Introduction to the Theory of Infinitesimals

Contents......Page all_14340_to_00343.cpc0007.djvu
Preface......Page all_14340_to_00343.cpc0011.djvu
Acknowledgments......Page all_14340_to_00343.cpc0015.djvu
Part 1 Classical Infinitesimals......Page all_14340_to_00343.cpc0017.djvu
1. Introduction: What Are Infinitesimals?......Page all_14340_to_00343.cpc0019.djvu
2.1 A Free Ultrafilter U on a Countable Set......Page all_14340_to_00343.cpc0023.djvu
2.2 An Ultrapower of the Rational Numbers......Page all_14340_to_00343.cpc0024.djvu
2.3 Some Calculus of Polynomials......Page all_14340_to_00343.cpc0026.djvu
2.4 The Exponential Function......Page all_14340_to_00343.cpc0030.djvu
2.5 Peano\'s Existence Theorem......Page all_14340_to_00343.cpc0032.djvu
2.6 Summary......Page all_14340_to_00343.cpc0034.djvu
3.1 Introduction......Page all_14340_to_00343.cpc0036.djvu
3.2 Definition of a Superstructure......Page all_14340_to_00343.cpc0039.djvu
3.4 Nonstandard Models of Superstructures......Page all_14340_to_00343.cpc0041.djvu
3.5 The Formal Language......Page all_14340_to_00343.cpc0046.djvu
3.6 Interpretations of the Formal Language......Page all_14340_to_00343.cpc0051.djvu
3.7 Models of a Superstructure......Page all_14340_to_00343.cpc0052.djvu
3.8 Nonstandard Ultrapower Models......Page all_14340_to_00343.cpc0053.djvu
3.9 Bounded Formal Sentences......Page all_14340_to_00343.cpc0058.djvu
3.10 Embedding X^J in Set Theory......Page all_14340_to_00343.cpc0059.djvu
3.11 *-Transforms of Categories......Page all_14340_to_00343.cpc0062.djvu
3.12 Postscript to Chapter 3......Page all_14340_to_00343.cpc0063.djvu
4.1 Addition, Multiplication, and Order in *R......Page all_14340_to_00343.cpc0065.djvu
4.2 Some Simplifications of the Notation......Page all_14340_to_00343.cpc0067.djvu
4.3 *R Is Non-Archimedean......Page all_14340_to_00343.cpc0068.djvu
4.4 Infinite, Infinitesimal, and Finite Numbers......Page all_14340_to_00343.cpc0069.djvu
4.5 Some External Entities......Page all_14340_to_00343.cpc0072.djvu
4.6 Further Simplification of Notation and Classical Functions......Page all_14340_to_00343.cpc0074.djvu
4.7 Hypercomplex Numbers......Page all_14340_to_00343.cpc0076.djvu
A.1 Terminology......Page all_14340_to_00343.cpc0078.djvu
A.2 Ordered Rings and Fields......Page all_14340_to_00343.cpc0080.djvu
A.3 Archimedean Totally Ordered Fields......Page all_14340_to_00343.cpc0083.djvu
5.1 Continuity and Limits......Page all_14340_to_00343.cpc0086.djvu
5.2 Uniform Continuity......Page all_14340_to_00343.cpc0093.djvu
5.3 Basic Definitions of Calculus......Page all_14340_to_00343.cpc0094.djvu
5.4 The Mean Value Theorem......Page all_14340_to_00343.cpc0101.djvu
5.5 The Fundamental Theorem of Calculus......Page all_14340_to_00343.cpc0104.djvu
5.6 Landau\'s \"Oh-Calculus,\"......Page all_14340_to_00343.cpc0106.djvu
5.7 Differential Vector Calculus......Page all_14340_to_00343.cpc0108.djvu
5.8 Integral Vector Calculus......Page all_14340_to_00343.cpc0126.djvu
5.9 Calculus on Manifolds......Page all_14340_to_00343.cpc0142.djvu
6.1 Peano\'s Existence Theorem Revisited......Page all_14340_to_00343.cpc0158.djvu
6.2 Interchanging Limits......Page all_14340_to_00343.cpc0161.djvu
6.3 Euler\'s Product for the Sine Function......Page all_14340_to_00343.cpc0163.djvu
6.4 Robinson\'s Lemma and Generalized Limits......Page all_14340_to_00343.cpc0166.djvu
6.5 Dynamical Systems......Page all_14340_to_00343.cpc0171.djvu
6.6 Geometry of the Unit Ball and Boundary Behavior......Page all_14340_to_00343.cpc0175.djvu
Part 2 Infinitesimals in Functional Analysis......Page all_14340_to_00343.cpc0189.djvu
7.1 Countable Ultrapowers......Page all_14340_to_00343.cpc0191.djvu
7.2 Enlargements......Page all_14340_to_00343.cpc0192.djvu
7.4 Saturated Models......Page all_14340_to_00343.cpc0196.djvu
7.5 Ultralimits......Page all_14340_to_00343.cpc0199.djvu
7.6 Properties of Polysaturated Models......Page all_14340_to_00343.cpc0203.djvu
7.7 The Isomorphism Property of Ultralimits......Page all_14340_to_00343.cpc0204.djvu
8.1 Monads with Respect to a Ring of Sets......Page all_14340_to_00343.cpc0211.djvu
8.2 Chromatic Sets......Page all_14340_to_00343.cpc0214.djvu
8.3 Topological Aspects of Monad Theory......Page all_14340_to_00343.cpc0215.djvu
8.4 Uniform Infinitesimal Relations and Finite Points......Page all_14340_to_00343.cpc0221.djvu
8.5 Topological Infinitesimals at Remote Points......Page all_14340_to_00343.cpc0237.djvu
9.1 Discrete Èech-Stone Compactification of N......Page all_14340_to_00343.cpc0244.djvu
9.2 Measurable infinitesimals......Page all_14340_to_00343.cpc0247.djvu
9.3 The Samuel Compactification of the Hyperbolic Plane......Page all_14340_to_00343.cpc0251.djvu
9.4 Normal Meromorphic Functions and Analytic Disks......Page all_14340_to_00343.cpc0254.djvu
9.5 The Fatou-Lindelöf Boundary......Page all_14340_to_00343.cpc0258.djvu
9.6 Bounded Holomorphic Functions and Gleason Parts......Page all_14340_to_00343.cpc0260.djvu
9.7 Fixed Points of Analytic Maps on M......Page all_14340_to_00343.cpc0268.djvu
9.8 The Bohr Group......Page all_14340_to_00343.cpc0275.djvu
10.1 Basic Theory......Page all_14340_to_00343.cpc0284.djvu
10.2 Hilbert Spaces......Page all_14340_to_00343.cpc0301.djvu
10.3 Banach Spaces......Page all_14340_to_00343.cpc0308.djvu
10.4 Distributions......Page all_14340_to_00343.cpc0315.djvu
10.5 Mixed Spaces......Page all_14340_to_00343.cpc0322.djvu
10.6 (HM)-Spaces......Page all_14340_to_00343.cpc0328.djvu
10.7 Postscript to Chapter 10......Page all_14340_to_00343.cpc0330.djvu
References......Page all_14340_to_00343.cpc0332.djvu
Index......Page all_14340_to_00343.cpc0339.djvu