Real analysis / by Edward James McShane and Truman Arthur Botts.
Series The university series in undergraduate mathematicsEditor: Princeton, N. J. : Van Nostrand, c1959Descripción: ix, 272 p. ; 24 cmOtra clasificación: 26-01CONTENTS CHAPTER 0—PRELIMINARIES SECTION PAGE 1. Sets [1] 2. Functions and Relations [3] 3. Natural Numbers and Integers [4] 4. Disclaimer [7] CHAPTER I—REAL NUMBERS 1. Fields [9] 2. Associativity, Commutativity, Distributivity [12] 3. Ordered Fields [14] 4. Isomorphism of Ordered Fields [18] 5. Complete Ordered Fields [20] 6. Uniqueness of Complete Ordered Fields [22] 7. The Real Number System [24] 8. Partially Ordered Sets [27] 9. The Maximality Principle [29] CHAPTER II—CONVERGENCE 1. Convergence of Real-Valued Functions [32] 2. Elementary Properties [35] 3. Subdirected Functions [37] 4. Topological Spaces [39] 5. Convergence in Topological Spaces [40] 6. Product Spaces [42] 7. Neighborhoods and Open Sets [44] 8. Closed Sets [46] 9. Relative Topologies [49] 10. Compact Sets [50] 11. Order-Convergence [54] 12. Metric Spaces [59] 13. Convergence in R* [62] CHAPTER III—CONTINUITY 1. Definition of Continuity [37] 2. Functions on Topological Spaces [68] 3. Semicontinuity [73] 4. Uniform Continuity [77] 5. Double Limits [80] 6. Uniform Convergence [84] 7. The Stone-Weierstrass Approximation Theorem [87] 8. Ascoli’s Theorem [92] 9. Extensions of Functions [94] CHAPTER IV—BOUNDED VARIATION, ABSOLUTE CONTINUITY, DIFFERENTIATION 1. Monotone Functions [98] 2. Functions of Bounded Variation [99] 3. Absolutely Continuous Functions [102] 4. Functions of Intervals [105] 5. Derivatives [110] 6. Continuous Nowhere-Differentiable Functions [113] 7. Taylor’s Formula [116] 8. Differentials [118] 9. The Implicit Functions Theorem [121] CHAPTER V—LEBESGUE-STIELTJES INTEGRATION 1. U-Functions and L-Functions [126] 2. Properties of U-Functions and of L-Functions [131] 3. The Integral [133] 4. Convergence Theorems and Sets of Measure Zero [137] 5. Summability and Intervals of Continuity [141] 6. Fubini’s Theorem [143] 7. Measurable Functions [146] 8. Measurable Sets [153] 9. Integrals from Measures [160] 10. Inequalities [161] 11. The Riemann and Riemann-Stieltjes Integrals [165] CHAPTER VI—THE INTEGRAL AS A FUNCTION OF SETS 1. Abstract Measure Theory [173] 2. Hahn Decomposition [175] 3. Absolute Continuity [177] 4. Lebesgue Decomposition [178] 5. The Radon-Nikodym Theorem [180] 6. Differentiability of the Lebesgue Integral [183] 7. Consequences of the Differentiation Theorem [187] 8. Integration by Parts; Substitution [191] 9. Substitution Theory for Multiple Integrals [196] CHAPTER VII—THE Lp SPACES 1. Function Spaces [202] 2. Topology of the Lp Spaces [206] 3. Linear Functionals on Lp Spaces [209] 4. The Hahn-Banach Theorem [214] 5. The Riesz Representation Theorem [216] 6. Geometry of Hilbert Space [219] 7. Orthonormal Systems and Fourier Series [227] 8. The Fourier Transform [233] 9. Bounded Hermitian Operators on Hilbert Space [241] APPENDIX I—THE PRINCIPLE OF INDUCTIVE DEFINITION [249] APPENDIX II—THE MAXIMALITY PRINCIPLE [251] 1. Six Equivalent Formulations [251] 2. Well-Ordering and Transfinite Induction [252] 3. Choice, Well-Ordering, and Maximality [255] APPENDIX III—TYCHONOFF’S THEOREM [260] Bibliography [261] Symbols [263] Index [265]
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| 26 L781-3 The elements of the theory of real functions. | 26 L863 Advanced calculus / | 26 M165 The generalized Riemann integral / | 26 M175 Real analysis / | 26 M175e Análisis real / | 26 M175e Análisis real / | 26 M271 Séries adhérentes. Régularisation des suites. Applications / |
CONTENTS --
CHAPTER 0—PRELIMINARIES --
SECTION PAGE --
1. Sets [1] --
2. Functions and Relations [3] --
3. Natural Numbers and Integers [4] --
4. Disclaimer [7] --
CHAPTER I—REAL NUMBERS --
1. Fields [9] --
2. Associativity, Commutativity, Distributivity [12] --
3. Ordered Fields [14] --
4. Isomorphism of Ordered Fields [18] --
5. Complete Ordered Fields [20] --
6. Uniqueness of Complete Ordered Fields [22] --
7. The Real Number System [24] --
8. Partially Ordered Sets [27] --
9. The Maximality Principle [29] --
CHAPTER II—CONVERGENCE --
1. Convergence of Real-Valued Functions [32] --
2. Elementary Properties [35] --
3. Subdirected Functions [37] --
4. Topological Spaces [39] --
5. Convergence in Topological Spaces [40] --
6. Product Spaces [42] --
7. Neighborhoods and Open Sets [44] --
8. Closed Sets [46] --
9. Relative Topologies [49] --
10. Compact Sets [50] --
11. Order-Convergence [54] --
12. Metric Spaces [59] --
13. Convergence in R* [62] --
CHAPTER III—CONTINUITY --
1. Definition of Continuity [37] --
2. Functions on Topological Spaces [68] --
3. Semicontinuity [73] --
4. Uniform Continuity [77] --
5. Double Limits [80] --
6. Uniform Convergence [84] --
7. The Stone-Weierstrass Approximation Theorem [87] --
8. Ascoli’s Theorem [92] --
9. Extensions of Functions [94] --
CHAPTER IV—BOUNDED VARIATION, ABSOLUTE CONTINUITY, DIFFERENTIATION --
1. Monotone Functions [98] --
2. Functions of Bounded Variation [99] --
3. Absolutely Continuous Functions [102] --
4. Functions of Intervals [105] --
5. Derivatives [110] --
6. Continuous Nowhere-Differentiable Functions [113] --
7. Taylor’s Formula [116] --
8. Differentials [118] --
9. The Implicit Functions Theorem [121] --
CHAPTER V—LEBESGUE-STIELTJES INTEGRATION --
1. U-Functions and L-Functions [126] --
2. Properties of U-Functions and of L-Functions [131] --
3. The Integral [133] --
4. Convergence Theorems and Sets of Measure Zero [137] --
5. Summability and Intervals of Continuity [141] --
6. Fubini’s Theorem [143] --
7. Measurable Functions [146] --
8. Measurable Sets [153] --
9. Integrals from Measures [160] --
10. Inequalities [161] --
11. The Riemann and Riemann-Stieltjes Integrals [165] --
CHAPTER VI—THE INTEGRAL AS A FUNCTION OF SETS --
1. Abstract Measure Theory [173] --
2. Hahn Decomposition [175] --
3. Absolute Continuity [177] --
4. Lebesgue Decomposition [178] --
5. The Radon-Nikodym Theorem [180] --
6. Differentiability of the Lebesgue Integral [183] --
7. Consequences of the Differentiation Theorem [187] --
8. Integration by Parts; Substitution [191] --
9. Substitution Theory for Multiple Integrals [196] --
CHAPTER VII—THE Lp SPACES --
1. Function Spaces [202] --
2. Topology of the Lp Spaces [206] --
3. Linear Functionals on Lp Spaces [209] --
4. The Hahn-Banach Theorem [214] --
5. The Riesz Representation Theorem [216] --
6. Geometry of Hilbert Space [219] --
7. Orthonormal Systems and Fourier Series [227] --
8. The Fourier Transform [233] --
9. Bounded Hermitian Operators on Hilbert Space [241] --
APPENDIX I—THE PRINCIPLE OF INDUCTIVE --
DEFINITION [249] --
APPENDIX II—THE MAXIMALITY PRINCIPLE [251] --
1. Six Equivalent Formulations [251] --
2. Well-Ordering and Transfinite Induction [252] --
3. Choice, Well-Ordering, and Maximality [255] --
APPENDIX III—TYCHONOFF’S THEOREM [260] --
Bibliography [261] --
Symbols [263] --
Index [265] --
MR, 22 #84
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