Introduction to Analysis, An, 4th edition

  • William R. Wade

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Hallmark features of this title

  • Flexible presentation: Uniform writing style and notation covers material in small sections, allowing instructors to adapt it to their syllabus.
  • Early introduction of fundamental goals of analysis: Examines how a limit operation interacts with algebraic operation.
  • More than 200 worked examples and 600 exercises help students test comprehension while using techniques in other contexts.
  • Optional appendices and enrichment sections help students understand the material and allow instructors to tailor their courses.
  • An alternate chapter on metric spaces allows instructors to cover either chapter independently.
  • Separate coverage of topology and analysis presents purely computational material first, followed by topological material in alternate chapters.

Published by Pearson (July 1st 2022) - Copyright © 2023

ISBN-13: 9780137981663

Subject: Advanced Math

Category: Real Analysis

Table of contents




1. The Real Number System

1.1 Introduction

1.2 Ordered field axioms

1.3 Completeness Axiom

1.4 Mathematical Induction

1.5 Inverse functions and images

1.6 Countable and uncountable sets


2. Sequences in R

2.1 Limits of sequences

2.2 Limit theorems

2.3 Bolzano-Weierstrass Theorem

2.4 Cauchy sequences

*2.5 Limits supremum and infimum


3. Continuity on R

3.1 Two-sided limits

3.2 One-sided limits and limits at infinity

3.3 Continuity

3.4 Uniform continuity


4. Differentiability on R

4.1 The derivative

4.2 Differentiability theorems

4.3 The Mean Value Theorem

4.4 Taylor's Theorem and l'Hôpital's Rule

4.5 Inverse function theorems


5 Integrability on R

5.1 The Riemann integral

5.2 Riemann sums

5.3 The Fundamental Theorem of Calculus

5.4 Improper Riemann integration

*5.5 Functions of bounded variation

*5.6 Convex functions


6. Infinite Series of Real Numbers

6.1 Introduction

6.2 Series with nonnegative terms

6.3 Absolute convergence

6.4 Alternating series

*6.5 Estimation of series

*6.6 Additional tests


7. Infinite Series of Functions

7.1 Uniform convergence of sequences

7.2 Uniform convergence of series

7.3 Power series

7.4 Analytic functions

*7.5 Applications




8. Euclidean Spaces

8.1 Algebraic structure

8.2 Planes and linear transformations

8.3 Topology of Rn

8.4 Interior, closure, boundary


9. Convergence in Rn

9.1 Limits of sequences

9.2 Heine-Borel Theorem

9.3 Limits of functions

9.4 Continuous functions

*9.5 Compact sets

*9.6 Applications


10. Metric Spaces

10.1 Introduction

10.2 Limits of functions

10.3 Interior, closure, boundary

10.4 Compact sets

10.5 Connected sets

10.6 Continuous functions

10.7 Stone-Weierstrass Theorem


11. Differentiability on Rn

11.1 Partial derivatives and partial integrals

11.2 The definition of differentiability

11.3 Derivatives, differentials, and tangent planes

11.4 The Chain Rule

11.5 The Mean Value Theorem and Taylor's Formula

11.6 The Inverse Function Theorem

*11.7 Optimization


12. Integration on Rn

12.1 Jordan regions

12.2 Riemann integration on Jordan regions

12.3 Iterated integrals

12.4 Change of variables

*12.5 Partitions of unity

*12.6 The gamma function and volume


13. Fundamental Theorems of Vector Calculus

13.1 Curves

13.2 Oriented curves

13.3 Surfaces

13.4 Oriented surfaces

13.5 Theorems of Green and Gauss

13.6 Stokes's Theorem


*14. Fourier Series

*14.1 Introduction

*14.2 Summability of Fourier series

*14.3 Growth of Fourier coefficients

*14.4 Convergence of Fourier series

*14.5 Uniqueness



A. Algebraic laws

B. Trigonometry

C. Matrices and determinants

D. Quadric surfaces

E. Vector calculus and physics

F. Equivalence relations



Answers and Hints to Exercises

Subject Index

Symbol Index


*Enrichment section

Your questions answered

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