Publisher: John Wiley (2008)Details: 307 pages, Hardcover
Series: Pure and Applied Mathematics
Price: $100.00
ISBN: 9780470146194
Category: Textbook
Topics: Functional Analysis
[Reviewed by Mihaela Poplicher, on 10/09/2008]
S. David Promislow
Publisher: John Wiley (2008)
Details: 307 pages, Hardcover
Series: Pure and Applied Mathematics
Price: $100.00
ISBN: 9780470146194
Category: Textbook
Topics: Functional Analysis
[Reviewed by Mihaela Poplicher, on 10/09/2008]
This book was written by S. David Promislow, Professor Emeritus of Mathematics at York University in Toronto, Canada, as a text-book for a one-semester introductory course in functional analysis. It is based on the author’s experience in teaching such a course for upper-undergraduate and graduate students with very diverse backgrounds in mathematics, statistics, and engineering.
The prerequisites needed for reading/using the book are a knowledge of basic real analysis (sequences and series, continuity, uniform convergence of functions, topology of the real line, elements of metric spaces), as well as elementary linear algebra (linear independence, bases, matrix manipulation).
The book treats the most important topics in a first functional analysis course: linear spaces and operators, normed linear spaces, major Banach space theorems, Hilbert spaces, Hahn-Banach theorem, duality, topological linear spaces, compact operators. There are also chapters about the spectrum, applications to integral and differential equations, spectral theorem for bounded self-adjoint operators.
Most of the topics included in the book do not require knowledge beyond the basics of real analysis and linear algebra, but for people who want to study more, or to see the proofs of some of the results used and not proven (in the main chapters), there are appendices on measure and integration, Zorn’s lemma, the Stone-Weierstrass theorem, and Tychonoff’s theorem. A good list of references helps the readers who want to further their studies even deeper.
What sets this book apart is the way the proofs are presented, outlining the logic behind the steps, explaining the development of the arguments and discussing the connections between the concepts. Each section concludes with a set of exercises of different levels of difficulty which are very helpful for readers in their quest for understanding of the material.
In short, I think this is an excellent text for reaching students of diverse backgrounds and majors, as well as scientists from other disciplines (physics, economics, finance, and engineering) who want an introduction in functional analysis.
Mihaela Poplicher is an associate professor of mathematics at the University of Cincinnati. Her research interests include functional analysis, harmonic analysis, and complex analysis. She is also interested in the teaching of mathematics. Her email address is Mihaela.Poplicher@uc.edu.