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Elementary Real Analysis

Elementary Real Analysis - 01 edition

ISBN13: 978-0130190758

Cover of Elementary Real Analysis 01 (ISBN 978-0130190758)
ISBN13: 978-0130190758
ISBN10: 0130190756
Cover type:
Edition/Copyright: 01
Publisher: Prentice Hall, Inc.
Published: 2001
International: No

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Elementary Real Analysis - 01 edition

ISBN13: 978-0130190758

Brien S. Thomson, Judith B. Bruckner and Andrew M. Bruckner

ISBN13: 978-0130190758
ISBN10: 0130190756
Cover type:
Edition/Copyright: 01
Publisher: Prentice Hall, Inc.

Published: 2001
International: No
Summary

For undergraduate courses in Advanced Calculus and Real Analysis, as well as for beginning graduate students in mathematics. For a one or two semester sequence.

Elementary Real Analysis is written in a reader friendly style with motivational and historical material that emphasizes the ''big picture'' and makes proofs seem natural rather than mysterious. It introduces key concepts such as point set theory, uniform continuity of functions and uniform convergence of sequences of functions. It is designed to prepare students for graduate work in mathematical analysis.

Author Bio

Thomson, Brian S. : Simon Frasier University

Bruckner, Andrew M. : University of California-Santa Barbara

Table of Contents

1. Properties of the Real Numbers

Introduction
The Real Number System
Algebraic Structure
Order Structure
Bounds
Sups and Infs
The Archimedean Property
Inductive Property of IN
The Rational Numbers Are Dense
The Metric Structure of R
Challenging Problems for Chapter 1


2. Sequences

Introduction
Sequences
Countable Sets
Convergence
Divergence
Boundedness Properties of Limits
Algebra of Limits
Order Properties of Limits
Monotone Convergence Criterion
Examples of Limits
Subsequences
Cauchy Convergence Criterion
Upper and Lower Limits
Challenging Problems for Chapter 2


3. Infinite Sums

Introduction
Finite Sums
Infinite Unordered Sums
Ordered Sums: Series
Criteria for Convergence
Tests for Convergence
Rearrangements
Products of Series
Summability Methods
More on Infinite Sums
Infinite Products
Challenging Problems for Chapter 3


4. Sets of Real Numbers

Introduction
Points
Sets
Elementary Topology
Compactness Arguments
Countable Sets
Challenging Problems for Chapter 4


5. Continuous Functions

Introduction to Limits
Properties of Limits
Limits Superior and Inferior
Continuity
Properties of Continuous Functions
Uniform Continuity
Extremal Properties
Darboux Property
Points of Discontinuity
Challenging Problems for Chapter 5


6. More on Continuous Functions and Sets

Introduction
Dense Sets
Nowhere Dense Sets
The Baire Category Theorem
Cantor Sets
Borel Sets
Oscillation and Continuity
Sets of Measure Zero
Challenging Problems for Chapter 6


7. Differentiation

Introduction
The Derivative
Computations of Derivatives
Continuity of the Derivative?
Local Extrema
Mean Value Theorem
Monotonicity Dini Derivatives
The Darboux Property of the Derivative
Convexity
L'Hopital's Rule
Taylor Polynomials
Challenging Problems for Chapter 7


8. The Integral

Introduction
Cauchy's First Method
Properties of the Integral
Cauchy's Second Method
Cauchy's Second Method (Continued)
The Riemann Integral
Properties of the Riemann Integral
The Improper Riemann Integral
More on the Fundamental Theorem of Calculus
Challenging Problems for Chapter 8


9. Sequences and Series of Functions

Introduction
Pointwise Limits
Uniform Limits
Uniform Convergence and Continuity
Uniform Convergence and the Integral
Uniform Convergence and Derivatives
Pompeiu's Function
Continuity and Pointwise Limits
Challenging Problems for Chapter 9


10. Power Series

Introduction Power Series: Convergence
Uniform Covergence
Functions Represented by Power Series
The Taylor Series
Products of Power Series
Composition of Power Series
Trigonometric Series


11. The Euclidean Spaces Rn

The Algebraic Structure of Rn
The Metric Structure of Rn
Elementary Topology of Rn
Sequences in Rn
Functions and Mappings
Limits of Functions from Rn to Rm
Continuity of Functions from Rn to Rm
Compact Sets in Rn
Continuous Functions on Compact Sets
Additional Remarks


12. Differentiation on Rn

Introduction Partial and Directional Derivatives
Integrals Depending on a Parameter
Differentiable Functions
Chain Rules
Implicit Function Theorems
Functions from R to Rm
Functions from Rn to Rm


13. Metric Spaces

Introduction
Metric Spaces--Specific Examples
Convergence
Sets in a Metric Space
Functions
Separable Spaces
Complete Spaces
Contraction Maps
Applications of Contraction Maps (I)
Applications of Contraction Maps (II)
Compactness
Baire Category Theorem
Applications of the Baire Category Theorem
Challenging Problems for Chapter 13

Appendix A: Background

Should I Read This Chapter?
Notation
What Is Analysis?
Why Proofs?
Indirect Proof
Contraposition
Counterexamples
Induction Quantifiers

Appendix B:

Hints for Selected Exercises
Subject Index

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