Real analysis / H. L. Royden.
Editor: New York : London : Macmillan ; Collier-Macmillan, c1963Descripción: xvi, 284 p. ; 24 cmOtra clasificación: 26-01 (28-01 46-01)1 Set Theory [1] Introduction, [1] Functions, [3] Unions, intersections, and complements, [6] Algebras of sets, [11] The axiom of choice and infinite direct products, [13] Countable sets, [13] Relations and equivalences, [16] Partial orderings and the maximal principle, [18] Part One THEORY OF FUNCTIONS OF A REAL VARIABLE 2 The Real Number System [21] 1 Axioms for the real numbers, [21] 2 The natural and rational numbers as subsets of R, [24] 3 The extended real numbers, [26] 4 Sequences of real numbers, [26] 5 Open and closed sets of real numbers, [30] 6 Continuous functions, [36] 7 Borel sets, [41] 3 Lebesgue Measure [43] 1 Introduction, [43] 2 Outer measure, [44] 3 Measurable sets and Lebesgue measure, [47] 4 A nonmeasurable set, [52] 5 Measurable functions, [54] 6 Littlewood’s three principles, [59] 4 The Lebesgue Integral [61] 1 The Riemann integral, [61] 2 The Lebesgue integral of a bounded function over a set of finite measure, [63] 3 The integral of a nonnegative function, [70] 4 The general Lebesgue integral, [75] 5 Convergence in measure, [78] 5 Differentiation and Integration [80] 1 Differentiation of monotone functions, [80] 2 Functions of bounded variation, [84] 3 Differentiation of an integral, [86] 4 Absolute continuity, [90] 6 The Classical Banach Spaces [93] 1 The Lp spaces, [93] 2 The Holder and Minkowski inequalities, [94] 3 Convergence and completeness, [97] 4 Bounded linear functionals on the Lp spaces, [101] Epilogue to Part One [106] Part Two ABSTRACT SPACES 7 Metric Spaces [109] 1 Introduction, [109] 2 Open and closed sets, [111] 3 Continuous functions and homeomorphisms, [113] 4 Convergence and completeness, [115] 5 Uniform continuity and uniformity, [117] 6 Subspaces, [119] 7 Baire category, [121] 8 Topological Spaces [124] 1 Fundamental notions, [124] 2 Bases and countability, [127] 3 The separation axioms and continuous real-valued functions, [130] 4 Connectedness, [133] 5 Nets, [134] 9 Compact Spaces [136] 1 Basic properties, [136] 2 Countable compactness and the Bolzano-Weierstrass property, [138] 3 Compact metric spaces, [141] 4 Products of compact spaces, [143] 5 Locally compact spaces, [146] 6 The Stone-Weierstrass theorem, [147] 7 The Ascoli theorem, [153] 10 Banach Spaces [157] 1 Introduction, [157] 2 Linear operators, [160] 3 Linear functionals and the Hahn-Banach theorem, [162] 4 The closed graph theorem, [169] 5 Weak topologies, [172] 6 Convexity, [175] 7 Hilbert space, [183] Epilogue to Part Two [188] Part Three GENERAL MEASURE AND INTEGRATION THEORY 11 Measure and Integration [191] 1 Measure spaces, [191] 2 Measurable functions, [195] 3 Integration, [196] 4 Signed measures, [202] 5 The Radon-Nikodym theorem, [207] 12 Measure and Outer Measure [216] 1 Outer measure and measurability, [216] 2 The extension theorem, [219] 3 The Lebesgue-Stieltjes integral, [225] 4 Product measures, [229] 5 Carathéodory outer measure, [235] 15 The Daniell Integral [238] 1 Introduction, [238] 2 The extension theorem, [239] 3 Measurability, [244] 4 Uniqueness, [247] 5 Measure and topology, [250] 6 Bounded linear functionals on C(X), [254] 14 Mappings of Measure Spaces [260] 1 Point mappings and set mappings, [260] 2 Measure algebras, [262] 3 Borel equivalences, [267] 4 Set mappings and point mappings on the unit interval, [271] 5 The isometries of Lp, [273] Epilogue to Part Three [278] Bibliography [279] Index of Symbols [280] Subject Index [282]
| Item type | Home library | Shelving location | Call number | Materials specified | Status | Date due | Barcode | Course reserves |
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Libros
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Instituto de Matemática, CONICET-UNS | Libros ordenados por tema | 26 R888 (Browse shelf) | Available | A-1650 |
1 Set Theory [1] --
Introduction, [1] --
Functions, [3] --
Unions, intersections, and complements, [6] --
Algebras of sets, [11] --
The axiom of choice and infinite direct products, [13] --
Countable sets, [13] --
Relations and equivalences, [16] --
Partial orderings and the maximal principle, [18] --
Part One --
THEORY OF FUNCTIONS OF A REAL VARIABLE --
2 The Real Number System [21] --
1 Axioms for the real numbers, [21] --
2 The natural and rational numbers as subsets of R, [24] --
3 The extended real numbers, [26] --
4 Sequences of real numbers, [26] --
5 Open and closed sets of real numbers, [30] --
6 Continuous functions, [36] --
7 Borel sets, [41] --
3 Lebesgue Measure [43] --
1 Introduction, [43] --
2 Outer measure, [44] --
3 Measurable sets and Lebesgue measure, [47] --
4 A nonmeasurable set, [52] --
--
--
5 Measurable functions, [54] --
6 Littlewood’s three principles, [59] --
4 The Lebesgue Integral [61] --
1 The Riemann integral, [61] --
2 The Lebesgue integral of a bounded function over a set of finite measure, [63] --
3 The integral of a nonnegative function, [70] --
4 The general Lebesgue integral, [75] --
5 Convergence in measure, [78] --
5 Differentiation and Integration [80] --
1 Differentiation of monotone functions, [80] --
2 Functions of bounded variation, [84] --
3 Differentiation of an integral, [86] --
4 Absolute continuity, [90] --
6 The Classical Banach Spaces [93] --
1 The Lp spaces, [93] --
2 The Holder and Minkowski inequalities, [94] --
3 Convergence and completeness, [97] --
4 Bounded linear functionals on the Lp spaces, [101] --
Epilogue to Part One [106] --
Part Two --
ABSTRACT SPACES --
7 Metric Spaces [109] --
1 Introduction, [109] --
2 Open and closed sets, [111] --
3 Continuous functions and homeomorphisms, [113] --
4 Convergence and completeness, [115] --
5 Uniform continuity and uniformity, [117] --
6 Subspaces, [119] --
7 Baire category, [121] --
8 Topological Spaces [124] --
1 Fundamental notions, [124] --
2 Bases and countability, [127] --
3 The separation axioms and continuous real-valued functions, [130] --
4 Connectedness, [133] --
5 Nets, [134] --
9 Compact Spaces [136] --
1 Basic properties, [136] --
2 Countable compactness and the Bolzano-Weierstrass property, [138] --
3 Compact metric spaces, [141] --
4 Products of compact spaces, [143] --
5 Locally compact spaces, [146] --
6 The Stone-Weierstrass theorem, [147] --
7 The Ascoli theorem, [153] --
10 Banach Spaces [157] --
1 Introduction, [157] --
2 Linear operators, [160] --
3 Linear functionals and the Hahn-Banach theorem, [162] --
4 The closed graph theorem, [169] --
5 Weak topologies, [172] --
6 Convexity, [175] --
7 Hilbert space, [183] --
Epilogue to Part Two [188] --
Part Three --
GENERAL MEASURE AND INTEGRATION THEORY --
11 Measure and Integration [191] --
1 Measure spaces, [191] --
2 Measurable functions, [195] --
3 Integration, [196] --
4 Signed measures, [202] --
5 The Radon-Nikodym theorem, [207] --
12 Measure and Outer Measure [216] --
1 Outer measure and measurability, [216] --
2 The extension theorem, [219] --
3 The Lebesgue-Stieltjes integral, [225] --
4 Product measures, [229] --
5 Carathéodory outer measure, [235] --
15 The Daniell Integral [238] --
1 Introduction, [238] --
2 The extension theorem, [239] --
3 Measurability, [244] --
4 Uniqueness, [247] --
5 Measure and topology, [250] --
6 Bounded linear functionals on C(X), [254] --
14 Mappings of Measure Spaces [260] --
1 Point mappings and set mappings, [260] --
2 Measure algebras, [262] --
3 Borel equivalences, [267] --
4 Set mappings and point mappings on the unit interval, [271] --
5 The isometries of Lp, [273] --
Epilogue to Part Three [278] --
Bibliography [279] --
Index of Symbols [280] --
Subject Index [282] --
MR, 27 #1540
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