This book is a quick but precise and careful introduction to the subject of functional analysis. It covers the topics that can be found in a basic graduate analysis text and 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.

### Table of Contents

Preface

1. Fundamentals

2. Ode to the Dual

3. Hilbert Space

4. The Algebra of Operators

5. Banach Algebra Basics

6. Topological Vector Spaces

7. Distributions

8. Spectral

9. Convexity

10. Fixed-Point Theorems

Table of Notation

Glossary

Bibliography

Index

About the Author

### Excerpt: 1.1 What is Functional Analysis? (p. 1)

The mathematical analysts of the nineteenth century (Cauchy, Riemann, Weierstrass, and others) contented themselves with studying one function at a time. As a sterling instance, the Weierstrass nowhere differentiable function is a world-changing example of the real function theory of “one function at a time.” Some of Riemann’s examples in Fourier analysis give other instances. This was the world view 150 years ago. To be sure, Cauchy and others considered sequences and series of functions, but the end goal was to consider the single limit function.

A major paradigm shift took place, however, in the early twentieth century. For then people began to consider *spaces* of functions. By this we mean a linear space, equipped with a norm. The process began slowly. At first people considered very specific spaces, such as the square-integrable real functions on the unit circle. Much later, people branched out to more general classes of spaces. An important feature of the spaces under study was that they must be complete. For we want to pass to limits, and completeness guarantees that this process is reliable.

Thus was born the concepts of Hilbert space and Banach space.

### About the Author

**Steven G. Krantz** was born in San Francisco, California in 1951. He received a B.A. degree from the University of California at Santa Cruz in 1971 and the Ph.D. from Princeton University in 1974. Krantz has taught at UCLA, Penn State, Princeton University, and Washington University in St. Louis. He served as Chair of the latter department for five years. Krantz has published more than 60 books and more than 160 scholarly papers. He is the recipient of the Chauvenet Prize and the Beckenbach Book Award of the MAA. He has received the UCLA Alumni Foundation Distinguished Teaching Award and the Kemper Award. He has directed 18 Ph.D. theses and 9 Masters theses.

### All MAA Guides

1. A Guide to Complex Variables

2. A Guide to Advanced Real Analysis

3. A Guide to Real Variables

4. A Guide to Topology

5. A Guide to Elementary Number Theory

6. A Guide to Advanced Linear Algebra

7. A Guide to Plane Algebraic Curves

8. A Guide to Groups, Rings, and Fields

9. A Guide to Functional Analysis