Real analysis / H. L. Royden.
Editor: New York : London : Macmillan ; Collier Macmillan, c1989Edición: 3rd edDescripción: xvii, 444 p. ; 25 cmISBN: 0029466202 (International ed.); 0024041513 (hardcover)Tema(s): Functions of real variables | Functional analysis | Measure theoryOtra clasificación: 00A05 (26-01 28-01 46-01)Prologue to the Student [1] 1 Set Theory [6] 1 Introduction [6] 2 Functions [9] 3 Unions, intersections, and complements [12] 4 Algebras of sets [17] 5 The axiom of choice and infinite direct products [19] 6 Countable sets [20] 7 Relations and equivalences [23] 8 Partial orderings and the maximal principle [24] 9 Well ordering and the countable ordinals [26] Part One THEORY OF FUNCTIONS OF A REAL VARIABLE 2 The Real Number System [31] 1 Axioms for the real numbers [31] 2 The natural and rational numbers as subsets of R [34] 3 The extended real numbers [36] 4 Sequences of real numbers [37] 5 Open and closed sets of real numbers [40] 6 Continuous functions [47] 7 Borel sets [52] 3 Lebesgue Measure [54] 1 Introduction [54] 2 Outer measure [56] 3 Measurable sets and Lebesgue measure [58] *4 A nonmeasurable set [64] 5 Measurable functions [66] 6 Littlewood’s three principles [72] 4 The Lebesgue Integral [75] 1 The Riemann integral [75] 2 The Lebesgue integral of a bounded function over a set of finite measure [77] 3 The integral of a nonnegative function [85] 4 The general Lebesgue integral [89] *5 Convergence in measure [95] 5 Differentiation and Integration [97] 1 Differentiation of monotone functions [97] 2 Functions of bounded variation [102] 3 Differentiation of an integral [104] 4 Absolute continuity [108] 5 Convex functions [113] 6 The Classical Banach Spaces [118] 1 The Lp spaces [118] 2 The Minkowski and Holder inequalities [119] 3 Convergence and completeness [123] 4 Approximation in Lp [127] 5 Bounded linear functionals on the Lp spaces [130] Part Two ABSTRACT SPACES 7 Metric Spaces [139] 1 Introduction [139] 2 Open and closed sets [141] 3 Continuous functions and homeomorphisms [144] 4 Convergence and completeness [146] 5 Uniform continuity and uniformity [148] 6 Subspaces [151] 7 Compact metric spaces [152] 8 Baire category [158] 9 Absolute G؏'s [164] 10 The Ascoli-Arzela Theorem [167] 8 Topological Spaces [171] 1 Fundamental notions [171] 2 Bases and countability [175] 3 The separation axioms and continuous real-valued functions [178] 4 Connectedness [182] 5 Products and direct unions of topological spaces [184] *6 Topological and uniform properties [187] *7 Nets [188] 9 Compact and Locally Compact Spaces [190] 1 Compact spaces [190] 2 Countable compactness and the Bolzano-Weierstrass property [193] 3 Products of compact spaces [196] 4 Locally compact spaces [199] 5 (7-compact spaces [203] *6 Paracompact spaces [204] 7 Manifolds [206] *8 The Stone-Čech compactification [209] 9 The Stone-Weierstrass Theorem [210] 10 Banach Spaces [217] 1 Introduction [217] 2 Linear operators [220] 3 Linear functionals and the Hahn-Banach Theorem [222] 4 The Closed Graph Theorem [224] 5 Topological vector spaces [233] 6 Weak topologies [236] 7 Convexity [239] 8 Hilbert space [245] Part Three GENERAL MEASURE AND INTEGRATION THEORY 11 Measure and Integration [253] 1 Measure spaces [253] 2 Measurable functions [259] 3 Integration [263] 4 General Convergence Theorems [268] 5 Signed measures [270] 6 The Radon-Nikodym Theorem [276] 7 The Lp-spaces [282] 12 Measure and Outer Measure [288] 1 Outer measure and measurability [288] 2 The Extension Theorem [291] 3 The Lebesgue-Stieltjes integral [299] 4 Product measures [303] 5 Integral operators [313] *6 Inner measure [317] *7 Extension by sets of measure zero [325] 8 Caratheodory outer measure [326] 9 Hausdorff measure [329] 13 Measure and Topology [331] 1 Baire sets and Borel sets [331] 2 The regularity of Baire and Borel measures [337] 3 The construction of Borel measures [345] 4 Positive linear functionals and Borel measures [352] 5 Bounded linear functionals on C(X) [355] 14 Invariant Measures [361] 1 Homogeneous spaces [361] 2 Topological equicontinuity [362] 3 The existence of invariant measures [365] 4 Topological groups [370] 5 Group actions and quotient spaces [376] 6 Unicity of invariant measures [378] 7 Groups of diffeomorphisms [388] 15 Mappings of Measure Spaces [392] 1 Point mappings and set mappings [392] 2 Boolean σ-algebras [394] 3 Measure algebras [398] 4 Borel equivalences [401] 5 Borel measures on complete separable metric spaces [406] 6 Set mappings and point mappings on complete separable metric spaces [412] 7 The isometries of Lp [415] 16 The Daniell Integral [419] Introduction [419] 2 The Extension Theorem [422] 3 Uniqueness [427] 4 Measurability and measure [429] Bibliography [435] Index of Symbols [437] Subject Index [439]
| Item type | Home library | Shelving location | Call number | Materials specified | Status | Date due | Barcode | Course reserves |
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Incluye referencias bibliográficas (p. 435-436) e índices.
Prologue to the Student --
[1] --
1 Set Theory [6] --
1 Introduction [6] --
2 Functions [9] --
3 Unions, intersections, and complements [12] --
4 Algebras of sets [17] --
5 The axiom of choice and infinite direct products [19] --
6 Countable sets [20] --
7 Relations and equivalences [23] --
8 Partial orderings and the maximal principle [24] --
9 Well ordering and the countable ordinals [26] --
Part One --
THEORY OF FUNCTIONS OF A REAL VARIABLE --
2 The Real Number System --
[31] --
1 Axioms for the real numbers [31] --
2 The natural and rational numbers as subsets of R [34] --
3 The extended real numbers [36] --
4 Sequences of real numbers [37] --
5 Open and closed sets of real numbers [40] --
6 Continuous functions [47] --
7 Borel sets [52] --
3 Lebesgue Measure [54] --
1 Introduction [54] --
2 Outer measure [56] --
3 Measurable sets and Lebesgue measure [58] --
*4 A nonmeasurable set [64] --
5 Measurable functions [66] --
6 Littlewood’s three principles [72] --
4 The Lebesgue Integral [75] --
1 The Riemann integral [75] --
2 The Lebesgue integral of a bounded function over a set of finite measure [77] --
3 The integral of a nonnegative function [85] --
4 The general Lebesgue integral [89] --
*5 Convergence in measure [95] --
5 Differentiation and Integration [97] --
1 Differentiation of monotone functions [97] --
2 Functions of bounded variation [102] --
3 Differentiation of an integral [104] --
4 Absolute continuity [108] --
5 Convex functions [113] --
6 The Classical Banach Spaces [118] --
1 The Lp spaces [118] --
2 The Minkowski and Holder inequalities [119] --
3 Convergence and completeness [123] --
4 Approximation in Lp [127] --
5 Bounded linear functionals on the Lp spaces [130] --
Part Two --
ABSTRACT SPACES --
7 Metric Spaces [139] --
1 Introduction [139] --
2 Open and closed sets [141] --
3 Continuous functions and homeomorphisms [144] --
4 Convergence and completeness [146] --
5 Uniform continuity and uniformity [148] --
6 Subspaces [151] --
7 Compact metric spaces [152] --
8 Baire category [158] --
9 Absolute G؏'s [164] --
10 The Ascoli-Arzela Theorem [167] --
8 Topological Spaces [171] --
1 Fundamental notions [171] --
2 Bases and countability [175] --
3 The separation axioms and continuous real-valued functions [178] --
4 Connectedness [182] --
5 Products and direct unions of topological spaces [184] --
*6 Topological and uniform properties [187] --
*7 Nets [188] --
9 Compact and Locally Compact Spaces [190] --
1 Compact spaces [190] --
2 Countable compactness and the Bolzano-Weierstrass property [193] --
3 Products of compact spaces [196] --
4 Locally compact spaces [199] --
5 (7-compact spaces [203] --
*6 Paracompact spaces [204] --
7 Manifolds [206] --
*8 The Stone-Čech compactification [209] --
9 The Stone-Weierstrass Theorem [210] --
10 Banach Spaces [217] --
1 Introduction [217] --
2 Linear operators [220] --
3 Linear functionals and the Hahn-Banach Theorem [222] --
4 The Closed Graph Theorem [224] --
5 Topological vector spaces [233] --
6 Weak topologies [236] --
7 Convexity [239] --
8 Hilbert space [245] --
Part Three --
GENERAL MEASURE AND INTEGRATION THEORY --
11 Measure and Integration [253] --
1 Measure spaces [253] --
2 Measurable functions [259] --
3 Integration [263] --
4 General Convergence Theorems [268] --
5 Signed measures [270] --
6 The Radon-Nikodym Theorem [276] --
7 The Lp-spaces [282] --
12 Measure and Outer Measure [288] --
1 Outer measure and measurability [288] --
2 The Extension Theorem [291] --
3 The Lebesgue-Stieltjes integral [299] --
4 Product measures [303] --
5 Integral operators [313] --
*6 Inner measure [317] --
*7 Extension by sets of measure zero [325] --
8 Caratheodory outer measure [326] --
9 Hausdorff measure [329] --
13 Measure and Topology [331] --
1 Baire sets and Borel sets [331] --
2 The regularity of Baire and Borel measures [337] --
3 The construction of Borel measures [345] --
4 Positive linear functionals and Borel measures [352] --
5 Bounded linear functionals on C(X) [355] --
14 Invariant Measures [361] --
1 Homogeneous spaces [361] --
2 Topological equicontinuity [362] --
3 The existence of invariant measures [365] --
4 Topological groups [370] --
5 Group actions and quotient spaces [376] --
6 Unicity of invariant measures [378] --
7 Groups of diffeomorphisms [388] --
15 Mappings of Measure Spaces [392] --
1 Point mappings and set mappings [392] --
2 Boolean σ-algebras [394] --
3 Measure algebras [398] --
4 Borel equivalences [401] --
5 Borel measures on complete separable metric spaces [406] --
6 Set mappings and point mappings on complete separable metric spaces [412] --
7 The isometries of Lp [415] --
16 The Daniell Integral [419] --
Introduction [419] --
2 The Extension Theorem [422] --
3 Uniqueness [427] --
4 Measurability and measure [429] --
Bibliography [435] --
Index of Symbols [437] --
Subject Index [439] --
MR, 90g:00004
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