Basic Real Analysis

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Springer Science & Business Media, Jun 3, 2003 - Mathematics - 559 pages
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One of the bedrocks of any mathematics education, the study of real analysis introduces students both to mathematical rigor and to the deep theorems and counterexamples that arise from such rigor: for instance, the construction of number systems, the Cantor Set, the Weierstrass nowhere differentiable function, and the Weierstrass approximation theorem. Basic Real Analysis is a modern, systematic text that presents the fundamentals and touchstone results of the subject in full rigor, but in a style that requires little prior familiarity with proofs or mathematical language.

Key features include:

* A broad view of mathematics throughout the book

* Treatment of all concepts for real numbers first, with extensions to metric spaces later, in a separate chapter

* Elegant proofs

* Excellent choice of topics

* Numerous examples and exercises to enforce methodology; exercises integrated into the main text, as well as at the end of each chapter

* Emphasis on monotone functions throughout

* Good development of integration theory

* Special topics on Banach and Hilbert spaces and Fourier series, often not included in many courses on real analysis

* Solid preparation for deeper study of functional analysis

* Chapter on elementary probability

* Comprehensive bibliography and index

* Solutions manual available to instructors upon request

By covering all the basics and developing rigor simultaneously, this introduction to real analysis is ideal for senior undergraduates and beginning graduate students, both as a classroom text or for self-study. With its wide range of topics and its view of real analysis in a larger context, the book will be appropriate for more advanced readers as well.

 

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Contents

Set Theory
1
12 Relations and Functions
5
13 Basic Algebra Counting and Arithmetic
16
14 Infinite Direct Products Axiom of Choice and Cardinal Numbers
28
15 Problems
33
Sequences and Series of Real Numbers
37
22 Sequences in ℝ
47
23 Infinite Series
55
72 Some Classes of Integrable Functions
258
73 Sets of Measure Zero and Lebesgues Integrability Criterion
264
74 Properties of the Riemann Integral
271
75 Fundamental Theorem of Calculus
279
76 Functions of Bounded Variation
283
77 Problems
288
Sequences and Series of Functions
297
82 Pointwise and Uniform Convergence
301

24 Unordered Series and Summability
69
25 Problems
76
Limits of Functions
85
31 Bounded and Monotone Functions
86
32 Limits of Functions
88
33 Properties of Limits
90
34 Onesided Limits and Limits Involving Infinity
94
35 Indeterminate Forms Equivalence Landaus Little oh and Big Oh
102
36 Problems
109
Topology of ℝ and Continuity
113
41 Compact and Connected Subsets of ℝ
114
42 The Cantor Set
118
43 Continuous Functions
122
44 Onesided Continuity Discontinuity and Monotonicity
127
45 Extreme Value and Intermediate Value Theorems
132
46 Uniform Continuity
137
47 Approximation by Step Piecewise Linear and Polynomial Functions
144
48 Problems
150
Metric Spaces
157
51 Metrics and Metric Spaces
158
52 Topology of a Metric Space
162
53 Limits Cauchy Sequences and Completeness
166
54 Continuity
172
55 Uniform Continuity and Continuous Extensions
179
56 Compact Metric Spaces
186
57 Connected Metric Spaces
194
58 Problems
200
The Derivative
209
61 Differentiability
210
62 Derivatives of Elementary Functions
214
63 The Differential Calculus
216
64 Mean Value Theorems
221
65 LHôpitals Rule
226
66 Higher Derivatives and Taylors Formula
230
67 Convex Functions
240
68 Problems
244
The Riemann Integral
251
83 Uniform Convergence and Limit Theorems
307
84 Power Series
312
85 Elementary Transcendental Functions
322
86 Fourier Series
327
87 Problems
343
Normed and Function Spaces
351
92 Banach Spaces
357
93 Hilbert Spaces
366
94 Function Spaces
378
95 Problems
385
The Lebesgue Integral F Rieszs Approach
395
101 Improper Riemann Integrals
396
102 Step Functions and Their Integrals
399
103 Convergence Almost Everywhere
401
104 The Lebesgue Integral
405
105 Convergence Theorems
411
106 The Banach Space L¹
421
107 Problems
425
Lebesgue Measure
431
111 Measurable Functions
432
112 Measurable Sets and Lebesgue Measure
434
113 Measurability Lebesgues Definition
439
114 The Theorems of Egorov Lusin and Steinhaus
443
115 Regularity of Lebesgue Measure
447
116 Lebesgues Outer and Inner Measures
451
117 The Hilbert Spaces L²E 𝔽
458
118 Problems
460
General Measure and Probability
467
122 Measurable Functions
481
123 Integration
484
124 Probability
495
125 Problems
513
Construction of Real Numbers
531
References
537
Index
543
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