标题：超实讲义
图书信息
出版社: 世界图书出版公司; 第1版 (2011年4月1日)
外文书名: Lectures on the Hyperreals:An Introduction to Nonstandard Analysis
平装: 289页
正文语种: 英语
开本: 24
ISBN: 9787510032981
条形码: 9787510032981
尺寸: 22.2 x 15.2 x 1.4 cm
重量: 399 g
作者简介
作者：（新西兰）哥德布拉特（Robert Goldblatt）
内容简介
《超实讲义》是一部讲述非标准分析的入门教程，是由作者数年教学讲义发展并扩充而成。具备基本分析知识的高年级本科生，研究生以及自学人员都可以完全读懂。非标准分析理论不仅是研究无限大和无限小的强有力的理论，也是一种截然不同于标准数学概念和结构的方法，更是新的结构，目标和证明的源泉，推理原理的新起点。书中是从超实数系统开始，从非标准的角度讲述单变量积分，分析和拓扑，着重强调变换原理作为一个重要的数学工具的重要作用。数学宇宙的讲述为全面研究非标准方法论提供了基础保证。最后一章着眼于应用，将这些理论应用于loeb 测度理论及其与lebesgue 的一些关系，ramsey 定理，p-进数的非标准结构和幂级数，boolean 代数的stone 表示定理的非标准证明和hahn-banach 定理。《超实讲义：英文(影印版)》的最大特点尽早引入内集，外集，超有限集，以及集理论扩展方法，较常规的建立在超结构基础上，这样的方式更加显而易见。
读者对象：数学专业的高年级本科生，研究生和科研人员。
目录
i foundations 
1 what are the hyperreals? 
1.1 infinitely small and large 
1.2 historical background 
1.3 what is a real number? 
1.4 historical references 
2 large sets 
2.1 infinitesimals as variable quantities 
2.2 largeness 
2.3 filters 
2.4 examples of filters 
2.5 facts about filters 
2.6 zorn's lemma 
2.7 exercises on filters 
3 ultrapower construction of the hyperreals 
3.1 the ring of real-valued sequences 
3.2 equivalence modulo an ultrafilter 
3.3 exercises on almost-everywhere agreement 
3.4 a suggestive logical notation 
3.5 exercises on statement values 
3.6 the ultrapower 
3.7 including the reals in the hyperreals 
3.8 infinitesimals and unlimited numbers 
3.9 enlarging sets 
3.10 exercises on enlargement 
3.11 extending functions 
3.12 exercises on extensions 
3.13 partial functions and hypersequences 
3.14 enlarging relations 
3.15 exercises on enlarged relations 
3.16 is the hyperreal system unique? 
4 the transfer principle 
4.1 transforming statements 
4.2 relational structures 
4.3 the language of a relational structure 
4.4 *-transforms 
4.5 the transfer principle 
4.6 justifying transfer 
4.7 extending transfer 
5 hyperreals great and small 
5.1 (un)limited, infinitesimal, and appreciable numbers 
5.2 arithmetic of hyperreals 
5.3 on the use of "finite" and "infinite" 
5.4 halos, galaxies, and real comparisons 
5.5 exercises on halos and galaxies 
5.6 shadows 
5.7 exercises on infinite closeness 
5.8 shadows and completeness 
5.9 exercise on dedekind completeness 
5.10 the hypernaturals 
5.11 exercises on hyperintegers and primes 
5.12 on the existence of infinitely many primes 
ii basic analysis 
6 convergence of sequences and series 
6.1 convergence 
6.2 monotone convergence 
6.3 limits 
6.4 boundedness and divergence 
6.5 cauchy sequences 
6.6 cluster points 
6.7 exercises on limits and cluster points 
6.8 limits superior and inferior 
6.9 exercises on lim sup and lim inf 
6.10 series 
6.11 exercises on convergence of series 
7 continuous functions 
7.1 cauchy's account of continuity 
7.2 continuity of the sine function 
7.3 limits of functions 
7.4 exercises on limits 
7.5 the intermediate value theorem 
7.6 the extreme value theorem 
7.7 uniform continuity 
7.8 exercises on uniform continuity 
7.9 contraction mappings and fixed points 
7.10 a first look at permanence 
7.11 exercises on permanence of functions 
7.12 sequences of functions 
7.13 continuity of a uniform limit 
7.14 continuity in the extended hypersequence 
7.15 was cauchy right? 
8 differentiation 
8.1 the derivative 
8.2 increments and differentials 
8.3 rules for derivatives 
8.4 chain rule 
8.5 critical point theorem 
8.6 inverse function theorem 
8.7 partial derivatives 
8.8 exercises on partial derivatives 
8.9 taylor series 
8.10 incremental approximation by taylor's formula 
8.11 extending the incremental equation 
8.12 exercises on increments and derivatives 
9 the riemann integral 
9.1 riemann sums 
9.2 the integral as the shadow of riemann sums 
9.3 standard properties of the integral 
9.4 differentiating the area function 
9.5 exercise on average function values 
10 topology of the reals 
10.1 interior, closure, and limit points 
10.2 open and closed sets 
10.3 compactness 
10.4 compactness and (uniform) continuity 
10.5 topologies on the hyperreals 
iii internal and external entities 
11 internal and external sets 
11.1 internal sets 
11.2 algebra of internal sets 
11.3 internal least number principle and induction 
11.4 the overflow principle 
11.5 internal order-completeness 
11.6 external sets 
11.7 defining internal sets 
11.8 the underflow principle 
11.9 internal sets and permanence 
11.10 saturation of internal sets 
11.11 saturation creates nonstandard entities 
11.12 the size of an internal set 
11.13 closure of the shadow of an internal set 
11.14 interval topology and hyper-open sets 
12 internal functions and hyperfinite sets 
12.1 internal functions 
12.2 exercises on properties of internal functions 
12.3 hyperfinite sets 
12.4 exercises on hyperfiniteness 
12.5 counting a hyperfinite set 
12.6 hyperfinite pigeonhole principle 
12.7 integrals as hyperflnite sums 
iv nonstandard frameworks 
13 universes and frameworks 
13.1 what do we need in the mathematical world? 
13.2 pairs are enough 
13.3 actually, sets are enough 
13.4 strong transitivity 
13.5 universes 
13.6 superstructures 
13.7 the language of a universe 
13.8 nonstandard frameworks 
13.9 standard entities 
13.10 internal entities 
13.11 closure properties of internal sets 
13.12 transformed power sets 
13.13 exercises on internal sets and functions 
13.14 external images are external 
13.15 internal set definition principle 
13.16 internal function definition principle 
13.17 hyperfiniteness 
13.18 exercises on hyperfinite sets and sizes 
13.19 hyperfinite summation 
13.20 exercises on hyperfinite sums 
14 the existence of nonstandard entities 
14.1 enlargements 
14.2 concurrence and hyperfinite approximation 
14.3 enlargements as ultrapowers 
14.4 exercises on the ultrapower construction 
15 permanence, comprehensiveness, saturation 
15.1 permanence principles 
15.2 robinson's sequential lemma 
15.3 uniformly converging sequences of functions 
15.4 comprehensiveness 
15.5 saturation 
v applications 
16 loeb measure 
16.1 rings and algebras 
16.2 measures 
16.3 outer measures 
16.4 lebesgue measure 
16.5 loeb measures 
16.6 μ-approximability 
16.7 loeb measure as approximability 
16.8 lebesgue measure via loeb measure 
17 ramsey theory 
17.1 colourings and monochromatic sets 
17.2 a nonstandard approach 
17.3 proving p, amsey's theorem 
17.4 the finite ramsey theorem 
17.5 the paris-harrington version 
17.6 reference 
18 completion by enlargement 
18.1 completing the rationals 
18.2 metric space completion 
18.3 nonstandard hulls 
18.4 p-adic integers 
18.5 p-adic numbers 
18.6 power series 
18.7 hyperfinite expansions in base p 
18.8 exercises 
19 hyperfinite approximation 
19.1 colourings and graphs 
19.2 boolean algebras 
19.3 atomic algebras 
19.4 hyperfinite approximating algebras 
19.5 exercises on generation of algebras 
19.6 connecting with the stone representation 
19.7 exercises on filters and lattices 
19.8 hyperfinite-dimensional vector spaces 
19.9 exercises on (hyper) real suhspaces 
19.10 the hahn-banach theorem 
19.11 exercises on (hyper) linear functionals 
20 books on nonstandard analysis 
index