A Guide to Functional Analysis: (Series Number 49 Dolciani Mathematical Expositions)

Functional analysis is an abstract and powerful modern theory that occupies a central role in mathematics. This book provides a quick but precise introduction to the subject, covering everything that a beginning graduate student needs to know. The subject has its roots in the theory of infinite-dimensional vector spaces which is where the book begins, with preliminaries from the theory of normed linear spaces, Hilbert spaces, operator algebras and distributions. The reader will then encounter more advanced topics such as spectral theory, convexity and fixed-point theorems. It contains plenty of examples and exercises, making it an ideal basis for advanced undergraduate and graduate level courses in a subject that has become an essential part of every analyst's toolkit.

Table of Contents:
Preface; 1. Fundamentals; 2. Ode to the dual space; 3. Hilbert space; 4. The algebra of operators; 5. Banach algebra basics; 6. Topological vector spaces; 7. Distributions; 8. Spectral theory; 9. Convexity; 10. Fixed-point theorems; Table of notation; Glossary; Bibliography; Index.

About the Author :
Steven G. Krantz earned a BA from the University of California, Santa Cruz and a PhD from Princeton University. He has taught at the University of California, Los Angeles, Pennsylvania State University, and Washington University. 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.

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