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

    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'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

    Biography

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