Completely continuous operators in l2;523
11.2.11;172. Integral operators in L2;524
11.2.12;173· The conjugate operator;525
11.2.13;174. Completely continuous operators;527
11.2.14;175. Spectral functions;529
11.2.15;176. The spectral function (continued);530
11.2.16;177. Unitary transformations in L2;532
11.2.17;178. Fourier transformations;535
11.2.18;179. Fourier transformations and Hermitian functions;539
11.2.19;180· The operation of multiplication;540
11.2.20;181· Kernels that depend on a difference;542
11.2.21;182. Weak convergence;546
11.2.22;183. Other concrete forms of space H;546
11.3;§ 3. Unbounded operators;548
11.3.1;184. Closed operators;548
11.3.2;185. Conjugate operators;550
11.3.3;186. The graph of an operator;553
11.3.4;187. Symmetric and self-conjugate operators;555
11.3.5;188. Examples of unbounded operators;558
11.3.6;189· The gpectrum of a self-conjugate operator;568
11.3.7;190· The case of a point spectrum;571
11.3.8;191· Invariant subspaces and the reducibility of an operator;573
11.3.9;192. Resolutions of the identity. The Stieltjes integral;577
11.3.10;193. Continuous functions of a self-conjugate operator;583
11.3.11;194· The resolvent;584
11.3.12;195. Eigenvalues;586
11.3.13;196. The case of a mixed spectrum;587
11.3.14;197. Functions of a self-conjugate operator;589
11.3.15;198. Small perturbations of the spectrum;592
11.3.16;199. The operator of multiplication;594
11.3.17;200· Integral operators;598
11.3.18;201. The extension of a closed symmetric operator;601
11.3.19;202. Deficiency indices;605
11.3.20;203. The conjugate operator;608
11.3.21;204· Maximal operators;610
11.3.22;205. Extension of symmetric semi-bounded operators;612
11.3.23;206. The comparison of semi-bounded operators;617
11.3.24;207. Examples on the theory of extensions;619
11.3.25;208. The spectrum of a symmetric operator;621
11.3.26;209. Some theorems on extensions and their spectra;624
11.3.27;210· The independence of the deficiency indices on .;627
11.3.28;211. The invariance of the continuous part of the spectral kernel in the case of symmetric extensions;629
11.3.29;212. The spectra of self-conjugate extensions;630
11.3.30;213. Examples;631
11.3.31;214· Infinite matrices;631
11.3.32;215· Jacobian matrices;633
11.3.33;216. Matrices and operators;638
11.3.34;217· The unitary equivalence of C-matrices;641
11.3.35;218. The existence of the spectral function;644
12;INDEX;648
13;VOLUMES PUBLISHED IN THIS SERIES;652