今后一段时间，我们将介绍（科普）现代无穷小分析的基础知识，理论内容（目录）请见本文附件。
特此，敬告读者。
袁萌 陈启清 4月8日
附件：
Chapter 1. Excursus into the History of Calculus 1
§ 1.1. G. W. Leibniz and I. Newton ................................ 2
§ 1.2. L. Euler .................................. 5
§ 1.3. G. Berkeley .................................................. 5
§ 1.4. J. D’Alembert and L. Carnot ................................ 6
§ 1.5. B. Bolzano, A. Cauchy, and K. Weierstrass ................... 7
§ 1.6. N. N. Luzin .................................................. 7
§ 1.7. A. Robinson ................................................. 9
Chapter 2. Naive Foundations of In?nitesimal Analysis 10
§ 2.1. The Concept of Set in In?nitesimal Analysi.. 10
§ 2.2. Preliminaries on Standard and Nonstandard Reals . 16
§ 2.3. Basics of Calculus on the Real Axis .......................... 23
Chapter 3. Set-Theoretic Formalisms of In?nitesimal Analysis 35
§ 3.1. The Language of Set Theory ................. 37
§ 3.2. Zermelo–Fraenkel
Set Theo.. 47
§ 3.3. Nelson Internal Set Theory....... 64
§ 3.4. External Set Theories ........................................ 72
iv Contents
§ 3.5. Credenda of In?nitesimal Analysis ........................... 80
§ 3.6. Von Neumann–G¨odel–Bernays Theory ....................... 85
§ 3.7. Nonstandard Class Theory ...... 94
§ 3.8. Consistency of NCT ......................................... 101
§ 3.9. Relative Internal Set Theory ................ 106
Chapter 4. Monads in General Topology 116
§ 4.1. Monads and Filters .......................................... 116
§ 4.2. Monads and Topological Spaces .............................. 123
§ 4.3. Nearstandardness and Compactness .......................... 126
§ 4.4. In?nite Proximity in Uniform Space .......................... 129
§ 4.5. Prenearstandardness, Compactness, and Total Boundedness .. 133
§ 4.6. Relative Monads ............................................. 140
§ 4.7. Compactness and Subcontinuity ............................. 148
§ 4.8. Cyclic and Extensional Filters ............................... 151
§ 4.9. Essential and Proideal Points of Cyclic Monads .............. 156
§ 4.10. Descending Compact and Precompact Spaces ............... 159
§ 4.11. Proultra?lters and Extensional Filters ....................... 160
Chapter 5. In?nitesimals and Subdi?erentials 166
§ 5.1. Vector Topology ............................................. 166
§ 5.2. Classical Approximating and Regularizing Cones ............. 170
§ 5.3. Kuratowski and Rockafellar Limits ........................... 180
§ 5.4. Approximation Given a Set of In?nitesimals .................. 189
§ 5.5. Approximation to Composites ................................ 199
§ 5.6. In?nitesimal Subdi?erentials ................................. 204
§ 5.7. In?nitesimal Optimality ..................................... 219
Contents v
Chapter 6. Technique of Hyperapproximation 223
§ 6.1. Nonstandard Hulls ........................................... 224
§ 6.2. Discrete Approximation in Banach Space ..... 233
§ 6.3. Loeb Measure ............................................... 242
§ 6.4. Hyperapproximation of Measure Space ....................... 252
§ 6.5. Hyperapproximation of Integral Operators ................... 262
§ 6.6. Pseudointegral Operators and Random Loeb Measures ....... 272
Chapter 7. In?nitesimals in Harmonic Analysis 281
§ 7.1. Hyperapproximation of the Fourier Transform on the Reals ... 281
§ 7.2. A Nonstandard Hull of a Hyper?nite Group .................. 294
§ 7.3. The Case of a Compact Nonstandard Hull ................... 307
§ 7.4. Hyperapproximation of Locally Compact Abelian Groups .... 316
§ 7.5. Examples of Hyperapproximation ............................ 327
§ 7.6. Discrete Approximation of Function Spaces on a Locally Compact Abelian Group ......................... 340
§ 7.7. Hyperapproximation of Pseudodi?erential Operators ......... 355
Chapter 8. Exercises and Unsolved Problems 367
§ 8.1. Nonstandard Hulls and Loeb Measures ....................... 367
§ 8.2. Hyperapproximation and Spectral Theory .................... 369
§ 8.3. Combining Nonstandard Methods ............................ 371
§ 8.4. Convex Analysis and Extremal Problems ..................... 374
§ 8.5. Miscellany ...................................................376
Appendix 380
References 385
Notation Index 414
Subject Index