312 Pages 41 B/W Illustrations
    by Chapman & Hall

    312 Pages 41 B/W Illustrations
    by Chapman & Hall

    Foundations of Analysis covers the basics of real analysis for a one- or two-semester course. In a straightforward and concise way, it helps students understand the key ideas and apply the theorems. The book’s accessible approach will appeal to a wide range of students and instructors.

    Each section begins with a boxed introduction that familiarizes students with the upcoming topics and sets the stage for the work to be done. Each section ends with several questions that ask students to review what they have just learned. The text is also scattered with notes pointing out places where different pieces of terminology seem to conflict with each other or where different ideas appear not to fit together properly. In addition, many remarks throughout help put the material in perspective.

    As with any real analysis text, exercises are powerful and effective learning tools. This book is no exception. Each chapter generally contains at least 50 exercises that build in difficulty, with an exercise set at the end of every section. This allows students to more easily link the exercises to the material in the section.

    Number Systems
    The Real Numbers
    The Complex Numbers

    Convergence of Sequences
    Limsup and Liminf
    Some Special Sequences

    Series of Numbers
    Convergence of Series
    Elementary Convergence Tests
    Advanced Convergence Tests
    Some Special Series
    Operations on Series

    Basic Topology
    Open and Closed Sets
    Further Properties of Open and Closed Sets
    Compact Sets
    The Cantor Set
    Connected and Disconnected Sets
    Perfect Sets

    Limits and Continuity of Functions
    Basic Properties of the Limit of a Function
    Continuous Functions
    Topological Properties and Continuity
    Classifying Discontinuities and Monotonicity

    Differentiation of Functions
    The Concept of Derivative
    The Mean Value Theorem and Applications
    More on the Theory of Differentiation

    The Integral
    Partitions and the Concept of Integral
    Properties of the Riemann Integral

    Sequences and Series of Functions
    Convergence of a Sequence of Functions
    More on Uniform Convergence
    Series of Functions
    The Weierstrass Approximation Theorem

    Elementary Transcendental Functions
    Power Series
    More on Power Series: Convergence Issues
    The Exponential and Trigonometric Functions
    Logarithms and Powers of Real Numbers

    Appendix I: Elementary Number Systems
    Appendix II: Logic and Set Theory

    Table of Notation





    Steven G. Krantz is a professor of mathematics at Washington University in St. Louis. He has previously taught at UCLA, Princeton University, and Pennsylvania State University. He has written more than 75 books and more than 195 scholarly papers and is the founding editor of the Journal of Geometric Analysis and Complex Analysis and its Synergies. An AMS Fellow, Dr. Krantz has been a recipient of the Chauvenet Prize, Beckenbach Book Award, and Kemper Prize. He received a Ph.D. from Princeton University.

    "… there is a good set of exercises in each section … . If real analysis is to be dealt with in a one-semester course, this book appears to provide a reasonable text for the course."
    Mathematical Reviews, April 2015