Author	Sohrab, Houshang H. author
Title	Basic Real Analysis [electronic resource] / by Houshang H. Sohrab
Imprint	Boston, MA : Birkhรคuser Boston : Imprint: Birkhรคuser, 2003
Connect to	http://dx.doi.org/10.1007/978-0-8176-8232-3
Descript	XIII, 559 p. online resource

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

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, Landauโ{128}{153}s Little โ{128}{156}ohโ{128}{157} and Big โ{128}{156}Ohโ{128}{157} -- 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 Lโ{128}{153}Hรดpitalโ{128}{153}s Rule -- 6.6 Higher Derivatives and Taylorโ{128}{153}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 Lebesgueโ{128}{153}s Integrability Criterion -- 7.4 Properties of the Riemann Integral -- 7.5 Fundamental Theorem of Calculus -- 7.6 Functions of Bounded Variation -- 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. Rieszโ{128}{153}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 (Lebesgueโ{128}{153}s Definition) -- 11.4 The Theorems of Egorov, Lusin, and Steinhaus -- 11.5 Regularity of Lebesgue Measure -- 11.6 Lebesgueโ{128}{153}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