Basic Real and Abstract Analysis

PrefaceChapter 1. Orientation 1.1 Real Numbers 1.2 Sets 1.3 Mathematical Induction 1.4 Logic 1.5 Completion 1.6 Dedekind Cuts 1.7 Geometry 1.8 Decimal, Ternary, and Binary Representations 1.9 Absolute Value and Inequalities 1.10 Complex NumbersChapter 2. Sets and Spaces 2.1 Equivalent Sets 2.2 Infinite Sets 2.3 Sequences of Sets 2.4 Metric Spaces 2.5 Open Sets 2.6 Compact Sets 2.7 Properties in Ek 2.8 Perfect Sets in Ek 2.9 Cardinals 2.10 Connected Sets 2.11 Distance and Relative Properties 2.12 Ek as a Vector SpaceChapter 3. Sequences and Series 3.1 The Extended Real Number System 3.2 Limits Inferior and Superior 3.3 Limit of a Real Sequence 3.4 Sequences in Metric Spaces 3.5 Sequences of Complex Numbers 3.6 Series 3.7 Series of Real Numbers 3.8 Rearranging and Grouping 3.9 Absolute and Conditional Convergence 3.10 The Space ℓ2 3.11 Double Sequences and Series 3.12 Power SeriesChapter 4. Measure and Integration 4.1 Outer Lebesgue Measure in E1 4.2 Outer Measures and Measurability 4.3 Measurable Functions 4.4 Rings and Additivity 4.5 Lebesgue Integration 4.6 Real ℓ2 Spaces 4.7 Complex ℓ2 SpacesChapter 5. Measure Theory 5.1 Metric Outer Measure 5.2 Properties of Lebesgue Measure 5.3 σ-Algebras 5.4 Lebesgue Outer k-Measure 5.5 Fubini's Theorem 5.6 Outer Ordinate Sets 5.7 Ergodic TheoryChapter 6. Continuity 6.1 Limits and Continuity of Functions 6.2 Relative Openness and Continuity 6.3 Uniformity 6.4 Weierstrass Approximation Theorem 6.5 Absolute Continuity 6.6 Equicontinuity 6.7 Semicontinuity 6.8 Discontinuities 6.9 Approximate Continuity 6.10 Continuous Linear FunctionalsChapter 7. Derivatives 7.1 Dini Derivatives 7.2 Mean Values 7.3 Trigonometry 7.4 Fourier Series 7.5 Derivatives Almost Everywhere 7.6 Bounded Variation 7.7 Derivatives and Integrals 7.8 Change of Variable 7.9 Exponents and Logarithms 7.10 Taylor's TheoremChapter 8. Stieltjes Integrals 8.1 Riemann-Stieltjes Integrals 8.2 Darboux-Stieltjes Integrals 8.3 Riemann Integrals 8.4 Integrators of Bounded Variation 8.5 Lebesgue Integral Relations 8.6 Lebesgue-Stieltjes Integrals 8.7 Lebesgue Decomposition and Radon-Nikodym TheoremsBibliographyIndex