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Real Analysis through Modern Infinitesimals

Book Description

Real Analysis Through Modern Infinitesimals provides a course on mathematical analysis based on Internal Set Theory (IST) introduced by Edward Nelson in 1977. After motivating IST through an ultrapower construction, the book provides a careful development of this theory representing each external class as a proper class. This foundational discussion, which is presented in the first two chapters, includes an account of the basic internal and external properties of the real number system as an entity within IST. In its remaining fourteen chapters, the book explores the consequences of the perspective offered by IST as a wide range of real analysis topics are surveyed. The topics thus developed begin with those usually discussed in an advanced undergraduate analysis course and gradually move to topics that are suitable for more advanced readers. This book may be used for reference, self-study, and as a source for advanced undergraduate or graduate courses.

Table of Contents

  1. Cover
  2. Half Title
  3. Title Page
  4. Copyright
  5. Contents
  6. Preface
  7. Introduction
    1. 0.1 Infinite sets and the continuum
    2. 0.2 An analytic model of the straight line
    3. 0.3 The rise and fall of infinitesimals
    4. 0.4 The return of infinitesimals
    5. 0.5 Ultrafilters and ultrapowers
    6. 0.6 What is internal set theory?
    7. 0.7 Internal, external, and standard sets
  8. Part I: Elements of Real Analysis
    1. 1. Internal set theory
      1. 1.1 The basic language of IST
      2. 1.2 Exercises
      3. 1.3 Classes
      4. 1.4 Basic concepts and axioms of IST
      5. 1.5 Exercises
      6. 1.6 Relations and functions
      7. 1.7 Exercises
      8. 1.8 The replacement axiom
      9. 1.9 The regularity and infinity axioms
      10. 1.10 The transfer axiom
      11. 1.11 Exercises
    2. 2. The real number system
      1. 2.1 Ordered field properties of R
      2. 2.2 Integers, rationals, and irrationals
      3. 2.3 Exercises
      4. 2.4 The supremum principle
      5. 2.5 Exercises
      6. 2.6 Ordering properties of the integers
      7. 2.7 Exercises
      8. 2.8 Absolute value and intervals
      9. 2.9 Exercises
      10. 2.10 Finite and infinite sets
      11. 2.11 Exercises
      12. 2.12 Idealization axiom
      13. 2.13 Nonstandard numbers
      14. 2.14 Exercises
      15. 2.15 Standardization axiom
      16. 2.16 Exercises
      17. 2.17 Standard finite sets
      18. 2.18 Exercises
    3. 3. Sequences and series
      1. 3.1 Convergence of sequences
      2. 3.2 Exercises
      3. 3.3 Monotone sequences
      4. 3.4 Exercises
      5. 3.5 Subsequences
      6. 3.6 Exercises
      7. 3.7 Cauchy convergence criterion
      8. 3.8 Exercises
      9. 3.9 Infinite series
      10. 3.10 Exercises
      11. 3.11 Power and logarithmic functions
      12. 3.12 More on the exponential function
      13. 3.13 Exercises
    4. 4. The topology of R
      1. 4.1 Open and closed sets
      2. 4.2 Exercises
      3. 4.3 The Heine–Borel theorem
      4. 4.4 Exercises
    5. 5. Limits and continuity
      1. 5.1 Limit of a function at a point
      2. 5.2 Exercises
      3. 5.3 One-sided limits and infinite limits
      4. 5.4 Exercises
      5. 5.5 Continuous functions
      6. 5.6 Exercises
      7. 5.7 Properties of continuous functions
      8. 5.8 Exercises
      9. 5.9 Uniform continuity
      10. 5.10 Exercises
      11. 5.11 Monotone functions
      12. 5.12 Oscillation of a function
      13. 5.13 Exercises
    6. 6. Differentiation
      1. 6.1 Definitions and basic properties
      2. 6.2 Exercises
      3. 6.3 Differentiability and local properties
      4. 6.4 Exercises
      5. 6.5 The mean value theorem
      6. 6.6 Exercises
      7. 6.7 L’Hospital’s rule
      8. 6.8 Exercises
      9. 6.9 Higher-order derivatives
      10. 6.10 Taylor polynomials
      11. 6.11 Exercises
      12. 6.12 Convex functions
      13. 6.13 Exercises
      14. 6.14 Darboux’s theorem
      15. 6.15 Exercises
    7. 7. Integration
      1. 7.1 Definition of the Riemann integral
      2. 7.2 Basic properties of the Riemann integral
      3. 7.3 Exercises
      4. 7.4 The Lebesgue–Riemann theorem
      5. 7.5 Exercises
      6. 7.6 The fundamental theorem of calculus
      7. 7.7 Exercises
      8. 7.8 Improper Riemann integration
      9. 7.9 Exercises
    8. 8. Sequences and series of functions
      1. 8.1 Pointwise and uniform convergence
      2. 8.2 Exercises
      3. 8.3 Applications of uniform convergence
      4. 8.4 Exercises
    9. 9. Infinite series
      1. 9.1 Upper and lower limits of sequences
      2. 9.2 Exercises
      3. 9.3 Absolute and conditional convergence
      4. 9.4 Exercises
      5. 9.5 Power series
      6. 9.6 Functions defined by power series
      7. 9.7 Exercises
  9. Part II: Elements of Abstract Analysis
    1. 10. Point set topology
      1. 10.1 Topological spaces
      2. 10.2 Monads in topological spaces
      3. 10.3 Exercises
      4. 10.4 Continuous functions
      5. 10.5 Exercises
      6. 10.6 Compactness
      7. 10.7 Local compactness
      8. 10.8 Connectedness
      9. 10.9 Monads of filters
      10. 10.10 Exercises
    2. 11. Metric spaces
      1. 11.1 The metric topology
      2. 11.2 Normed vector spaces
      3. 11.3 Metric space properties of Rn
      4. 11.4 Exercises
      5. 11.5 Standard hulls of classes and functions
      6. 11.6 Exercises
      7. 11.7 The Peano existence theorem
    3. 12. Complete metric spaces
      1. 12.1 Completeness
      2. 12.2 Total boundedness
      3. 12.3 Compactness in metric spaces
      4. 12.4 Uniform and Lipschitz continuity
      5. 12.5 Products of metric spaces
      6. 12.6 The completion of a metric space
      7. 12.7 Exercises
    4. 13. Some applications of completeness
      1. 13.1 Baire category theorem
      2. 13.2 Two extension theorems
      3. 13.3 Banach fixed point theorem
    5. 14. Linear operators
      1. 14.1 The p-norm of a linear operator
      2. 14.2 The operator norm
      3. 14.3 Invertible operators
      4. 14.4 Integral operators
      5. 14.5 Exercises
    6. 15. Differential calculus on Rn
      1. 15.1 First-order differentials
      2. 15.2 Directional and partial derivatives
      3. 15.3 Exercises
      4. 15.4 Higher-order differentials
      5. 15.5 Inverse and implicit function theorems
      6. 15.6 Exercises
    7. 16. Function space topologies
      1. 16.1 The K-convergence topology
      2. 16.2 Metrization of K-convergence topology
      3. 16.3 The K-open topology
      4. 16.4 The Ascoli theorem
      5. 16.5 The Stone–Weierstrass theorem
      6. 16.6 Exercises
  10. Appendix A Vector spaces
  11. Appendix B The b-adic representation of numbers
    1. B.1 b-adic representation of integers
    2. B.2 b-adic representation of real numbers
    3. B.3 Exercises
    4. B.4 Some examples of proofs by induction
    5. B.5 Existence of roots
  12. Appendix C Finite, denumerable, and uncountable sets
    1. C.1 Finite sets
    2. C.2 Exercises
    3. C.3 Denumerable sets
    4. C.4 Uncountable sets
    5. C.5 Exercises
  13. Appendix D The syntax of mathematical languages
    1. D.1 Constituents of mathematical statements
    2. D.2 Logical and non-logical symbols
    3. D.3 Terms and formulas
    4. D.4 Exercises
  14. References
  15. Index