It is generally believed that solving problems is the most important part of the learning process in mathematics because it forces students to truly understand the definitions, comb through the theorems and proofs, and think at length about the mathematics. The purpose of this book is to complement the existing literature in introductory real and functional analysis at the graduate level with a variety of conceptual problems (1,457 in total), ranging from easily accessible to thought provoking, mixing the practical and the theoretical aspects of the subject. Problems are grouped into ten chapters covering the main topics usually taught in courses on real and functional analysis. Each of these chapters opens with a brief reader’s guide stating the needed definitions and basic results in the area and closes with a short description of the problems.
The Problem chapters are accompanied by Solution chapters, which include solutions to two-thirds of the problems. Students can expect the solutions to be written in a direct language that they can understand; usually the most “natural” rather than the most elegant solution is presented.
Alberto Torchinsky, Indiana University, Bloomington, IN
Preface Part 1. Problems Chapter 1. Set Theory and Metric Spaces Problems Chapter 2. Measures Problems Chapter 3. Lebesgue Measure Problems Chapter 4. Measurable and Integrable Functions Problems Chapter 5. Lp Spaces Problems Chapter 6. Sequences of Functions Problems Chapter 7. Product Measures Problems Chapter 8. Normed Linear Spaces. Functionals Problems Chapter 9. Normed Linear Spaces. Linear Operators Problems Chapter 10. Hilbert Spaces Problems Part 2. Solutions Chapter 11. Set Theory and Metric Spaces Solutions Chapter 12. Measures Solutions Chapter 13. Lebesgue Measure Solutions Chapter 14. Measurable and Integrable Functions Solutions Chapter 15. Lp Spaces Solutions Chapter 16. Sequences of Functions Solutions Chapter 17. Product Measures Solutions Chapter 18. Normed Linear Spaces. Functionals Solutions Chapter 19. Normed Linear Spaces. Linear Operators Solutions Chapter 20. Hilbert Spaces Solutions