## Infinitesimal AnalysisInfinitesimal analysis, once a synonym for calculus, is now viewed as a technique for studying the properties of an arbitrary mathematical object by discriminating between its standard and nonstandard constituents. Resurrected by A. Robinson in the early 1960's with the epithet 'nonstandard', infinitesimal analysis not only has revived the methods of infinitely small and infinitely large quantities, which go back to the very beginning of calculus, but also has suggested many powerful tools for research in every branch of modern mathematics. The book sets forth the basics of the theory, as well as the most recent applications in, for example, functional analysis, optimization, and harmonic analysis. The concentric style of exposition enables this work to serve as an elementary introduction to one of the most promising mathematical technologies, while revealing up-to-date methods of monadology and hyperapproximation. This is a companion volume to the earlier works on nonstandard methods of analysis by A.G. Kusraev and S.S. Kutateladze (1999), ISBN 0-7923-5921-6 and Nonstandard Analysis and Vector Lattices edited by S.S. Kutateladze (2000), ISBN 0-7923-6619-0 |

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### Contents

Excursus into the History of Calculus | 1 |

11 G W Leibniz and I Newton | 2 |

12 L Euler | 5 |

14 J DAlembert and L Carnot | 6 |

15 B Bolzano A Cauchy and K Weierstrass | 7 |

17 A Robinson | 9 |

Naive Foundations of Infinitesimal Analysis | 10 |

22 Preliminaries on Standard and Nonstandard Reals | 16 |

Infinitesimals and Subdifferentials | 166 |

52 Classical Approximating and Regularizing Cones | 170 |

53 Kuratowski and Rockafellar Limits | 180 |

54 Approximation Given a Set of Infinitesimals | 189 |

55 Approximation to Composites | 199 |

56 Infinitesimal Subdifferentials | 204 |

57 Infinitesimal Optimality | 219 |

Technique of Hyperapproximation | 223 |

23 Basics of Calculus on the Real Axis | 23 |

SetTheoretic Formalisms of Infinitesimal Analysis | 35 |

31 The Language of Set Theory | 37 |

32 ZermeloFraenkel Set Theory | 47 |

33 Nelson Internal Set Theory | 64 |

34 External Set Theories | 72 |

35 Credenda of Infinitesimal Analysis | 80 |

36 Von NeumannGodelBernays Theory | 85 |

37 Nonstandard Class Theory | 94 |

38 Consistency of NCT | 101 |

39 Relative Internal Set Theory | 106 |

Monads in General Topology | 116 |

42 Monads and Topological Spaces | 123 |

Nearstandardness and Compactness | 126 |

44 Infinite Proximity in Uniform Space | 129 |

45 Prenearstandardness Compactness and Total Boundedness | 133 |

46 Relative Monads | 140 |

47 Compactness and Subcontinuity | 148 |

48 Cyclic and Extensional Filters | 151 |

49 Essential and Proideal Points of Cyclic Monads | 156 |

410 Descending Compact and Precompact Spaces | 159 |

411 Proultrafilters and Extensional Filters | 160 |

61 Nonstandard Hulls | 224 |

62 Discrete Approximation in Banach Space | 233 |

63 Loeb Measure | 242 |

64 Hyperapproximation of Measure Space | 252 |

65 Hyperapproximation of Integral Operators | 262 |

Infinitesimals in Harmonic Analysis | 281 |

72 A Nonstandard Hull of a Hyperfinite Group | 294 |

73 The Case of a Compact Nonstandard Hull | 307 |

74 Hyperapproximation of Locally Compact Abelian Groups | 316 |

75 Examples of Hyperapproximation | 327 |

76 Discrete Approximation of Function Spaces on a Locally Compact Abelian Group | 340 |

77 Hyperapproximation of Pseudodifferential Operators | 355 |

Exercises and Unsolved Problems | 367 |

82 Hyperapproximation and Spectral Theory | 369 |

83 Combining Nonstandard Methods | 371 |

84 Convex Analysis and Extremal Problems | 374 |

85 Miscellany | 376 |

Appendix | 380 |

385 | |

Notation Index | 414 |

417 | |