Functional Analysis

1. Front Matter

   Pages i-xii

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2. Preliminaries

   • Sergei Ovchinnikov

   Pages 1-17

3. Metric Spaces

   • Sergei Ovchinnikov

   Pages 19-46

4. Special Spaces

   • Sergei Ovchinnikov

   Pages 47-73

5. Normed Spaces

   • Sergei Ovchinnikov

   Pages 75-99

6. Linear Functionals

   • Sergei Ovchinnikov

   Pages 101-130

7. Fundamental Theorems

   • Sergei Ovchinnikov

   Pages 131-148

8. Hilbert Spaces

   • Sergei Ovchinnikov

   Pages 149-191

9. Back Matter

   Pages 193-205

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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

“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)

• Department of Mathematics, San Francisco State University, San Francisco, USA

  Sergei Ovchinnikov

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). 

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