Introduction to the Theory of Infiniteseimals (eBook)

Front Cover 1
INTRODUCTION TO THE THEORY OF INFINITESIMALS 4
Pure and Applied Mathematics 3
Copyright Page 5
CONTENTS 8
Preface 12
Acknowledgments 16
PART 1: CLASSICAL INFINITESIMALS 18
CHAPTER 1. INTRODUCTION: WHAT ARE INFINITESIMALS? 20
CHAPTER 2. A FIRST LOOK AT ULTRAPOWERS: A MODEL OF RATIONAL ANALYSIS 24
2.1 A Free Ultrafilter u on a Countable Set J 24
2.2 An Ultrapower of the Rational Numbers 25
2.3 Some Calculus of Polynomials 27
2.4 The Exponential Function 31
2.5 Peano's Existence Theorem 33
2.6 Summary 35
CHAPTER 3. SUPERSTRUCTURES AND THEIR NONSTANDARD MODELS 37
3.1 Introduction 37
3.2 Definition of a Superstructure 40
3.3 Superstructures Are Big Enough 42
3.4 Nonstandard Models of Superstructures 42
3.5 The Formal Language 47
3.6 Interpretations of the Formal Language 52
3.7 Models of a Superstructure 53
3.8 Nonstandard Ultrapower Models 54
3.9 Bounded Formal Sentences 59
3.10 Embedding xj in Set Theory 60
3.11 *-Transforms of Categories 63
3.12 Postscript to Chapter 64
CHAPTER 4. SOME BASIC FACTS ABOUT HYPERREAL NUMBERS 66
4.1 Addition, Multiplication, and Order in 'R 66
4.2 Some Simplifications of the Notation 68
4.3 'R Is Non-Archimedean 69
4.4 Infinite, Infinitesimal, and Finite Numbers 70
4.5 Some External Entities 73
4.6 Further Simplification of Notation and Classical Functions 75
4.7 Hypercomplex Numbers 77
APPENDIX A. PRELIMINARY RESULTS ON ORDERED RINGS AND FIELDS 79
A.l Terminology 79
A.2 Ordered Rings and Fields 81
A.3 Archimedean Totally Ordered Fields 84
CHAPTER 5. FOUNDATIONS OF INFINITESIMAL CALCULUS 87
5.1 Continuity and Limits 87
5.2 Uniform Continuity 94
5.3 Basic Definitions of Calculus 95
5.4 The Mean Value Theorem 102
5.5 The Fundamental Theorem of Calculus 105
5.6 Landau's "Oh-Calculus" 107
5.7 Differential Vector Calculus 109
5.8 Integral Vector Calculus 127
5.9 Calculus on Manifolds 143
CHAPTER 6. TOPICS IN INFINITESIMAL CALCULUS 159
6.1 Peano's Existence Theorem Revisited 159
6.2 Interchanging Limits 162
6.3 Euler's Product for the Sine Function 164
6.4 Robinson's Lemma and Generalized Limits 167
6.5 Dynamical Systems 172
6.6 Geometry of the Unit Ball and Boundary Behavior 176
PART 2: INFINITESIMALS IN FUNCTIONAL ANALYSIS 190
CHAPTER 7. MORE TOOLS FROM MODEL THEORY 192
7.1 Countable Ultrapowers 192
7.2 Enlargements 193
7.3 Comprehensive Models 197
7.4 Saturated Models 197
7.5 Ultralimits 200
7.6 Properties of Polysaturated Models 204
7.7 The Isomorphism Property of Ultralimits 205
CHAPTER 8. THE GENERAL THEORY OF MONADS AND INFINITESIMALS 212
8.1 Monads with Respect to a Ring of Sets 212
8.2 Chromatic Sets 215
8.3 Topological Aspects of Monad Theory 216
8.4 Uniform Infinitesimal Relations and Finite Points 222
8.5 Topological lnfinitesimals at Remote Points 238
CHAPTER 9. COMPACTIFICATIONS 245
9.1 Discrete Cech–Stone Compactification of N 245
9.2 Measurable Infinitesimals 248
9.3 The Samuel Compactification of the Hyperbolic Plane 252
9.4 Normal Meromorphic Functions and Analytic Disks 255
9.5 The Fatou–Lindelöf Boundary 259
9.6 Bounded Holomorphic Functions and Gleason Parts 261
9.7 Fixed Points of Analytic Maps on A M 269
9.8 The Bohr Group 276
CHAPTER 10. LINEAR INFINITESIMALS AND THE LOCALLY CONVEX HULL 285
10.1 Basic Theory 285
10.2 Hilbert Spaces 302
10.3 Banach Spaces 309
10.4 Distributions 316
10.5 Mixed Spaces 323
10.6 (HM)-Spaces 329
10.7 Postscript to Chapter 10 331
References 333
Index 340