Functional Analysis: An Elementary Introduction

This book sneaks up on functional analysis. It assumes little background in analysis or linear algebra and develops the needed material in the first half of the book. (There is also a series of appendices that provide even more mathematical background.) It then starts working on problems with a functional-theoretic flavor, such as integral equations, but they are tackled in an ad-hoc manner. The general theorems, such as Hahn-Banach, are not developed until the end of the book.

I like this approach, especially for the extra attention given to Baire’s category theorem and to the contraction mapping theorem. It is not strictly a historical approach, but it does have some echoes of the way in which functional analysis was developed in the 1920s to pull together many diverse threads of mathematics. The book has a very valuable historical overview of the development of functional analysis at the end. The book is in a graduate series, but appears to be aimed at upper-division undergraduates.

The book has numerous exercises at the end of each chapter, divided into three classes: (A) things left as an exercise for the reader, (B) the real exercises, and (C) extra credit and optional. They present a good mix of routine and challenging problems.

A similar though more concise book is Saxe’s Beginning Functional Analysis. It too assumes little knowledge of analysis and develops the needed background material from scratch. Both books have a lot of historical information. The big difference in the books is that Saxe proceeds more directly to the theorems of modern functional analysis and does the applications afterwards. Neither book is particularly abstract or theoretical, but Haase is more oriented to applications.

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Allen Stenger is a math hobbyist and retired software developer. He is an editor of the Missouri Journal of Mathematical Sciences. His mathematical interests are number theory and classical analysis.