Infinitesimal Analysis

Cover
Title page
Foreword
Chapter 1. Excursus into the History of Calculus
1.1. G. W. Leibniz and I. Newton
1.2. L . Euler
1.3. G . Berkeley
1.4. J. D'Alembert and L. Carnot
1.5. B. Bolzano, A. Cauchy, and K. Weierstrass
1.6. N. N. Luzin
1.7. A. Robinson
Chapter 2. Naive Foundations of Infinitesimal Analysis
2.1. The Concept of Set in Infinitesimal Analysis
2.2. Preliminaries on Standard and Nonstandard Reals
2.3. Basics of Calculus on the Real Axis
Chapter 3. Set-Theoretic Formalisms of Infinitesimal Analysis
3.1. The Language of Set Theory
3.2. Zermelo-Fraenkel Set Theory
3.3. Nelson InternaI Set Theory
3.4. External Set Theories
3.5. Credenda of Infinitesimal Analysis
3.6. Von Neumann-Gödel-Bernays Theory
3. 7. Nonstandard Class Theory
3.8. Consistency of NCT
3.9. Relative Internal Set Theory
Chapter 4. Monads in General Topology
4.1. Monads and Filters
4.2. Monads and Topological Spaces
4.3. Nearstandardness and Compactness
4.4. Infinite Proximity in Uniform Space
4.5. Prenearstandardness, Compactness, and Total Boundedness
4.6. Relative Monads
4.7. Compactness and Subcontinuity
4.8. Cyclic and Extensional Filters
4.9. Essential and Proideal Points of Cyclic Monads
4.10. Descending Compact and Precompact Spaces
4.11. Proultrafilters and Extensional Filters
Chapter 5. Infinitesimals and Subdifferentials
5.1. Vector Topology
5.2. Classical Approximating and Regularizing Cones
5.3. Kuratowski and Rockafellar Limits
5.4. Approximation Given a Set of Infinitesimals
5.5. Approximation to Composites
5.6. Infinitesimal Subdifferentials
5.7. Infinitesimal Optimality
Chapter 6. Technique of Hyperapproximation
6.1. Nonstandard Hulls
6.2. Discrete Approximation in Banach Space
6.3. Loeb Measure
6.4. Hyperapproximation of Measure Space
6.5. Hyperapproximation of Integral Operators
6.6. Pseudointegral Operators and Random Loeb Measures
Chapter 7. Infinitesimals in Harmonic Analysis
7.1. Hyperapproximation of the Fourier Transform on the Reals
7.2. A Nonstandard Hull of a Hyperfinite Group
7.3. The Case of a Compact Nonstandard Hull
7.4. Hyperapproximation of Locally Compact Abelian Groups
7.5. Examples of Hyperapproximation
7.6. Discrete Approximation of Function Spaces on a Locally Compact Abelian Group
7.7. Hyperapproximation of Pseudodifferential Operators
Chapter 8. Exercises and U nsolved Problems
8.1. N onstandard Huns and Loeb Measures
8.2. Hyperapproximation and Spectral Theory
8.3. Combining N onstandard Methods
8.4. Convex Analysis and Extremal Problems
8.5. Miscellany
Appendix
References
Notation Index
Subject Index