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.

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Inhalt
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.