# A Guide to Functional Analysis

### By Steven G. Krantz

Catalog Code: DOL-49
Print ISBN: 978-0-88385-357-3
Electronic ISBN: 978-1-61444-213-4
149 pp., Hardbound, 2013
List Price: $51.00 MAA Member:$38.25
Series: Dolciani Mathematical Expositions

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.

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

### 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.