© 2018

Functional Analysis

An Introductory Course


  • Provides an elementary treatment of the subject that establishes the foundation for further study

  • Enriches understanding of the theory with numerous examples and counterexamples

  • Includes many exercises for readers to practice techniques

  • Dissects proofs of difficult results into small steps to improve understanding


Part of the Universitext book series (UTX)

Table of contents

  1. Front Matter
    Pages i-xii
  2. Sergei Ovchinnikov
    Pages 1-17
  3. Sergei Ovchinnikov
    Pages 19-46
  4. Sergei Ovchinnikov
    Pages 47-73
  5. Sergei Ovchinnikov
    Pages 75-99
  6. Sergei Ovchinnikov
    Pages 101-130
  7. Sergei Ovchinnikov
    Pages 131-148
  8. Sergei Ovchinnikov
    Pages 149-191
  9. Back Matter
    Pages 193-205

About this book


This concise text provides a gentle introduction to functional analysis. Chapters cover essential topics such as special spaces, normed spaces, linear functionals, and Hilbert spaces. Numerous examples and counterexamples aid in the understanding of key concepts, while exercises at the end of each chapter provide ample opportunities for practice with the material. Proofs of theorems such as the Uniform Boundedness Theorem, the Open Mapping Theorem, and the Closed Graph Theorem are worked through step-by-step, providing an accessible avenue to understanding these important results. The prerequisites for this book are linear algebra and elementary real analysis, with two introductory chapters providing an overview of material necessary for the subsequent text.
Functional Analysis offers an elementary approach ideal for the upper-undergraduate or beginning graduate student. Primarily intended for a one-semester introductory course, this text is also a perfect resource for independent study or as the basis for a reading course. 


Functional analysis textbook Introductory functional analysis Normed space Banach space Inner product space Hilbert space Summable families Fundamental theorems of functional analysis Hahn-Banach theorem

Authors and affiliations

  1. 1.Department of MathematicsSan Francisco State UniversitySan FranciscoUSA

About the authors

Sergei Ovchinnikov is Professor Emeritus of Mathematics at San Francisco State University. His other Universitext books are Measure, Integral, Derivative: a Course on Lebesgue's Theory (2013) and Graphs and Cubes (2011). 

Bibliographic information

  • Book Title Functional Analysis
  • Book Subtitle An Introductory Course
  • Authors Sergei Ovchinnikov
  • Series Title Universitext
  • Series Abbreviated Title Universitext
  • DOI
  • Copyright Information Springer International Publishing AG, part of Springer Nature 2018
  • Publisher Name Springer, Cham
  • eBook Packages Mathematics and Statistics Mathematics and Statistics (R0)
  • Softcover ISBN 978-3-319-91511-1
  • eBook ISBN 978-3-319-91512-8
  • Series ISSN 0172-5939
  • Series E-ISSN 2191-6675
  • Edition Number 1
  • Number of Pages XII, 205
  • Number of Illustrations 13 b/w illustrations, 0 illustrations in colour
  • Topics Functional Analysis
  • Buy this book on publisher's site
Industry Sectors
Finance, Business & Banking


“This textbook is well organized and the proofs are carefully written. … Each chapter is concluded with an interesting note and several exercises, helping the reader to better understand the topics of the chapter. … it will be useful for upper-undergraduate and beginning graduate students.” (Mohammad Sal Moslehian, zbMATH 1398.46001, 2018)