Real Analysis 1st Edition by N. L. Carothers, ISBN-13: 978-0521497565

Description

Real Analysis 1st Edition by N. L. Carothers, ISBN-13: 978-0521497565

[PDF eBook eTextbook]

• Publisher: ‎ Cambridge University Press; 1st edition (August 15, 2000)
• Language: ‎ English
• 416 pages
• ISBN-10: ‎ 0521497566
• ISBN-13: ‎ 978-0521497565

A text for a first graduate course in real analysis for students in pure and applied mathematics, statistics, education, engineering, and economics.

This course in real analysis is directed at advanced undergraduates and beginning graduate students in mathematics and related fields. Presupposing only a modest background in real analysis or advanced calculus, the book offers something of value to specialists and nonspecialists alike. The text covers three major topics: metric and normed linear spaces, function spaces, and Lebesgue measure and integration on the line. In an informal, down-to-earth style, the author gives motivation and overview of new ideas, while still supplying full details and complete proofs. He provides a great many exercises and suggestions for further study.

Table of Contents:

Cover

Title

Copyright

Dedication

Contents

Preface

PART ONE METRIC SPACES

1 Calculus Review

The Real Numbers

Limits and Continuity

Notes and Remarks

2 Countable and Uncountable Sets

Equivalence and Cardinality

The Cantor Set

Monotone Functions

Notes and Remarks

3 Metrics and Norms

Metric Spaces

Normed Vector Spaces

More Inequalities

Limits in Metric Spaces

Notes and Remarks

4 Open Sets and Closed Sets

Open Sets

Closed Sets

The Relative Metric

Notes and Remarks

5 Continuity

Continuous Functions

Homeomorphisms

The Space of Continuous Functions

Notes and Remarks

6 Connectedness

Connected Sets

Notes and Remarks

7 Completeness

Totally Bounded Sets

Complete Metric Spaces

Fixed Points

Completions

Notes and Remarks

8 Compactness

Compact Metric Spaces

Uniform Continuity

Equivalent Metrics

Notes and Remarks

9 Category

Discontinuous Functions

The Baire Category Theorem

Notes and Remarks

PART TWO FUNCTION SPACES

10 Sequences of Functions

Historical Background

Pointwise and Uniform Convergence

Interchanging Limits

The Space of Bounded Functions

Notes and Remarks

11 The Space of Continuous Functions

The Weierstrass Theorem

Trigonometric Polynomials

Infinitely Differentiable Functions

Equicontinuity

Continuity and Category

Notes and Remarks

12 The Stone–Weierstrass Theorem

Algebras and Lattices

The Stone–Weierstrass Theorem

Notes and Remarks

13 Functions of Bounded Variation

Functions of Bounded Variation

Helly’s First Theorem

Notes and Remarks

14 The Riemann–Stieltjes Integral

Weights and Measures

The Riemann–Stieltjes Integral

The Space of Integrable Functions

Integrators of Bounded Variation

The Riemann Integral

The Riesz Representation Theorem

Other Definitions, Other Properties

Notes and Remarks

15 Fourier Series

Preliminaries

Dirichlet’s Formula

Fejér’s Theorem

Complex Fourier Series

Notes and Remarks

PART THREE LEBESGUE MEASURE AND INTEGRATION

16 Lebesgue Measure

The Problem of Measure

Lebesgue Outer Measure

Riemann Integrability

Measurable Sets

The Structure of Measurable Sets

A Nonmeasurable Set

Other Definitions

Notes and Remarks

17 Measurable Functions

Measurable Functions

Extended Real-Valued Functions

Sequences of Measurable Functions

Approximation of Measurable Functions

Notes and Remarks

18 The Lebesgue Integral

Simple Functions

Nonnegative Functions

The General Case

Lebesgue’s Dominated Convergence Theorem

Approximation of Integrable Functions

Notes and Remarks

19 Additional Topics

Convergence in Measure

The Lp Spaces

Approximation of Lp Functions

More on Fourier Series

Notes and Remarks

20 Differentiation

Lebesgue’s Differentiation Theorem

Absolute Continuity

Notes and Remarks

References

Symbol Index

Topic Index

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