# Basic Real Analysis

Springer Science & Business Media, Jun 3, 2003 - Mathematics - 559 pages

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.

### What people are saying -Write a review

We haven't found any reviews in the usual places.

### 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 Copyright