Elementary Real Analysis

by H. G. Eggleston

• ISBN13: 9780521098687
• ISBN10: 0521098688
• Edition: 1st
• Format: Paperback
• Copyright: 2008-12-18
• Publisher: Cambridge University Press
• More Book Details

Summary

This textbook covers all the theoretical aspects of real variable analysis which undergraduates reading mathematics are likely to require during the first two or three years of their course. It is based on lecture courses which the author has given in the universities of Wales, Cambridge and London. The subject is presented rigorously and without padding. Definitions are stated explicitly and the whole development of the subject is logical and self-contained. Complex numbers are used but the complex variable calculus is not. 'Applied analysis', such as differential equations and Fourier series, is not dealt with. A large number of examples is included, with hints for the solution of many of them. These will be of particular value to students working on their own.

Table of Contents

Preface

Notation and Conventions

Preliminaries

Enumerability and sequences

Bounds for sets of numbers

Bounds for functions and sequences

Limits of the sequences

Monotonic sequences; two important examples ; irrational powers of positive real numbers

Upper and lower limits of real sequences: the general principle of convergence

Convergence of series; absolute convergence

Conditional convergence

Rearrangement and multiplication of absolutely convergent series

Double series

Power series

Point set theory

The Bolzano-Weierstrass, Cantor and Heine-Borel theorems

Functions defined over real or complex numbers

Functions of a single real variable; limits and continuity

Monotonic functions; functions of bounded variation

Differentiation; mean-value theorems

The nth mean-value theorem: Taylor's theorem

Convex and concave functions

The elementary transcendental functions

Inequalities

The Riemann integral

Integration and differentiation

The Riemann-Stiltjes integral

Improper integrals; convergence of integrals

Further tests for the convergence of series

Uniform convergence

Functions of two real variables. Continuity and differentiability

Hints on the solution of exercises and answers to exercises

Appendix

Index

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