Introduction to Real Analysis

by William F. Trench

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This book introduces readers to a rigorous understanding of mathematical analysis and presents challenging math concepts as clearly as possible. This book is intended for those who want to gain an understanding of mathematical analysis and challenging mathematical concepts.

Description from LibreTexts.

Publisher
Trinity University
Subjects
Mathematics, Analysis
Licence
CC BY-NC-SA 3.0, as LibreTexts states it. Non-commercial use only.
Who can read
Tweens profiles and older, judged from its title and subjects

Contents

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  1. Front Matter
  2. TitlePage
  3. InfoPage
  4. Table of Contents
  5. Licensing
  6. Preface
  7. 1: The Real Numbers
  8. 1.1: The Real Number System
  9. 1.2: Mathematical Induction
  10. 1.3: The Real Line
  11. 2: Differential Calculus of Functions of One Variable
  12. 2.1: Functions and Limits
  13. 2.2: Continuity
  14. 2.3: Differentiable Functions of One Variable
  15. 2.4: L’Hospital’s Rule
  16. 2.5: Taylor’s Theorem
  17. 3: Integral Calculus of Functions of One Variable
  18. 3.1: Definition of the Integral
  19. 3.2: Existence of the Integral
  20. 3.3: Properties of the Integral
  21. 3.4: Improper Integrals
  22. 3.5: A More Advanced Look at the Existence of the Proper Riemann Integral
  23. 4: Infinite Sequences and Series
  24. 4.1: Sequences of Real Numbers
  25. 4.2: Earlier Topics Revisited With Sequences
  26. 4.3: Infinite Series of Constants
  27. 4.4: Sequences and Series of Functions
  28. 4.5: Power Series
  29. 5: Real-Valued Functions of Several Variables
  30. 5.1: Structure of Rn
  31. 5.2: Continuous Real-Valued Function of n Variables
  32. 5.3: Partial Derivatives and the Differential
  33. 5.4: The Chain Rule and Taylor’s Theorem
  34. 6: Vector-Valued Functions of Several Variables
  35. 6.1: Linear Transformations and Matrices
  36. 6.2: Continuity and Differentiability of Transformations
  37. 6.3: The Inverse Function Theorem
  38. 6.4: The Implicit Function Theorem
  39. 7: Integrals of Functions of Several Variables
  40. 7.1: Definition and Existence of the Multiple Integral
  41. 7.2: Iterated Integrals and Multiple Integrals
  42. 7.3: Change of Variables in Multiple Integrals
  43. 8: Metric Spaces
  44. 8.1: Introduction to Metric Spaces
  45. 8.2: Compact Sets in a Metric Space
  46. 8.3: Continuous Functions on Metric Spaces
  47. Back Matter
  48. Index
  49. Glossary
  50. Detailed Licensing
  51. Answers to Selected Exercises