Basic Real Analysis

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!+-
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