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A Guide to Functional Analysis (Dolciani Mathematical Expositions), by Steven G. Krantz
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This book is a quick but precise and careful introduction to the subject of functional analysis. It covers the basic topics that can be found in a basic graduate analysis text. But it also covers more sophisticated topics such as spectral theory, convexity, and fixed-point theorems. A special feature of the book is that it contains a great many examples and even some applications. It concludes with a statement and proof of Lomonosov's dramatic result about invariant subspaces.
- Sales Rank: #2975170 in Books
- Published on: 2013-04-05
- Original language: English
- Number of items: 1
- Dimensions: 8.98" h x .55" w x 5.98" l, .80 pounds
- Binding: Hardcover
- 149 pages
Review
This book is a short introduction to functional analysis along with a brief description of the most well known examples. It opens with a chapter on the mathematical fundamentals and then moves on to the primary examples. It is written at a level where the practicing mathematician or upper level graduate student can use it as a stand-alone resource to learn the basics of functional analysis. Proofs of all of the main theorems and propositions are included. Krantz has won many awards for his skill at expository writing and with this book he once again demonstrates that he deserves all of the accolades he receives. --Charles Ashbacher, Journal of Recreational Mathematics
This book (barely over 100 pages of text) is very short...but nevertheless addresses most or all of the standard topics that one would expect to see in an introductory graduate-level semester in functional analysis, and perhaps even one or two things that might not get mentioned. ...
Like other books in the Guide series, this one contains a good selection of examples, which I think is crucial. Another particularly nice feature of this book is the attention paid to applications of functional analysis, which even longer books frequently overlook. As some (non-exhaustive) examples, we see here, for example, the Uniform Boundedness theorem used to prove the existence of a broad class of functions with divergent Fourier series, the Hahn-Banach theorem invoked to establish the existence of the Green's function for smoothly bounded domains in the plane, and the contraction mapping principle used both to establish an existence-uniqueness theorem for differential equations and to give an elegant proof of the implicit function theorem. ...
This book continues the tradition of high-quality expositions that have characterized every other Guide that I have looked at. This series in general, and this book in particular, deserve, and I hope will get, a wide audience. --Mark Hunacek, MAA Reviews
This is a brief monograph (135 pages) in the MAA Guides series. The series, is, to quote the description given by the MAA, "meant for students, especially graduate students, and faculty who would like an overview of the subject. They will be useful to those preparing for qualifying exams." It takes care of the "fundamentals," the so-called three big results and their applications in the 26 pages of Chapter 1 and then proceeds to cover bounded operators, Banach algebra basics, topological vector spaces, distributions, convexity, fixed point theorems, and spectral theory in the remaining eight chapters. It is a nice review book for somebody who has seen the stuff before or is sufficiently sophisticated in another field of mathematics. --Heydar Radjavi, CMS Notes
About the Author
Steven G. Krantz was born in San Francisco. He earned a B.A. degree from the University of California at Santa Cruz and a Ph.D. from Princeton University. He has taught at UCLA, Penn State, and Washington University, also serving as Chair of the latter department. Over the course of his career he has written over 70 books and over 180 scholarly papers. He is a fellow of the American Mathematical Society and the holder of the Chauvenet Prize, the Beckenbach Book Award, and the UCLA Alumni Foundation Teaching Award.
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