Basic Real Analysis

"Basic Real Analysis" demonstrates the richness of real analysis, giving students an introduction both to mathematical rigor and to the deep theorems and counterexamples that arise from such rigor: for example, the construction of real numbers, the Cantor set, the Weierstrass nowhere differentiable function, and the Weierstrass approximation theorem. In this modern, systematic text, all the touchstone results and fundamentals are carefully presented, but in a style that requires little prior familiarity with proofs or mathematical language. Basic topics---metric spaces, integration, series, and topology---are methodically developed; advanced material on Banach and Hilbert spaces and Fourier series is also included. Different types of convergence, monotone functions, and applications to probability and other areas of mathematics are featured as well. With its many examples and exercises and broad view of analysis, this work is ideal for senior undergraduates and beginning graduate students, either in the classroom or for self-study.

Table of Contents:
1 Set Theory.- 1.1 Rings and Algebras of Sets.- 1.2 Relations and Functions.- 1.3 Basic Algebra, Counting, and Arithmetic.- 1.4 Infinite Direct Products, Axiom of Choice, and Cardinal Numbers.- 1.5 Problems.- 2 Sequences and Series of Real Numbers.- 2.1 Real Numbers.- 2.2 Sequences in ?.- 2.3 Infinite Series.- 2.4 Unordered Series and Summability.- 2.5 Problems.- 3 Limits of Functions.- 3.1 Bounded and Monotone Functions.- 3.2 Limits of Functions.- 3.3 Properties of Limits.- 3.4 One-sided Limits and Limits Involving Infinity.- 3.5 Indeterminate Forms, Equivalence, Landau’s Little “oh” and Big “Oh”.- 3.6 Problems.- 4 Topology of ? and Continuity.- 4.1 Compact and Connected Subsets of ?.- 4.2 The Cantor Set.- 4.3 Continuous Functions.- 4.4 One-sided Continuity, Discontinuity, and Monotonicity.- 4.5 Extreme Value and Intermediate Value Theorems.- 4.6 Uniform Continuity.- 4.7 Approximation by Step, Piecewise Linear, and Polynomial Functions.- 4.8 Problems.- 5 Metric Spaces.- 5.1 Metrics and Metric Spaces.- 5.2 Topology of a Metric Space.- 5.3 Limits, Cauchy Sequences, and Completeness.- 5.4 Continuity.- 5.5 Uniform Continuity and Continuous Extensions.- 5.6 Compact Metric Spaces.- 5.7 Connected Metric Spaces.- 5.8 Problems.- 6 The Derivative.- 6.1 Differentiability.- 6.2 Derivatives of Elementary Functions.- 6.3 The Differential Calculus.- 6.4 Mean Value Theorems.- 6.5 L’Hôpital’s Rule.- 6.6 Higher Derivatives and Taylor’s Formula.- 6.7 Convex Functions.- 6.8 Problems.- 7 The Riemann Integral.- 7.1 Tagged Partitions and Riemann Sums.- 7.2 Some Classes of Integrable Functions.- 7.3 Sets of Measure Zero and Lebesgue’s Integrability Criterion.- 7.4 Properties of the Riemann Integral.- 7.5 Fundamental Theorem of Calculus.- 7.6 Functions of BoundedVariation.- 7.7 Problems.- 8 Sequences and Series of Functions.- 8.1 Complex Numbers.- 8.2 Pointwise and Uniform Convergence.- 8.3 Uniform Convergence and Limit Theorems.- 8.4 Power Series.- 8.5 Elementary Transcendental Functions.- 8.6 Fourier Series.- 8.7 Problems.- 9 Normed and Function Spaces.- 9.1 Norms and Normed Spaces.- 9.2 Banach Spaces.- 9.3 Hilbert Spaces.- 9.4 Function Spaces.- 9.5 Problems.- 10 The Lebesgue Integral (F. Riesz’s Approach).- 10.1 Improper Riemann Integrals.- 10.2 Step Functions and Their Integrals.- 10.3 Convergence Almost Everywhere.- 10.4 The Lebesgue Integral.- 10.5 Convergence Theorems.- 10.6 The Banach Space L1.- 10.7 Problems.- 11 Lebesgue Measure.- 11.1 Measurable Functions.- 11.2 Measurable Sets and Lebesgue Measure.- 11.3 Measurability (Lebesgue’s Definition).- 11.4 The Theorems of Egorov, Lusin, and Steinhaus.- 11.5 Regularity of Lebesgue Measure.- 11.6 Lebesgue’s Outer and Inner Measures.- 11.7 The Hilbert Spaces L2(E, % MathType!MTEF!2!1!+- % feaagaart1ev2aaatCvAUfKttLearuqr1ngBPrgarmWu51MyVXguY9 % gCGievaerbd9wDYLwzYbWexLMBbXgBcf2CPn2qVrwzqf2zLnharyav % P1wzZbItLDhis9wBH5garqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC % 0xbbL8F4rqqrFfpeea0xe9Lq-Jc9vqaqpepm0xbba9pwe9Q8fs0-yq % aqpepae9pg0FirpepeKkFr0xfr-xfr-xb9adbaqaaeGaciGaaiaabe % qaamaaeaqbaaGcbaWefv3ySLgznfgDOjdarCqr1ngBPrginfgDObcv % 39gaiyaacqWFfcVraaa!47BC! $$ \mathbb{F} $$).- 11.8 Problems.- 12 General Measure and Probability.- 12.1 Measures and Measure Spaces.- 12.2 Measurable Functions.- 12.3 Integration.- 12.4 Probability.- 12.5 Problems.- A Construction of Real Numbers.- References.

Review :
"Students who find Goffman's 'Real Functions' (1953), Halmos's 'Measure Theory' (1950), Hewitt and Stromberg's 'Real and Abstract Analysis' (1965), Lang's (1969) or Royden's 'Real Analysis' (1963), or Rudin's (1973) or Yosida's 'Functional Analysis' (1965) to be too hard, or too easy, may find Sohrab's presentation just right. Problems and exercises abound; an appendix constructs the reals as the Cauchy (sequential) completion of the rationals; references are copious and judiciously chosen; and a detailed index brings up the rear... Recommended." -CHOICE "This book is intended as a text for a one-year course for senior undergraduates or beginning graduate students, though it seems to the reviewer that it contains more than enough material for one year's study... The quality of the exposition is good: strong and complete versions of theorems are preferred, and the material is organised so that all the proofs are of easily manageable length; motivational comments are helpful, and there are plenty of illustrative examples. The reader is strongly encouraged to learn by doing: exercises are sprinkled liberally throughout the text and each chapter ends with a set of problems, about 650 in all, some of which are of considerable intrinsic interest." -MATHEMATICAL REVIEWS "The book is a clear and well structured introduction to real analysis aimed at senior undergraduate and beginning graduate students... The author managed to confine within a reasonable size book, all the basic concepts in real analysis and also some developed topics ... The text contains carefully worked out examples which contribute motivating and helping to understand the theory. There is also an excellent selection of exercises within the text and problem sections at the end of each chapter. In fact this textbook can serve as a source of examples and exercises in real analysis... This book can be highly recommended as a good reference on real analysis." -ZENTRALBLATT MATH