A course of higher mathematics. Volume V, Integration and functional analysis /
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 St...
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| Format: | eBook |
| Language: | English |
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Oxford :
Pergamon Press,
1964.
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| Series: | International series of monographs in pure and applied mathematics ;
62. |
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| Online Access: | Connect to the full text of this electronic book |
Table of Contents:
- Front Cover; Higher Mathematics; Copyright Page ; Table of Contents; INTRODUCTION; PREFACE; CHAPTER 1. 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 continuoue fonctione15. General form of the functional in space C; 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. ConvoIution theorem.
- 44. The limit of a measurable function45. The C properly; 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. Hilbert function space; 58· Orthogonal systems of functions; 59. The space l2; 60. Lineals in L2; 61. Examples of closed systems.
- 62. The Holder and Minhkoskii inequalities63. 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 3. SET FUNCTIONS. ABSOLUTE CONTINUITY GENERALIZATION OF THE INTEGRAL; 72. Additive set functions; 73. Siogular function; 74· The case of one variable; 75. Absolutely continuous set functions; 76. Example.