© 2017

Real Analysis

Series, Functions of Several Variables, and Applications


  • Corresponds to a second course in real analysis to follow the authors' book Real Analysis: Foundations and Functions of One Variable

  • Motivates ideas and results in analysis by exploring concepts and applications

  • Showcases a comprehensive collection of exercises, allowing students to develop proficiency over a broad range of problems


Part of the Undergraduate Texts in Mathematics book series (UTM, volume 3)

Table of contents

  1. Front Matter
    Pages i-ix
  2. Miklós Laczkovich, Vera T. Sós
    Pages 1-65
  3. Miklós Laczkovich, Vera T. Sós
    Pages 67-94
  4. Miklós Laczkovich, Vera T. Sós
    Pages 95-121
  5. Miklós Laczkovich, Vera T. Sós
    Pages 123-153
  6. Miklós Laczkovich, Vera T. Sós
    Pages 155-192
  7. Miklós Laczkovich, Vera T. Sós
    Pages 193-228
  8. Miklós Laczkovich, Vera T. Sós
    Pages 229-301
  9. Miklós Laczkovich, Vera T. Sós
    Pages 303-360
  10. Miklós Laczkovich, Vera T. Sós
    Pages 361-381
  11. Back Matter
    Pages 383-392

About this book


This book develops the theory of multivariable analysis, building on the single variable foundations established in the companion volume, Real Analysis: Foundations and Functions of One Variable. Together, these volumes form the first English edition of the popular Hungarian original, Valós Analízis I & II, based on courses taught by the authors at Eötvös Loránd University, Hungary, for more than 30 years. Numerous exercises are included throughout, offering ample opportunities to master topics by progressing from routine to difficult problems. Hints or solutions to many of the more challenging exercises make this book ideal for independent study, or further reading.

Intended as a sequel to a course in single variable analysis, this book builds upon and expands these ideas into higher dimensions. The modular organization makes this text adaptable for either a semester or year-long introductory course. Topics include: differentiation and integration of functions of several variables; infinite numerical series; sequences and series of functions; and applications to other areas of mathematics. Many historical notes are given and there is an emphasis on conceptual understanding and context, be it within mathematics itself or more broadly in applications, such as physics. By developing the student’s intuition throughout, many definitions and results become motivated by insights from their context.


Continuity of functions History of Fourier series History of infinite series Limit of functions Real analysis Multivariable real analysis Multivariable real analysis textbook adoption

Authors and affiliations

  1. 1.Department of AnalysisEötvös Loránd University—ELTEBudapestHungary
  2. 2.Alfréd Rényi Institute of MathematicsHungarian Academy of SciencesBudapestHungary

About the authors

Miklós Laczkovich is a member of the Hungarian Academy of Sciences and professor emeritus at Eötvös Loránd University Budapest and University College London. He was awarded the Ostrowski Prize in 1993 and the Széchenyi Prize in 1998.

Vera T. Sós is a member of the Hungarian Academy of Sciences and professor emeritus at Eötvös Loránd Univesity Budapest and the Alfred Renyi Mathematical Institute of the Hungarian Academy of Sciences. She was awarded the Széchenyi Prize in 1997.

Bibliographic information

  • Book Title Real Analysis
  • Book Subtitle Series, Functions of Several Variables, and Applications
  • Authors Miklós Laczkovich
    Vera T. Sós
  • Series Title Undergraduate Texts in Mathematics
  • Series Abbreviated Title Undergraduate Texts Mathematics
  • DOI
  • Copyright Information Springer Science+Business Media LLC 2017
  • Publisher Name Springer, New York, NY
  • eBook Packages Mathematics and Statistics Mathematics and Statistics (R0)
  • Hardcover ISBN 978-1-4939-7367-5
  • Softcover ISBN 978-1-4939-8464-0
  • eBook ISBN 978-1-4939-7369-9
  • Series ISSN 0172-6056
  • Series E-ISSN 2197-5604
  • Edition Number 1
  • Number of Pages IX, 392
  • Number of Illustrations 44 b/w illustrations, 0 illustrations in colour
  • Additional Information First English Edition, translated from 2nd Hungarian edition by Gergely Bálint
  • Topics Analysis
  • Buy this book on publisher's site
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“This book develops a thorough treatment of multivariate derivatives, viewing them both as linear transformations and as partial derivatives. … The book is consistent in addressing the classical analysis of real functions of several variables. This volume will appeal to students in pure and applied mathematics, as well as scientists looking to acquire a firm footing in mathematical analysis.” (Teodora-Liliana Rădulescu, zbMATH 1407.26001, 2019)

“The explicit intent of the authors is to present material gradually, and to develop precision based on intuition with the help of well-designed examples. The book contains 600 exercises. In my opinion this interesting book represents an alternative viewpoint in real analysis.” (Zbigniew Grande, Mathematical Reviews, August, 2018)​