A Basic Course in Real Analysis: 1st Edition (Hardback) book cover

A Basic Course in Real Analysis

1st Edition

By Ajit Kumar, S. Kumaresan

Chapman and Hall/CRC

322 pages | 101 B/W Illus.

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pub: 2014-01-10
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Description

Based on the authors’ combined 35 years of experience in teaching, A Basic Course in Real Analysis introduces students to the aspects of real analysis in a friendly way. The authors offer insights into the way a typical mathematician works observing patterns, conducting experiments by means of looking at or creating examples, trying to understand the underlying principles, and coming up with guesses or conjectures and then proving them rigorously based on his or her explorations.

With more than 100 pictures, the book creates interest in real analysis by encouraging students to think geometrically. Each difficult proof is prefaced by a strategy and explanation of how the strategy is translated into rigorous and precise proofs. The authors then explain the mystery and role of inequalities in analysis to train students to arrive at estimates that will be useful for proofs. They highlight the role of the least upper bound property of real numbers, which underlies all crucial results in real analysis. In addition, the book demonstrates analysis as a qualitative as well as quantitative study of functions, exposing students to arguments that fall under hard analysis.

Although there are many books available on this subject, students often find it difficult to learn the essence of analysis on their own or after going through a course on real analysis. Written in a conversational tone, this book explains the hows and whys of real analysis and provides guidance that makes readers think at every stage.

Reviews

"… there are some unique features that put this book aside. … a welcome addition to the library of teachers and student alike."

Zentralblatt MATH 1308

"… this book describes the basic results of analysis in an extremely clear, straightforward, and well-motivated way. … if you’re looking for a text on the easy end of the spectrum for a course in real analysis, then this book is certainly worth a serious look …"

MAA Reviews, October 2014

Table of Contents

Real Number System

Algebra of the Real Number System

Upper and Lower Bounds

LUB Property and Its Applications

Absolute Value and Triangle Inequality

Sequences and Their Convergence

Sequences and Their Convergence

Cauchy Sequences

Monotone Sequences

Sandwich Lemma

Some Important Limits

Sequences Diverging to

Subsequences

Sequences Defined Recursively

Continuity

Continuous Functions

Definition of Continuity

Intermediate Value Theorem .

Extreme Value Theorem

Monotone Functions

Limits

Uniform Continuity

Continuous Extensions

Differentiation

Differentiability of Functions

Mean Value Theorems

L'Hospital's Rules

Higher-order Derivatives

Taylor's Theorem

Convex Functions

Cauchy's Form of the Remainder

Infinite Series

Convergence of an Infinite Series

Abel's Summation by Parts

Rearrangements of an Infinite Series

Cauchy Product of Two Infinite Series

Riemann Integration

Darboux Integrability

Properties of the Integral

Fundamental Theorems of Calculus

Mean Value Theorems for Integrals

Integral Form of the Remainder

Riemann's Original Definition

Sum of an Infinite Series as an Integral

Logarithmic and Exponential Functions

Improper Riemann Integrals

Sequences and Series of Functions

Pointwise Convergence

Uniform Convergence

Consequences of Uniform Convergence

Series of Functions

Power Series

Taylor Series of a Smooth Function

Binomial Series

Weierstrass Approximation Theorem

A Quantifiers

B Limit Inferior and Limit Superior

C Topics for Student Seminars

D Hints for Selected Exercises

Bibliography

Index

Subject Categories

BISAC Subject Codes/Headings:
MAT037000
MATHEMATICS / Functional Analysis