Basic Real and Abstract Analysis

1st Edition - January 1, 1968

Author: John F. Randolph

eBook ISBN:

9 7 8 - 1 - 4 8 3 2 - 7 2 7 5 - 7

Basic Real and Abstract Analysis focuses on the processes, methodologies, and approaches involved in the process of abstraction of mathematical problems. The book first offers… Read more

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Basic Real and Abstract Analysis focuses on the processes, methodologies, and approaches involved in the process of abstraction of mathematical problems. The book first offers information on orientation and sets and spaces, including equivalent and infinite sets, metric spaces, cardinals, distance and relative properties, real numbers, and absolute value and inequalities. The text then takes a look at sequences and series and measure and integration. Topics include rings and additivity, Lebesgue integration, outer measures and measurability, extended real number system, sequences in metric spaces, and series of real numbers. The publication ponders on measure theory, continuity, derivatives, and Stieltjes integrals. Discussions focus on integrators of bounded variation, Lebesgue integral relations, exponents and logarithms, bounded variation, mean values, trigonometry, and Fourier series. The manuscript is a valuable reference for mathematicians and researchers interested in the process of abstraction of mathematical equations.

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