关于现代数学的前沿课题
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LECTURE NOTES ON NONSTANDARD ANALYSIS UCLA SUMMER SCHOOL IN LOGIC
ISAAC GOLDBRING
Contents
1. The hyperreals 3 1.1. Basic facts about the ordered real eld 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. Dierentiation 33
5.1. The derivative 33
5.2. Continuous dierentiability 35
5.3. Problems 36
6. Riemann Integration 38
6.1. Hypernite 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.
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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. Hypernite-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
Nonstandard analysis was invented by Abraham Robinson in the 1960s as a way to rescue the na¨ve use of innitesimal and innite elements favored by mathematicians such as Leibniz and Euler before the advent of the rigorous methods introduced by Cauchy and Weierstrauss. Indeed, Robinson realized that the compactness theorem of rst-order logic could be used to provide elds that “logically behaved” like the ordered real eld while containing “ideal” elements such as innitesimal and innite elements.
LECTURE NOTES ON NONSTANDARD ANALYSIS 3
Since its origins,
nonstandard analysis has become a powerful mathematical tool, not
only for yielding easier denitions for standard concepts and
providing slick proofs of well-known mathematical theorems, but for
also providing mathematicians with amazing new tools to prove
theorems, e.g. hypernite approximation. In addition, by providing
useful mathematical heuristics a precise language to be discussed,
many mathematical ideas have been elucidated greatly. In these
notes, we try and cover a wide spectrum of applications of
nonstandard methods. In the rst part of these notes, we explain
what a nonstandard extension is and we use it to reprove some basic
facts from calculus. We then broaden our nonstandard framework to
handle more sophisticated mathematical situations and begin
studying metric space topology. We then enter functional analysis
by discussing Banach and Hilbert spaces. Here we prove our rst
serious theorems: the Spectral Theorem for compact hermitian
operators and the Bernstein-Robinson Theorem on invariant
subspaces; this latter theorem was the rst major theorem whose
rst proof was nonstandard. We then end by briey discussing Loeb
measure and using it to give a slick proof of an important
combinatorial result, the Szemer´edi Regularity Lemma. Due to time
limitations, there are many beautiful subjects I had to skip. In
particular, I had to omit the nonstandard hull construction
(although this is briey introduced in the second weekend problem
set) as well as applications of nonstandard analysis to Lie theory
(e.g. Hilbert’s fth problem), geometric group theory (e.g.
asymptotic cones), and commutative algebra (e.g. bounds in the
theory of polynomial rings).
Wehaveborrowedmuchofourp
1. The hyperreals 1.1. Basic facts about the ordered real eld. The ordered eld of real numbers is the structure (R;+,·,0,1,<). We recall some basic properties:… …(全文很长,请见“无穷小微积分”网站,di点击“NSA-Notes”下载)