近日，国教教育部教学指导委员会表态，不反对非标准分析，只设立教学内容“底线”，不设立内容“上线”（自主办学）。
据此， 哥德布林非标准分析可供数学分析教学参考资料。
请见本文附件。
袁萌 陈启清 6月13日
附件：
NOTES ON NONSTANDARD ANALYSIS UCLA SUMMER SCHOOL IN LOGIC ISAAC GOLDBRING Contents 1. The hyperreals 3
1.1. Basic facts about the ordered real field 3
1.2. The nonstandard extension 4
1.3. Arithmetic in the hyperreals 5
1.4. The structure of N ∗ 7
1.5. More practice with transfer 8
1.6. Problems 9
2. Logical formalisms for nonstandard extensions 10
2.1. Approach 1: The compactness theorem 11
2.2. Approach 2: The ultrapower construction 12 2.3. Problems 16
3. Sequences and series 17 3.1. First results about sequences 17
3.2. Cluster points 19 3.3. Series 21 3.4. Problems 22
4. Continuity 23
4.1. First results about continuity 23
4.2. Uniform continuity 25
4.3. Sequences of functions 27
4.4. Problems 30
5. Differentiation 33 5.1. The derivative 33
5.2. Continuous differentiability 35 5.3. Problems 36
6. Riemann Integration 38
6.1. Hyperfinite Riemann sums and integrability 38 6.2. The Peano Existence Theorem 41
6.3. Problems 43
7. Weekend Problem Set #1 44
8. Many-sorted and Higher-Type Structures 47
8.1. Many-sorted structures 47 Date: November 10, 2014. 1 2 ISAAC GOLDBRING 8.2. Higher-type sorts 48
8.3. Saturation 51
8.4. Useful nonstandard principles 53 8.5. Recap: the nonstandard setting 54
8.6. Problems 54
9. Metric Space Topology 55
9.1. Open and closed sets, compactness, completeness 55 9.2. More about continuity 63
9.3. Compact maps 64 9.4. Problems 65
10. Banach Spaces 67
10.1. Normed spaces 67
10.2. Bounded linear maps 68 10.3. Finite-dimensional spaces and compact linear maps 69 10.4. Problems 71
11. Hilbert Spaces 73
11.1. Inner product spaces 73
11.2. Orthonormal bases and ` 2 75
11.3. Orthogonal projections 79 11.4. Hyperfinite-dimensional subspaces 82
11.5. Problems 83
12. Weekend Problem Set #2 85
13. The Spectral Theorem for compact hermitian operators 88
13.1. Problems 93
14. The Bernstein-Robinson Theorem 94
15. Measure Theory 101
15.1. General measure theory 101
15.2. Loeb measure 102
15.3. Product measure 103
15.4. Integration 104 15.5. Conditional expectation 104 15.6. Problems 105 16. Szemer´edi Regularity Lemma 106
16.1. Problems 108 References 110