- 实数(英文版)
- - 作者：(美)J.史迪威
  - 出版社：世界图书出版公司
  - ISBN：9787519219383
  - 出版日期：2017/01/01
  - 页数：264
- 售价：18

内容大纲
J.史迪威著的《实数（英文版）》是Undergraduate Texts in Mathematics系列丛书之一，与多数简述实数的教材不同，本书则论述了实数的方方面面，特别聚焦分析集合论，把阐述分析的精髓和介绍集合论完美的结合在在一起，书中还涉及数学史内容。
本书主要面向掌握微积分等基本数学知识的大学高年级本科生，也适用于研究生和数学工作者。
作者介绍
目录
Preface
1  The Fundamental Questions
  1.1  A Specific Question: Why Does ab = ba?
  1.2  What Are Numbers?
  1.3  What Is the Line?
  1.4  What Is Geometry?
  1.5  What Are Functions?
  1.6  What Is Continuity?
  1.7  What Is Measure?
    1.7.1  Area and Volume
  1.8  What Does Analysis Want from R?
  1.9  Historical Remarks
2  From Discrete to Continuous
  2.1  Counting and Induction
  2.2  Induction and Arithmetic
    2.2.1  Addition
    2.2.2  Multiplication
    2.2.3  The Law ab = ba Revisited
  2.3  From Rational to Real Numbers
  2.3.1  Visualizing Dedekind Cuts
  2.4  Arithmetic of Real Numbers
    2.4.1  The Square Root of 2
    2.4.2  The Equation X/2X/3 = vr6
  2.5  Order and Algebraic Properties
    2.5.1  Algebraic Properties of g
  2.6  Other Completeness Properties
  2.7  Continued Fractions
  2.8  Convergence of Continued Fractions
  2.9  Historical Remarks
    2.9.1  R as a Complete Ordered Field
3  Infinite Sets
  3.1  Countably Infinite Sets
    3.1.1  The Universal Library
  3.2  An Explicit Bijection Between lq and N2
  3.3  Sets Equinumerous with R
  3.4  The Cantor-Schrtder-Bernstein Theorem
    3.4.1  More Sets Equinumerous with R
    3.4.2  The Universal Jukebox
  3.5  The Uncountability of R
    3.5.1  The Diagonal Argument
    3.5.2  The Measure Argument
  3.6  Two Classical Theorems About Infinite Sets
  3.7  The Cantor Set
    3.7.1  Measure of the Cantor Set
  3.8  Higher Cardinalities
    3.8.1  The Continuum Hypothesis
    3.8.2  Extremely High Cardinalities
  3.9  Historical Remarks
4  Functions and Limits
  4.1  Convergence of Sequences and Series

    4.1.1  Divergent and Conditionally Convergent Series
  4.2  Limits and Continuity
  4.3  Two Properties of Continuous Functions
    4.3.1  The Devil's Staircase
  4.4  Curves
    4.4.1 - A Curve Without Tangents
    4.4.2  A Space-Filling Curve
  4.5  Homeomorphisms
  4.6  Uniform Convergence
  4.7  Uniform Continuity
  4.8  The Riemann Integral
    4.8.1  The Fundamental Theorem of Calculus
  4.9  Historical Remarks
5  Open Sets and Continuity
  5.1  Open Sets
  5.2  Continuity via Open Sets
    5.2.1  The General Concept of Open Set
  5.3  Closed Sets
  5.4  Compact Sets
  5.5  Perfect Sets
    5.5.1  Beyond Open and Closed Sets
  5.6  Open Subsets of the Irrationals
    5.6.1  Encoding Open Subsets of At" by Elements of N
  5.7  Historical Remarks
6  Ordinals
  6.1  Counting Past Infinity
  6.2  What Are Ordinals?
    6.2.1  Finite Ordinals
    6.2.2  Infinite Ordinals: Successor and Least Upper Bound
    6.2.3  Uncountable ordinals
  6.3  Well-ordering and Transfinite Induction
  6.4  The Cantor-Bendixson Theorem
  6.5  The ZF Axioms for Set Theory
  6.6  Finite Set Theory and Arithmetic
  6.7  The Rank Hierarchy
    6.7.1  Cardinality
  6.8  Large Sets
  6.9  Historical Remarks
7  The Axiom of Choice
  7.1  Some Naive Questions About Infinity
  7.2  The Full Axiom of Choice and Well-Ordering
    7.2.1  Cardinal Numbers
  7.3  The Continuum Hypothesis
  7.4  Filters and Ultrafilters
  7.5  Games and Winning Strategies
  7.6  Infinite Games
    7.6.1  Strategies
  7.7  The Countable Axiom of Choice
  7.8  Zorn's Lemma
  7.9  Historical Remarks

  7.9.1  AC, AD, and the Natural Numbers
8  Borel Sets
  8.1  Borel Sets
  8.2  Borel Sets and Continuous Functions
  8.3  Universal ]~a Sets
  8.4  The Borel Hierarchy
  8.5  Bake Functions
  8.6  The Number of Borel Sets
  8.7  Historical Remarks
9  Measure Theory
  9.1  Measure of Open Sets
  9.2  Approximation and Measure
  9.3  Lebesgue Measure
  9.4  Functions Continuous Almost Everywhere
    9.4.1  Uniform a-Continuity
  9.5  Riemann Integrable Functions
  9.6  Vitali's Nonmeasurable Set
  9.7  Ultrafilters and Nonmeasurable Sets
  9.8  Historical Remarks
10  Reflections
  10.1  What Are Numbers?
  10.2  What Is the Line?
  10.3 What Is Geometry?
  10.4  What Are Functions?
  10.5 What Is Continuity?
  10.6  What Is Measure?
  10.7  What Does Analysis Want from R?
  10.8  Further Reading
    10.8.1  Greek Mathematics
    10.8.2  The Number Concept
    10.8.3  Analysis
    10.8.4  Set Theory
    10.8.5  Axiom of Choice
References
Index

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作者介绍

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