Exercises in Functional Analysis

The understanding of results and notions for a student in mathematics requires solving ex­ ercises. The exercises are also meant to test the reader's understanding of the text material, and to enhance the skill in doing calculations. This book is written with these three things in mind. It is a collection of more than 450 exercises in Functional Analysis, meant to help a student understand much better the basic facts which are usually presented in an introductory course in Functional Analysis. Another goal of this book is to help the reader to understand the richness of ideas and techniques which Functional Analysis offers, by providing various exercises, from different topics, from simple ones to, perhaps, more difficult ones. We also hope that some of the exercises herein can be of some help to the teacher of Functional Analysis as seminar tools, and to anyone who is interested in seeing some applications of Functional Analysis. To what extent we have managed to achieve these goals is for the reader to decide.

I: Normed spaces.- 1. Open, closed, and bounded sets in normed spaces.- 2. Linear and continuous operators on normed spaces.- 3. Linear and continuous functionals. Reflexive spaces.- 4. The distance between sets in Banach spaces.- 5. Compactness in Banach spaces. Compact operators.- 6. The Uniform Boundedness Principle.- 7. The Hahn—Banach theorem.- 8. Applications for the Hahn—Banach theorem.- 9. Baire’s category. The open mapping and closed graph theorems.- II: Hilbert spaces.- 10. Hilbert spaces, general theory.- 11. The projection in Hilbert spaces.- 12. Linear and continuous operators on Hilbert spaces.- III: General topological spaces.- 13. Linear topological and locally convex spaces.- 14. The weak topologies.- List of Symbols.