Elementary Real Analysis
For undergraduate courses in Advanced Calculus and Real Analysis, as well as for beginning graduate students in mathematics. For a one or two semester sequence.Elementary Real Analysis is written in a reader friendly style with motivational and historical material that emphasizes the "big picture" and makes proofs seem natural rather than mysterious. It introduces key concepts such as point set theory, uniform continuity of functions and uniform convergence of sequences of functions. It is designed to prepare students for graduate work in mathematical analysis.
- Hardback | 677 pages
- 180.3 x 236.2 x 33mm | 1,247.39g
- 01 Oct 2000
- Pearson Education (US)
- United States
Table of contents
1. Properties of the Real Numbers. Introduction. The Real Number System. Algebraic Structure. Order Structure. Bounds. Sups and Infs. The Archimedean Property. Inductive Property of IN. The Rational Numbers Are Dense. The Metric Structure of R. Challenging Problems for Chapter 1.2. Sequences. Introduction. Sequences. Countable Sets. Convergence. Divergence. Boundedness Properties of Limits. Algebra of Limits. Order Properties of Limits. Monotone Convergence Criterion. Examples of Limits. Subsequences. Cauchy Convergence Criterion. Upper and Lower Limits. Challenging Problems for Chapter 2.3. Infinite Sums. Introduction. Finite Sums. Infinite Unordered Sums. Ordered Sums: Series. Criteria for Convergence. Tests for Convergence. Rearrangements. Products of Series. Summability Methods. More on Infinite Sums. Infinite Products. Challenging Problems for Chapter 3.4. Sets of Real Numbers. Introduction. Points. Sets. Elementary Topology. Compactness Arguments. Countable Sets. Challenging Problems for Chapter 4.5. Continuous Functions. Introduction to Limits. Properties of Limits. Limits Superior and Inferior. Continuity. Properties of Continuous Functions. Uniform Continuity. Extremal Properties. Darboux Property. Points of Discontinuity. Challenging Problems for Chapter 5.6. More on Continuous Functions and Sets. Introduction. Dense Sets. Nowhere Dense Sets. The Baire Category Theorem. Cantor Sets. Borel Sets. Oscillation and Continuity. Sets of Measure Zero. Challenging Problems for Chapter 6.7. Differentiation. Introduction. The Derivative. Computations of Derivatives. Continuity of the Derivative? Local Extrema. Mean Value Theorem. Monotonicity. Dini Derivatives. The Darboux Property of the Derivative. Convexity. L'Hopital's Rule. Taylor Polynomials. Challenging Problems for Chapter 7.8. The Integral. Introduction. Cauchy's First Method. Properties of the Integral. Cauchy's Second Method. Cauchy's Second Method (Continued). The Riemann Integral. Properties of the Riemann Integral. The Improper Riemann Integral. More on the Fundamental Theorem of Calculus. Challenging Problems for Chapter 8.9. Sequences and Series of Functions. Introduction. Pointwise Limits. Uniform Limits. Uniform Convergence and Continuity. Uniform Convergence and the Integral. Uniform Convergence and Derivatives. Pompeiu's Function. Continuity and Pointwise Limits. Challenging Problems for Chapter 9.10. Power Series. Introduction. Power Series: Convergence. Uniform Covergence. Functions Represented by Power Series. The Taylor Series. Products of Power Series. Composition of Power Series. Trigonometric Series.11. The Euclidean Spaces Rn. The Algebraic Structure of Rn. The Metric Structure of Rn. Elementary Topology of Rn. Sequences in Rn. Functions and Mappings. Limits of Functions from Rn to Rm. Continuity of Functions from Rn to Rm. Compact Sets in Rn. Continuous Functions on Compact Sets. Additional Remarks.12. Differentiation on Rn. Introduction. Partial and Directional Derivatives. Integrals Depending on a Parameter. Differentiable Functions. Chain Rules. Implicit Function Theorems. Functions from R to Rm. Functions from Rn to Rm.13. Metric Spaces. Introduction. Metric Spaces-Specific Examples. Convergence. Sets in a Metric Space. Functions. Separable Spaces. Complete Spaces. Contraction Maps. Applications of Contraction Maps (I). Applications of Contraction Maps (II). Compactness. Baire Category Theorem. Applications of the Baire Category Theorem. Challenging Problems for Chapter 13.Appendix A: Background. Should I Read This Chapter? Notation. What Is Analysis? Why Proofs? Indirect Proof. Contraposition. Counterexamples. Induction. Quantifiers.Appendix B: Hints for Selected Exercises. Subject Index.