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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.

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Preface.- 1. Preliminaries.- 2. Metric Spaces.- 3. Special Spaces.- 4. Normed Spaces.- 5. Linear Functionals.- 6. Fundamental Theorems.- 7. Hilbert Spaces.- A. Hilbert Spaces L2(J).- References.- Index.

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"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)

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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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