Real Analysis and Applications

  • Fabio Silva Botelho

Table of contents

  1. Front Matter
    Pages i-xiii
  2. One Variable Real Analysis

    1. Front Matter
      Pages 1-1
    2. Fabio Silva Botelho
      Pages 3-49
    3. Fabio Silva Botelho
      Pages 51-64
    4. Fabio Silva Botelho
      Pages 65-96
    5. Fabio Silva Botelho
      Pages 97-140
    6. Fabio Silva Botelho
      Pages 141-185
    7. Fabio Silva Botelho
      Pages 187-209
    8. Fabio Silva Botelho
      Pages 211-270
  3. Multi-Variable Advanced Calculus

    1. Front Matter
      Pages 271-271
    2. Fabio Silva Botelho
      Pages 273-416
    3. Fabio Silva Botelho
      Pages 417-469
  4. Back Matter
    Pages 559-567

About this book


This textbook introduces readers to real analysis in one and n dimensions. It is divided into two parts: Part I explores real analysis in one variable, starting with key concepts such as the construction of the real number system, metric spaces, and real sequences and series. In turn, Part II addresses the multi-variable aspects of real analysis. Further, the book presents detailed, rigorous proofs of the implicit theorem for the vectorial case by applying the Banach fixed-point theorem and the differential forms concept to surfaces in Rn. It also provides a brief introduction to Riemannian geometry. 

With its rigorous, elegant proofs, this self-contained work is easy to read, making it suitable for undergraduate and beginning graduate students seeking a deeper understanding of real analysis and applications, and for all those looking for a well-founded, detailed approach to real analysis.


real analysis Implicit function theorem Banach's fixed point theorem metric spaces Arzela-Ascoli theorem Riemannian geometry

Authors and affiliations

  • Fabio Silva Botelho
    • 1
  1. 1.Department of MathematicsFederal University of Santa CatarinaFlorianópolisBrazil

Bibliographic information

  • DOI
  • Copyright Information Springer International Publishing AG, part of Springer Nature 2018
  • Publisher Name Springer, Cham
  • eBook Packages Mathematics and Statistics
  • Print ISBN 978-3-319-78630-8
  • Online ISBN 978-3-319-78631-5
  • Buy this book on publisher's site