超實講義

《超實講義》是2011年4月1日世界圖書出版公司出版的圖書，作者是（紐西蘭）哥德布拉特（Robert Goldblatt）

基本介紹

• ISBN：9787510032981
• 頁數：289頁
• 裝幀：平裝
• 開本：24

圖書信息

出版社: 世界圖書出版公司; 第1版 (2011年4月1日)

外文書名: Lectures on the Hyperreals:An Introduction to Nonstandard Analysis

平裝: 289頁

正文語種: 英語

開本: 24

ISBN: 9787510032981

條形碼: 9787510032981

尺寸: 22.2 x 15.2 x 1.4 cm

重量: 399 g

作者簡介

作者：（紐西蘭）哥德布拉特（Robert Goldblatt）

內容簡介

《超實講義》是一部講述非標準分析的入門教程，是由作者數年教學講義發展並擴充而成。具備基本分析知識的高年級本科生，研究生以及自學人員都可以完全讀懂。非標準分析理論不僅是研究無限大和無限小的強有力的理論，也是一種截然不同於標準數學概念和結構的方法，更是新的結構，目標和證明的源泉，推理原理的新起點。書中是從超實數系統開始，從非標準的角度講述單變數積分，分析和拓撲，著重強調變換原理作為一個重要的數學工具的重要作用。數學宇宙的講述為全面研究非標準方法論提供了基礎保證。最後一章著眼於套用，將這些理論套用於loeb 測度理論及其與lebesgue 的一些關係，ramsey 定理，p-進數的非標準結構和冪級數，boolean 代數的stone 表示定理的非標準證明和hahn-banach 定理。《超實講義：英文(影印版)》的最大特點儘早引入內集，外集，超有限集，以及集理論擴展方法，較常規的建立在超結構基礎上，這樣的方式更加顯而易見。

讀者對象：數學專業的高年級本科生，研究生和科研人員。

目錄

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1.4 historical references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