Mark Hunacek Reviews A Guide to Functional Analysis

Mark Hunacek reviews A Guide to Functional Analysis by Steven G. Krantz as part of MAA Reviews. "This book (barely over a hundred pages of text) is very short, even by the standards of this series, 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. More specifically, the first chapter starts with normed linear spaces, then defines Banach spaces and discusses the “big three” results typically associated with them (Uniform Boundedness, Open Mapping, Hahn-Banach). This is followed by chapters on the dual space, Hilbert space, the algebra of bounded linear operators on a Banach space (including a fairly lengthy section on compact operators), and Banach algebras. The author then generalizes things by discussing (chapter 6) arbitrary topological vector spaces. The four remaining chapters of the text discuss, in order, distributions, spectral theory (for bounded, particularly bounded normal, operators on a Hilbert space; some background in measure theory is needed for this chapter), convexity (including the Krein-Milman theorem), and fixed point theorems (the contraction mapping principle and the Schauder theorem)." Read the full review here. To purchase your copy today, visit the MAA Store or the MAA eBooks Store. Mark Hunacek (mhunacek@iastate.edu) teaches mathematics at Iowa State University.

Now Available in the MAA Store

A Guide to Functional Analysis  is now available in the MAA Store!   Steven G. Krantz describes his new book below. "Everyone knows that functional analysis is one of the most powerful tools of twentieth-century mathematics. The idea of studying entire spaces of functions, rather than just one function at a time, is both deep and original. Yet it is difficult to get a glimpse of what this subject is really about, or of how it works. The Guide to Functional Analysis takes the neophyte reader and shows him/her the basic concepts and rubric of this time-tested discipline. All the major ideas are illustrated with concrete examples and applications to other parts of mathematics. There are even some illustrations. This is a user-friendly, hands-on introduction to an otherwise austere and forbidding part of the mathematical lore. It will be helpful to students beginning the long journey down the path to mastery of analysis and also enlightening for any mathematician wanting to bone up on linear operators and their uses. This book is meant to be a guide, and we hope that it guides you to a pleasurable reading experience." Purchase a print copy in the MAA Store.

Steven G. Krantz earned his B.A. degree for the University of California at Santa Cruz and his Ph.D. from Princeton University. He has written over 70 books and over 180 scholarly papers. The MAA has awarded him both the Beckenbach Book Prize and the Chauvenet Prize.

New Book: A Guide to Functional Analysis

A Guide to Functional Analysis by Steven G. Krantz The purpose of A Guide to Functional Analysis is to introduce the reader with minimal background to the basic scripture of functional analysis. Readers should know some real analysis and some linear algebra. Measure theory rears its ugly head in some of the examples and also in the treatment of spectral theory. The latter is unavoidable and the former allows us to present a rich variety of examples. The nervous reader may safely skip any of the measure theory and still derive a lot from the rest of the book. Apart from this caveat, the book is almost completely self-contained; in a few instances we mention easily accessible references. A feature that sets this book apart from most other functional analysis texts is that it has a lot of examples and a lot of applications. This helps to make the material more concrete, and relates it to ideas that the reader has already seen. It also makes the book more accessible to a broader audience. Now available in the MAA eBooks Store Will be available in the MAA Store soon Check out these books also by Steven G. Krantz:  A Guide to Topology A Guide to Real Variables A Guide to Complex Variables A Mathematician Comes of Age An Episodic History of Mathematics Mathematical Apocrypha Mathematical Apocrypha Redux  Complex Analysis: The Geometric Viewpoint A Panorama of Harmonic Analysis