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Elements of Real Analysis

M. A. Al-Gwaiz and S. S. Elsanousi
Publisher: 
Chapman & Hall/CRC
Publication Date: 
2007
Number of Pages: 
436
Format: 
Hardcover
Series: 
Pure and Applied Mathematics 284
Price: 
89.95
ISBN: 
1584886617
Category: 
Textbook
[Reviewed by
Kai Brunkalla
, on
04/13/2007
]

Elements of Real Analysis is a monograph aimed at the senior undergraduate and first year graduate level introduction to real analysis. In 428 pages the authors expose the reader to topics from an introduction to real numbers and sequences and series to Lebesgue measure and integration. The material is divided into eleven chapters. The writing is informative without being too compelling.

The book will serve its intended purpose and audience well. It is accessible to advanced undergraduate students. The authors include enough exercises in each section to improve understanding of the material, though I think that some exercises could be omitted for more advanced problems. Unfortunately, some of the problems contain important definitions and concepts that should have been part of the main text. In some parts of the text there is a decided lack of examples for concepts like integration by parts. Some parts of the book can be challenging to students as the text contains minor mistakes, mainly concerning the interchangeable use of < and ≤.

In my opinion, the book is more of a theory of calculus than an introduction to real analysis because important concepts from the latter are omitted. The book lacks the construction of the real numbers, which are taken as given, and also introduces the idea of Lebesgue measure and integration only in the two last chapters. The book also lacks some of the historical comments that help to place the study of modern analysis in the context of calculus and mathematics in general. The book is recommended as a textbook for undergraduate courses in introduction to real analysis but the professor may have to supplement in some places.


Kai Brunkalla teaches at Walsh University in North Canton, Ohio.

 PREFACE
PRELIMINARIES
Sets
Functions
REAL NUMBERS
Field Axioms
Order Axioms
Natural Numbers, Integers, Rational Numbers
Completeness Axiom
Decimal Representation of Real Numbers
Countable Sets
SEQUENCES
Sequences and Convergence
Properties of Convergent Sequences
Monotonic Sequences
The Cauchy Criterion
Subsequences
Upper and Lower Limits
Open and Closed Sets
INFINITE SERIES
Basic Properties
Convergence Tests
LIMIT OF A FUNCTION
Limit of a Function
Basic Theorems
Some Extensions of the Limit
Monotonic Functions
CONTINUITY
Continuous Functions
Combinations of Continuous Functions
Continuity on an Interval
UniformContinuity
Compact Sets and Continuity
DIFFERENTIATION
The Derivative
TheMean Value Theorem
L'Hôpital's Rule
Taylor's Theorem
THE RIEMANN INTEGRAL
Riemann Integrability
Darboux's Theorem and Riemann Sums
Properties of the Integral
The Fundamental Theorem of Calculus
Improper Integrals
SEQUENCES AND SERIES OF FUNCTIONS
Sequences of Functions
Properties of Uniform Convergence
Series of Functions
Power Series
LEBESGUE MEASURE
Classes of Subsets of R
Lebesgue Outer Measure
Lebesgue Measure
Measurable Functions
LEBESGUE INTEGRATION
Definition of the Lebesgue Integral
Properties of the Lebesgue Integral
Lebesgue Integral and Pointwise Convergence
Lebesgue and Riemann Integrals
REFERENCES
NOTATION
INDEX

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