Elements of Real Analysis

Autorentext

David A. Sprecher

Klappentext

This classic text in introductory analysis delineates and explores the intermediate steps between the basics of calculus and the ultimate stage of mathematics: abstraction and generalization.
Since many abstractions and generalizations originate with the real line, the author has made it the unifying theme of the text, constructing the real number system from the point of view of a Cauchy sequence (a step which Dr. Sprecher feels is essential to learn what the real number system is).
The material covered in Elements of Real Analysis should be accessible to those who have completed a course in calculus. To help give students a sound footing, Part One of the text reviews the fundamental concepts of sets and functions and the rational numbers. Part Two explores the real line in terms of the real number system, sequences and series of number and the structure of point sets. Part Three examines the functions of a real variable in terms of continuity, differentiability, spaces of continuous functions, measure and integration, and the Fourier series.
An especially valuable feature of the book is the exercises which follow each section. There are over five hundred, ranging from the simple to the highly difficult, each focusing on a concept previously introduced.

Inhalt

Preface
Part I. Fundamental Concepts

1. Sets and Functions
2. Sets
3. The Algebra of Sets
4. Functions
5. Countability
6. The Rational Numbers
7. Alebraic Properties
8. Decimal Expansions
   Part II. The Real Line
9. The Real Number System
10. Cauchy Sequences and Their Equivalence Classes
11. The Real Number System
12. Completeness Properties of R
13. The Extended Real Line
14. Sequences and Series of Number
15. Sequences: Basic Limit Theorems
16. Upper and Lower Limits
17. Basic Properties of Series
18. Series with Nonnegative Terms
19. Alternating Series
20. Absolute Convergence
21. The Structure of Point Sets
22. Basic Notions
23. Closed Sets
24. Open Sets
25. Perfect Sets
26. Distance between Point Sets
27. Connected Sets
28. Functions of a Real Variable
29. Continuity; Limits of Functions
30. Continuous Functions
31. The Nature of Discontinuities
32. Monotonic Functions
33. Uniform Continuity
34. Differentiability
    28 The Derivative at a Point
35. A Continuous Nowhere Differentiable Function
36. Properties of the Derivative
37. Taylor's Theorem
38. Spaces of Continuous Functions
39. The Problems of Separability and Convergence
40. Uniform Convergence
41. Power Series
42. The Approximation of Functions
43. Equicontinuity
44. Summary
45. Measure and Integration
46. Measurable Sets
47. Properties of Measurable Sets
48. Measurable Functions
49. The Lebesgue Integral of Simple Functions
50. The Lebesgue Integral
51. Theorems on Limits under the Integral Sign
52. The Riemann Integral
53. Fourier Series
54. Basic Facts
55. The Space £ superscript 2
56. The Question of Convergence
    Bibliography; Glossary of Symbols; Index