Product Description

This introduction to the ideas and methods of linear functional analysis shows how familiar and useful concepts from finite-dimensional linear algebra can be extended or generalized to infinite-dimensional spaces. Aimed at advanced undergraduates in mathematics and physics, the book assumes a standard background of linear algebra, real analysis (including the theory of metric spaces), and Lebesgue integration, although an introductory chapter summarizes the requisite material. Highlights of the second edition include - a new chapter on the Hahn-Banach theorem and its applications to the theory of duality. This chapter also introduces the basic properties of projection operators on Banach spaces, and weak convergence of sequences in Banach spaces; - topics that have applications to both linear and nonlinear functional analysis; - extended coverage of the uniform boundedness theorem; - plenty of exercises, with solutions provided at the back of the book.

Product Details

• Amazon Sales Rank: #266944 in Books
• Published on: 2007-12-21
• Original language: English
• Number of items: 1
• Binding: Paperback
• 324 pages

Review
"This is an undergraduate introduction to functional analysis, with minimal prerequisites, namely linear algebra and some real analysis. … It is extensively cross-referenced, has a good index, a separate index of symbols (Very Good Feature), and complete solutions to all the exercises. It has numerous examples, and is especially good in giving both examples of objects that have a given property and objects that do not have the property." --Allen Stenger, MathDL, April, 2008

Customer Reviews

A good, accessible introduction at undergraduate level
Out of the several books on Functional Analysis available, this is the easiest and most accessible, and is suitable for undergraduates. But, you still need to know some prerequisite material, including linear algebra, analysis, measure theory and Lebesgue integration. In case you have forgotten, this is helpfully summarised in chapter 1. I had not studied measure theory before, but this book makes it sound so beautiful and fascinating, that I hope to investigate further. The next part of the book deals with Banach and Hilbert spaces, the fundamental ideas of Functional Analysis. Then it tells you about operators, which is where matters get interesting! An operator is basically a function from one space to another, but in some situations, the set of operators form their own space! Chapter 5 deals with linear operators on Hilbert spaces, and here things just get beautiful! It defines the spectrum of an operator, which is a subset of the complex numbers, simliar to the set of eigenvalues in linear algebra. So many wonderful things follow on so easily. The book goes on to deal with compact operators, a special type of operator. Finally, it explains how Functional Analysis can help you solve differential and/or integral equations, but the study of the subject is worthwhile in its own right. The exercises in this book are very challenging, which is good because it makes you really think about the material, and the authors have helpfully provided model answers. Having finished this book, the reader could then go onto a more advanced book on the topic, and a list of suggested reading is provided at the end. Ultimately, this is a very good book on Functional Analysis, full of useful information for the mathematician, and it shows you how far Analysis, as a subject, extends beyond the basic idea of "a more rigorous version of single-variable calculus". Heartily recommended!