1st Edition

# Elements of Real Analysis

By M.A. Al-Gwaiz, S.A. Elsanousi Copyright 2007
450 Pages 60 B/W Illustrations
by Chapman & Hall

436 Pages
by Chapman & Hall

Also available as eBook on:

Focusing on one of the main pillars of mathematics, Elements of Real Analysis provides a solid foundation in analysis, stressing the importance of two elements. The first building block comprises analytical skills and structures needed for handling the basic notions of limits and continuity in a simple concrete setting while the second component involves conducting analysis in higher dimensions and more abstract spaces.

Largely self-contained, the book begins with the fundamental axioms of the real number system and gradually develops the core of real analysis. The first few chapters present the essentials needed for analysis, including the concepts of sets, relations, and functions. The following chapters cover the theory of calculus on the real line, exploring limits, convergence tests, several functions such as monotonic and continuous, power series, and theorems like mean value, Taylor's, and Darboux's. The final chapters focus on more advanced theory, in particular, the Lebesgue theory of measure and integration.

Requiring only basic knowledge of elementary calculus, this textbook presents the necessary material for a first course in real analysis. Developed by experts who teach such courses, it is ideal for undergraduate students in mathematics and related disciplines, such as engineering, statistics, computer science, and physics, to understand the foundations of real analysis.

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'HÃ´pital'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

### Biography

M.A. Al-Gwaiz, S.A. Elsanousi