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.