Real Analysis 4th Edition by Halsey Royden, ISBN-13: 978-0131437470
[PDF eBook eTextbook]
- Publisher: Pearson; 4th edition (January 15, 2010)
- Language: English
- 544 pages
- ISBN-10: 013143747X
- ISBN-13: 978-0131437470
For all readers interested in real analysis.
Real Analysis, Fourth Edition, covers the basic material that every reader should know in the classical theory of functions of a real variable, measure and integration theory, and some of the more important and elementary topics in general topology and normed linear space theory. This text assumes a general background in mathematics and familiarity with the fundamental concepts of analysis.
Classical theory of functions, including the classical Banach spaces; General topology and the theory of general Banach spaces; Abstract treatment of measure and integration.
Table of Contents:
Preface
ix
I Lebesgue Integration for Functions of a Single Real Variable
1(180)
Preliminaries on Sets, Mappings, and Relations
3(1)
Unions and Intersections of Sets
3(2)
Equivalence Relations, the Axiom of Choice, and Zorn’s Lemma
5(2)
The Real Numbers: Sets, Sequences, and Functions
7(22)
The Field, Positivity, and Completeness Axioms
7(4)
The Natural and Rational Numbers
11(2)
Countable and Uncountable Sets
13(3)
Open Sets, Closed Sets, and Borel Sets of Real Numbers
16(4)
Sequences of Real Numbers
20(5)
Continuous Real-Valued Functions of a Real Variable
25(4)
Lebesgue Measure
29(25)
Introduction
29(2)
Lebesgue Outer Measure
31(3)
The &-Algebra of Lebesgue Measurable Sets
34(6)
Outer and Inner Approximation of Lebesgue Measurable Sets
40(3)
Countable Additivity, Continuity, and the Borel-Cantelli Lemma
43(4)
Nonmeasurable Sets
47(2)
The Cantor Set and the Cantor-Lebesgue Function
49(5)
Lebesgue Measurable Functions
54(14)
Sums, Products, and Compositions
54(6)
Sequential Pointwise Limits and Simple Approximation
60(4)
Littlewood’s Three Principles, Egoroff’s Theorem, and Lusin’s Theorem
64(4)
Lebesgue Integration
68(29)
The Riemann Integral
68(3)
The Lebesgue Integral of a Bounded Measurable Function over a Set of Finite Measure
71(8)
The Lebesgue Integral of a Measurable Nonnegative Function
79(6)
The General Lebesgue Integral
85(5)
Countable Additivity and Continuity of Integration
90(2)
Uniform Integrability: The Vitali Convergence Theorem
92(5)
Lebesgue Integration: Further Topics
97(10)
Uniform Integrability and Tightness: A General Vitali Convergence Theorem
97(2)
Convergence in Measure
99(3)
Characterizations of Riemann and Lebesgue Integrability
102(5)
Differentiation and Integration
107(28)
Continuity of Monotone Functions
108(1)
Differentiability of Monotone Functions: Lebesgue’s Theorem
109(7)
Functions of Bounded Variation: Jordan’s Theorem
116(3)
Absolutely Continuous Functions
119(5)
Integrating Derivatives: Differentiating Indefinite Integrals
124(6)
Convex Functions
130(5)
The Lp Spaces: Completeness and Approximation
135(20)
Normed Linear Spaces
135(4)
The Inequalities of Young, Holder, and Minkowski
139(5)
Lp Is Complete: The Riesz-Fischer Theorem
144(6)
Approximation and Separability
150(5)
The Lp Spaces: Duality and Weak Convergence
155(26)
The Riesz Representation for the Dual of Lp, 1 & p < &
155(7)
Weak Sequential Convergence in Lp
162(9)
Weak Sequential Compactness
171(3)
The Minimization of Convex Functionals
174(7)
II Abstract Spaces: Metric, Topological, Banach, and Hilbert Spaces
181(154)
Metric Spaces: General Properties
183(23)
Examples of Metric Spaces
183(4)
Open Sets, Closed Sets, and Convergent Sequences
187(3)
Continuous Mappings Between Metric Spaces
190(3)
Complete Metric Spaces
193(4)
Compact Metric Spaces
197(7)
Separable Metric Spaces
204(2)
Metric Spaces: Three Fundamental Theorems
206(16)
The Arzela-Ascoli Theorem
206(5)
The Baire Category Theorem
211(4)
The Banach Contraction Principle
215(7)
Topological Spaces: General Properties
222(17)
Open Sets, Closed Sets, Bases, and Subbases
222(5)
The Separation Properties
227(1)
Countability and Separability
228(2)
Continuous Mappings Between Topological Spaces
230(3)
Compact Topological Spaces
233(4)
Connected Topological Spaces
237(2)
Topological Spaces: Three Fundamental Theorems
239(14)
Urysohn’s Lemma and the Tietze Extension Theorem
239(5)
The Tychonoff Product Theorem
244(3)
The Stone-Weierstrass Theorem
247(6)
Continuous Linear Operators Between Banach Space
253(18)
Normed Linear Spaces
253(3)
Linear Operators
256(3)
Compactness Lost: Infinite Dimensional Normed Linear Spaces
259(4)
The Open Mapping and Closed Graph Theorems
263(5)
The Uniform Boundedness Principle
268(3)
Duality for Normed Linear Spaces
271(27)
Linear Functionals, Bounded Linear Functionals, and Weak Topologies
271(6)
The Hahn-Banach Theorem
277(5)
Reflexive Banach Spaces and Weak Sequential Convergence
282(4)
Locally Convex Topological Vector Spaces
286(4)
The Separation of Convex Sets and Mazur’s Theorem
290(5)
The Krein-Milman Theorem
295(3)
Compactness Regained: The Weak Topology
298(10)
Alaoglu’s Extension of Helley’s Theorem
298(2)
Reflexivity and Weak Compactness: Kakutani’s Theorem
300(2)
Compactness and Weak Sequential Compactness: The Eberlein-Smulian Theorem
302(3)
Metrizability of Weak Topologies
305(3)
Continuous Linear Operators on Hilbert Spaces
308(27)
The Inner Product and Orthogonality
309(4)
The Dual Space and Weak Sequential Convergence
313(3)
Bessel’s Inequality and Orthonormal Bases
316(3)
Adjoints and Symmetry for Linear Operators
319(5)
Compact Operators
324(2)
The Hilbert-Schmidt Theorem
326(3)
The Riesz-Schauder Theorem: Characterization of Fredholm Operators
329(6)
III Measure and Integration: General Theory
335(160)
General Measure Spaces: Their Properties and Construction
337(22)
Measures and Measurable Sets
337(5)
Signed Measures: The Hahn and Jordan Decompositions
342(4)
The Caratheodory Measure Induced by an Outer Measure
346(3)
The Construction of Outer Measures
349(3)
The Caratheodory-Hahn Theorem: The Extension of a Premeasure to a Measure
352(7)
Integration Over General Measure Spaces
359(35)
Measurable Functions
359(6)
Integration of Nonnegative Measurable Functions
365(7)
Integration of General Measurable Functions
372(9)
The Radon-Nikodym Theorem
381(7)
The Nikodym Metric Space: The Vitali-Hahn-Saks Theorem
388(6)
General Lp Spaces: Completeness, Duality, and Weak Convergence
394(20)
The Completeness of Lp (X, &), 1 & p & &
394(5)
The Riesz Representation Theorem for the Dual of Lp (X, &), 1 & p & &
399(5)
The Kantorovitch Representation Theorem for the Dual of L & (X, &)
404(3)
Weak Sequential Compactness in Lp (X, &), 1 < p < 1
407(2)
Weak Sequential Compactness in L1 (X, &): The Dunford-Pettis Theorem
409(5)
The Construction of Particular Measures
414(32)
Product Measures: The Theorems of Fubini and Tonelli
414(10)
Lebesgue Measure on Euclidean Space Rn
424(13)
Cumulative Distribution Functions and Borel Measures on R
437(4)
Caratheodory Outer Measures and Hausdorff Measures on a Metric Space
441(5)
Measure and Topology
446(31)
Locally Compact Topological Spaces
447(5)
Separating Sets and Extending Functions
452(2)
The Construction of Radon Measures
454(3)
The Representation of Positive Linear Functionals on Cc(X): The Riesz-Markov Theorem
457(5)
The Riesz Representation Theorem for the Dual of C(X)
462(8)
Regularity Properties of Baire Measures
470(7)
Invariant Measures
477(18)
Topological Groups: The General Linear Group
477(3)
Kakutani’s Fixed Point Theorem
480(5)
Invariant Borel Measures on Compact Groups: von Neumann’s Theorem
485(3)
Measure Preserving Transformations and Ergodicity: The Bogoliubov-Krilov Theorem
488(7)
Bibliography
495(2)
Index
497
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